probability.independence.basic ⟷ Mathlib.Probability.Independence.Basic

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -184,7 +184,7 @@ theorem indep_bot_right (m' : MeasurableSpace Ω) {m : MeasurableSpace Ω} {μ :
     [IsProbabilityMeasure μ] : Indep m' ⊥ μ :=
   by
   intro s t hs ht
-  rw [Set.mem_setOf_eq, MeasurableSpace.measurableSet_bot_iff] at ht 
+  rw [Set.mem_setOf_eq, MeasurableSpace.measurableSet_bot_iff] at ht
   cases ht
   · rw [ht, Set.inter_empty, measure_empty, MulZeroClass.mul_zero]
   · rw [ht, Set.inter_univ, measure_univ, mul_one]
@@ -267,7 +267,7 @@ theorem IndepSets.iUnion [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Se
     {μ : Measure Ω} (hyp : ∀ n, IndepSets (s n) s' μ) : IndepSets (⋃ n, s n) s' μ :=
   by
   intro t1 t2 ht1 ht2
-  rw [Set.mem_iUnion] at ht1 
+  rw [Set.mem_iUnion] at ht1
   cases' ht1 with n ht1
   exact hyp n t1 t2 ht1 ht2
 #align probability_theory.indep_sets.Union ProbabilityTheory.IndepSets.iUnion
@@ -278,7 +278,7 @@ theorem IndepSets.bUnion [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Se
     {μ : Measure Ω} {u : Set ι} (hyp : ∀ n ∈ u, IndepSets (s n) s' μ) :
     IndepSets (⋃ n ∈ u, s n) s' μ := by
   intro t1 t2 ht1 ht2
-  simp_rw [Set.mem_iUnion] at ht1 
+  simp_rw [Set.mem_iUnion] at ht1
   rcases ht1 with ⟨n, hpn, ht1⟩
   exact hyp n hpn t1 t2 ht1 ht2
 #align probability_theory.indep_sets.bUnion ProbabilityTheory.IndepSets.bUnion
@@ -473,7 +473,7 @@ theorem indepSets_piiUnionInter_of_disjoint [IsProbabilityMeasure μ] {s : ι 
     have hgm : ∀ i ∈ p1 ∪ p2, g i ∈ s i :=
       by
       intro i hi_mem_union
-      rw [Finset.mem_union] at hi_mem_union 
+      rw [Finset.mem_union] at hi_mem_union
       cases' hi_mem_union with hi1 hi2
       · have hi2 : i ∉ p2 := fun hip2 => set.disjoint_left.mp hST (hp1 hi1) (hp2 hip2)
         simp_rw [g, if_pos hi1, if_neg hi2, Set.inter_univ]
@@ -555,7 +555,7 @@ theorem indep_iSup_of_directed_le {Ω} {m : ι → MeasurableSpace Ω} {m' m0 :
   have h_pi_system_indep : indep_sets (⋃ n, p n) p' μ :=
     by
     refine' indep_sets.Union _
-    simp_rw [h_gen_n, h_gen'] at h_indep 
+    simp_rw [h_gen_n, h_gen'] at h_indep
     exact fun n => (h_indep n).IndepSets
   -- now go from π-systems to σ-algebras
   refine' indep_sets.indep (iSup_le h_le) h_le' hp_supr_pi hp'_pi _ h_gen' h_pi_system_indep
@@ -618,7 +618,7 @@ theorem iIndepSets.piiUnionInter_of_not_mem {π : ι → Set (Set Ω)} {a : ι}
     (hp_ind : iIndepSets π μ) (haS : a ∉ S) : IndepSets (piiUnionInter π S) (π a) μ :=
   by
   rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
-  rw [Finset.coe_subset] at hs_mem 
+  rw [Finset.coe_subset] at hs_mem
   classical
   let f n := ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
   have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n :=
@@ -861,7 +861,7 @@ theorem iIndepFun_iff_measure_inter_preimage_eq_mul {ι : Type _} {β : ι → T
     congr with (i x)
     simp only [Set.mem_iInter]
     constructor <;> intro h hi_mem <;> specialize h hi_mem
-    · rwa [h_preim i hi_mem] at h 
+    · rwa [h_preim i hi_mem] at h
     · rwa [h_preim i hi_mem]
   have h_right_eq : ∏ i in S, μ (setsΩ i) = ∏ i in S, μ (f i ⁻¹' setsβ i) :=
     by
@@ -954,7 +954,7 @@ theorem iIndepFun.indepFun_finset [IsProbabilityMeasure μ] {ι : Type _} {β :
       (Measurable.comap_le (measurable_pi_iff.mpr fun i => hf_meas i)) hπS_pi hπT_pi hπS_gen hπT_gen
       _
   rintro _ _ ⟨s, ⟨sets_s, hs1, hs2⟩, rfl⟩ ⟨t, ⟨sets_t, ht1, ht2⟩, rfl⟩
-  simp only [Set.mem_univ_pi, Set.mem_setOf_eq] at hs1 ht1 
+  simp only [Set.mem_univ_pi, Set.mem_setOf_eq] at hs1 ht1
   rw [← hs2, ← ht2]
   classical
   let sets_s' : ∀ i : ι, Set (β i) := fun i =>
@@ -978,7 +978,7 @@ theorem iIndepFun.indepFun_finset [IsProbabilityMeasure μ] {ι : Type _} {β :
     simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
     constructor <;> intro h
     · intro i hi; rw [h_sets_s'_eq hi]; exact h ⟨i, hi⟩
-    · rintro ⟨i, hi⟩; specialize h i hi; rw [h_sets_s'_eq hi] at h ; exact h
+    · rintro ⟨i, hi⟩; specialize h i hi; rw [h_sets_s'_eq hi] at h; exact h
   have h_eq_inter_T :
     (fun (ω : Ω) (i : ↥T) => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i :=
     by
@@ -986,8 +986,8 @@ theorem iIndepFun.indepFun_finset [IsProbabilityMeasure μ] {ι : Type _} {β :
     simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
     constructor <;> intro h
     · intro i hi; simp_rw [sets_t', dif_pos hi]; exact h ⟨i, hi⟩
-    · rintro ⟨i, hi⟩; specialize h i hi; simp_rw [sets_t', dif_pos hi] at h ; exact h
-  rw [Indep_fun_iff_measure_inter_preimage_eq_mul] at hf_Indep 
+    · rintro ⟨i, hi⟩; specialize h i hi; simp_rw [sets_t', dif_pos hi] at h; exact h
+  rw [Indep_fun_iff_measure_inter_preimage_eq_mul] at hf_Indep
   rw [h_eq_inter_S, h_eq_inter_T, hf_Indep S h_meas_s', hf_Indep T h_meas_t']
   have h_Inter_inter :
     ((⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i) =
@@ -1004,7 +1004,7 @@ theorem iIndepFun.indepFun_finset [IsProbabilityMeasure μ] {ι : Type _} {β :
   rw [h_Inter_inter, hf_Indep (S ∪ T)]
   swap
   · intro i hi_mem
-    rw [Finset.mem_union] at hi_mem 
+    rw [Finset.mem_union] at hi_mem
     cases hi_mem
     · rw [h_sets_t'_univ hi_mem, Set.inter_univ]; exact h_meas_s' i hi_mem
     · rw [h_sets_s'_univ hi_mem, Set.univ_inter]; exact h_meas_t' i hi_mem

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -325,7 +325,30 @@ section FromIndepToIndep
 
 #print ProbabilityTheory.iIndepSets.indepSets /-
 theorem iIndepSets.indepSets {s : ι → Set (Set Ω)} [MeasurableSpace Ω] {μ : Measure Ω}
-    (h_indep : iIndepSets s μ) {i j : ι} (hij : i ≠ j) : IndepSets (s i) (s j) μ := by classical
+    (h_indep : iIndepSets s μ) {i j : ι} (hij : i ≠ j) : IndepSets (s i) (s j) μ := by
+  classical
+  intro t₁ t₂ ht₁ ht₂
+  have hf_m : ∀ x : ι, x ∈ {i, j} → ite (x = i) t₁ t₂ ∈ s x :=
+    by
+    intro x hx
+    cases' finset.mem_insert.mp hx with hx hx
+    · simp [hx, ht₁]
+    · simp [finset.mem_singleton.mp hx, hij.symm, ht₂]
+  have h1 : t₁ = ite (i = i) t₁ t₂ := by simp only [if_true, eq_self_iff_true]
+  have h2 : t₂ = ite (j = i) t₁ t₂ := by simp only [hij.symm, if_false]
+  have h_inter :
+    (⋂ (t : ι) (H : t ∈ ({i, j} : Finset ι)), ite (t = i) t₁ t₂) =
+      ite (i = i) t₁ t₂ ∩ ite (j = i) t₁ t₂ :=
+    by simp only [Finset.set_biInter_singleton, Finset.set_biInter_insert]
+  have h_prod :
+    ∏ t : ι in ({i, j} : Finset ι), μ (ite (t = i) t₁ t₂) =
+      μ (ite (i = i) t₁ t₂) * μ (ite (j = i) t₁ t₂) :=
+    by
+    simp only [hij, Finset.prod_singleton, Finset.prod_insert, not_false_iff, Finset.mem_singleton]
+  rw [h1]
+  nth_rw 2 [h2]
+  nth_rw 4 [h2]
+  rw [← h_inter, ← h_prod, h_indep {i, j} hf_m]
 #align probability_theory.Indep_sets.indep_sets ProbabilityTheory.iIndepSets.indepSets
 -/
 
@@ -444,6 +467,41 @@ theorem indepSets_piiUnionInter_of_disjoint [IsProbabilityMeasure μ] {s : ι 
   by
   rintro t1 t2 ⟨p1, hp1, f1, ht1_m, ht1_eq⟩ ⟨p2, hp2, f2, ht2_m, ht2_eq⟩
   classical
+  let g i := ite (i ∈ p1) (f1 i) Set.univ ∩ ite (i ∈ p2) (f2 i) Set.univ
+  have h_P_inter : μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, μ (g n) :=
+    by
+    have hgm : ∀ i ∈ p1 ∪ p2, g i ∈ s i :=
+      by
+      intro i hi_mem_union
+      rw [Finset.mem_union] at hi_mem_union 
+      cases' hi_mem_union with hi1 hi2
+      · have hi2 : i ∉ p2 := fun hip2 => set.disjoint_left.mp hST (hp1 hi1) (hp2 hip2)
+        simp_rw [g, if_pos hi1, if_neg hi2, Set.inter_univ]
+        exact ht1_m i hi1
+      · have hi1 : i ∉ p1 := fun hip1 => set.disjoint_right.mp hST (hp2 hi2) (hp1 hip1)
+        simp_rw [g, if_neg hi1, if_pos hi2, Set.univ_inter]
+        exact ht2_m i hi2
+    have h_p1_inter_p2 :
+      ((⋂ x ∈ p1, f1 x) ∩ ⋂ x ∈ p2, f2 x) =
+        ⋂ i ∈ p1 ∪ p2, ite (i ∈ p1) (f1 i) Set.univ ∩ ite (i ∈ p2) (f2 i) Set.univ :=
+      by
+      ext1 x
+      simp only [Set.mem_ite_univ_right, Set.mem_inter_iff, Set.mem_iInter, Finset.mem_union]
+      exact
+        ⟨fun h i _ => ⟨h.1 i, h.2 i⟩, fun h =>
+          ⟨fun i hi => (h i (Or.inl hi)).1 hi, fun i hi => (h i (Or.inr hi)).2 hi⟩⟩
+    rw [ht1_eq, ht2_eq, h_p1_inter_p2, ← h_indep _ hgm]
+  have h_μg : ∀ n, μ (g n) = ite (n ∈ p1) (μ (f1 n)) 1 * ite (n ∈ p2) (μ (f2 n)) 1 :=
+    by
+    intro n
+    simp_rw [g]
+    split_ifs
+    · exact absurd rfl (set.disjoint_iff_forall_ne.mp hST _ (hp1 h) _ (hp2 h_1))
+    all_goals simp only [measure_univ, one_mul, mul_one, Set.inter_univ, Set.univ_inter]
+  simp_rw [h_P_inter, h_μg, Finset.prod_mul_distrib,
+    Finset.prod_ite_mem (p1 ∪ p2) p1 fun x => μ (f1 x), Finset.union_inter_cancel_left,
+    Finset.prod_ite_mem (p1 ∪ p2) p2 fun x => μ (f2 x), Finset.union_inter_cancel_right, ht1_eq, ←
+    h_indep p1 ht1_m, ht2_eq, ← h_indep p2 ht2_m]
 #align probability_theory.indep_sets_pi_Union_Inter_of_disjoint ProbabilityTheory.indepSets_piiUnionInter_of_disjoint
 -/
 
@@ -461,7 +519,7 @@ theorem iIndepSet.indep_generateFrom_of_disjoint [IsProbabilityMeasure μ] {s :
   · exact fun k => generate_from_le fun t ht => (Set.mem_singleton_iff.1 ht).symm ▸ hsm k
   · exact isPiSystem_piiUnionInter _ (fun k => IsPiSystem.singleton _) _
   · exact isPiSystem_piiUnionInter _ (fun k => IsPiSystem.singleton _) _
-  · classical
+  · classical exact indep_sets_pi_Union_Inter_of_disjoint (Indep.Indep_sets (fun n => rfl) hs) hST
 #align probability_theory.Indep_set.indep_generate_from_of_disjoint ProbabilityTheory.iIndepSet.indep_generateFrom_of_disjoint
 -/
 
@@ -476,7 +534,7 @@ theorem indep_iSup_of_disjoint [IsProbabilityMeasure μ] {m : ι → MeasurableS
       (generateFrom_piiUnionInter_measurableSet m T).symm _
   · exact isPiSystem_piiUnionInter _ (fun n => @is_pi_system_measurable_set Ω (m n)) _
   · exact isPiSystem_piiUnionInter _ (fun n => @is_pi_system_measurable_set Ω (m n)) _
-  · classical
+  · classical exact indep_sets_pi_Union_Inter_of_disjoint h_indep hST
 #align probability_theory.indep_supr_of_disjoint ProbabilityTheory.indep_iSup_of_disjoint
 -/
 
@@ -562,6 +620,35 @@ theorem iIndepSets.piiUnionInter_of_not_mem {π : ι → Set (Set Ω)} {a : ι}
   rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
   rw [Finset.coe_subset] at hs_mem 
   classical
+  let f n := ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
+  have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n :=
+    by
+    intro n hn_mem_insert
+    simp_rw [f]
+    cases' finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
+    · simp [hn_mem, ht2_mem_pia]
+    · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
+      simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
+  have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
+  have h_t1 : t1 = ⋂ n ∈ s, f n :=
+    by
+    suffices h_forall : ∀ n ∈ s, f n = ft1 n
+    · rw [ht1_eq]
+      congr with (n x)
+      congr with (hns y)
+      simp only [(h_forall n hns).symm]
+    intro n hnS
+    have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hnS)
+    simp_rw [f, if_pos hnS, if_neg hn_ne_a]
+  have h_μ_t1 : μ t1 = ∏ n in s, μ (f n) := by rw [h_t1, ← hp_ind s h_f_mem_pi]
+  have h_t2 : t2 = f a := by simp_rw [f]; simp
+  have h_μ_inter : μ (t1 ∩ t2) = ∏ n in insert a s, μ (f n) :=
+    by
+    have h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n := by
+      rw [h_t1, h_t2, Finset.set_biInter_insert, Set.inter_comm]
+    rw [h_t1_inter_t2, ← hp_ind (insert a s) h_f_mem]
+  have has : a ∉ s := fun has_mem => haS (hs_mem Membership)
+  rw [h_μ_inter, Finset.prod_insert has, h_t2, mul_comm, h_μ_t1]
 #align probability_theory.Indep_sets.pi_Union_Inter_of_not_mem ProbabilityTheory.iIndepSets.piiUnionInter_of_not_mem
 -/
 
@@ -571,6 +658,30 @@ theorem iIndepSets.iIndep [IsProbabilityMeasure μ] (m : ι → MeasurableSpace
     (h_le : ∀ i, m i ≤ m0) (π : ι → Set (Set Ω)) (h_pi : ∀ n, IsPiSystem (π n))
     (h_generate : ∀ i, m i = generateFrom (π i)) (h_ind : iIndepSets π μ) : iIndep m μ := by
   classical
+  refine' Finset.induction _ _
+  ·
+    simp only [measure_univ, imp_true_iff, Set.iInter_false, Set.iInter_univ, Finset.prod_empty,
+      eq_self_iff_true]
+  intro a S ha_notin_S h_rec f hf_m
+  have hf_m_S : ∀ x ∈ S, measurable_set[m x] (f x) := fun x hx => hf_m x (by simp [hx])
+  rw [Finset.set_biInter_insert, Finset.prod_insert ha_notin_S, ← h_rec hf_m_S]
+  let p := piiUnionInter π S
+  set m_p := generate_from p with hS_eq_generate
+  have h_indep : indep m_p (m a) μ :=
+    by
+    have hp : IsPiSystem p := isPiSystem_piiUnionInter π h_pi S
+    have h_le' : ∀ i, generate_from (π i) ≤ m0 := fun i => (h_generate i).symm.trans_le (h_le i)
+    have hm_p : m_p ≤ m0 := generateFrom_piiUnionInter_le π h_le' S
+    exact
+      indep_sets.indep hm_p (h_le a) hp (h_pi a) hS_eq_generate (h_generate a)
+        (h_ind.pi_Union_Inter_of_not_mem ha_notin_S)
+  refine' h_indep.symm (f a) (⋂ n ∈ S, f n) (hf_m a (Finset.mem_insert_self a S)) _
+  have h_le_p : ∀ i ∈ S, m i ≤ m_p := by
+    intro n hn
+    rw [hS_eq_generate, h_generate n]
+    exact le_generateFrom_piiUnionInter S hn
+  have h_S_f : ∀ i ∈ S, measurable_set[m_p] (f i) := fun i hi => (h_le_p i hi) (f i) (hf_m_S i hi)
+  exact S.measurable_set_bInter h_S_f
 #align probability_theory.Indep_sets.Indep ProbabilityTheory.iIndepSets.iIndep
 -/
 
@@ -733,6 +844,31 @@ theorem iIndepFun_iff_measure_inter_preimage_eq_mul {ι : Type _} {β : ι → T
   refine' ⟨fun h S sets h_meas => h _ fun i hi_mem => ⟨sets i, h_meas i hi_mem, rfl⟩, _⟩
   intro h S setsΩ h_meas
   classical
+  let setsβ : ∀ i : ι, Set (β i) := fun i =>
+    dite (i ∈ S) (fun hi_mem => (h_meas i hi_mem).some) fun _ => Set.univ
+  have h_measβ : ∀ i ∈ S, measurable_set[m i] (setsβ i) :=
+    by
+    intro i hi_mem
+    simp_rw [setsβ, dif_pos hi_mem]
+    exact (h_meas i hi_mem).choose_spec.1
+  have h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i :=
+    by
+    intro i hi_mem
+    simp_rw [setsβ, dif_pos hi_mem]
+    exact (h_meas i hi_mem).choose_spec.2.symm
+  have h_left_eq : μ (⋂ i ∈ S, setsΩ i) = μ (⋂ i ∈ S, f i ⁻¹' setsβ i) :=
+    by
+    congr with (i x)
+    simp only [Set.mem_iInter]
+    constructor <;> intro h hi_mem <;> specialize h hi_mem
+    · rwa [h_preim i hi_mem] at h 
+    · rwa [h_preim i hi_mem]
+  have h_right_eq : ∏ i in S, μ (setsΩ i) = ∏ i in S, μ (f i ⁻¹' setsβ i) :=
+    by
+    refine' Finset.prod_congr rfl fun i hi_mem => _
+    rw [h_preim i hi_mem]
+  rw [h_left_eq, h_right_eq]
+  exact h S h_measβ
 #align probability_theory.Indep_fun_iff_measure_inter_preimage_eq_mul ProbabilityTheory.iIndepFun_iff_measure_inter_preimage_eq_mul
 -/
 
@@ -821,6 +957,63 @@ theorem iIndepFun.indepFun_finset [IsProbabilityMeasure μ] {ι : Type _} {β :
   simp only [Set.mem_univ_pi, Set.mem_setOf_eq] at hs1 ht1 
   rw [← hs2, ← ht2]
   classical
+  let sets_s' : ∀ i : ι, Set (β i) := fun i =>
+    dite (i ∈ S) (fun hi => sets_s ⟨i, hi⟩) fun _ => Set.univ
+  have h_sets_s'_eq : ∀ {i} (hi : i ∈ S), sets_s' i = sets_s ⟨i, hi⟩ := by intro i hi;
+    simp_rw [sets_s', dif_pos hi]
+  have h_sets_s'_univ : ∀ {i} (hi : i ∈ T), sets_s' i = Set.univ := by intro i hi;
+    simp_rw [sets_s', dif_neg (finset.disjoint_right.mp hST hi)]
+  let sets_t' : ∀ i : ι, Set (β i) := fun i =>
+    dite (i ∈ T) (fun hi => sets_t ⟨i, hi⟩) fun _ => Set.univ
+  have h_sets_t'_univ : ∀ {i} (hi : i ∈ S), sets_t' i = Set.univ := by intro i hi;
+    simp_rw [sets_t', dif_neg (finset.disjoint_left.mp hST hi)]
+  have h_meas_s' : ∀ i ∈ S, MeasurableSet (sets_s' i) := by intro i hi; rw [h_sets_s'_eq hi];
+    exact hs1 _
+  have h_meas_t' : ∀ i ∈ T, MeasurableSet (sets_t' i) := by intro i hi;
+    simp_rw [sets_t', dif_pos hi]; exact ht1 _
+  have h_eq_inter_S :
+    (fun (ω : Ω) (i : ↥S) => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i :=
+    by
+    ext1 x
+    simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
+    constructor <;> intro h
+    · intro i hi; rw [h_sets_s'_eq hi]; exact h ⟨i, hi⟩
+    · rintro ⟨i, hi⟩; specialize h i hi; rw [h_sets_s'_eq hi] at h ; exact h
+  have h_eq_inter_T :
+    (fun (ω : Ω) (i : ↥T) => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i :=
+    by
+    ext1 x
+    simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
+    constructor <;> intro h
+    · intro i hi; simp_rw [sets_t', dif_pos hi]; exact h ⟨i, hi⟩
+    · rintro ⟨i, hi⟩; specialize h i hi; simp_rw [sets_t', dif_pos hi] at h ; exact h
+  rw [Indep_fun_iff_measure_inter_preimage_eq_mul] at hf_Indep 
+  rw [h_eq_inter_S, h_eq_inter_T, hf_Indep S h_meas_s', hf_Indep T h_meas_t']
+  have h_Inter_inter :
+    ((⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i) =
+      ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) :=
+    by
+    ext1 x
+    simp only [Set.mem_inter_iff, Set.mem_iInter, Set.mem_preimage, Finset.mem_union]
+    constructor <;> intro h
+    · intro i hi
+      cases hi
+      · rw [h_sets_t'_univ hi]; exact ⟨h.1 i hi, Set.mem_univ _⟩
+      · rw [h_sets_s'_univ hi]; exact ⟨Set.mem_univ _, h.2 i hi⟩
+    · exact ⟨fun i hi => (h i (Or.inl hi)).1, fun i hi => (h i (Or.inr hi)).2⟩
+  rw [h_Inter_inter, hf_Indep (S ∪ T)]
+  swap
+  · intro i hi_mem
+    rw [Finset.mem_union] at hi_mem 
+    cases hi_mem
+    · rw [h_sets_t'_univ hi_mem, Set.inter_univ]; exact h_meas_s' i hi_mem
+    · rw [h_sets_s'_univ hi_mem, Set.univ_inter]; exact h_meas_t' i hi_mem
+  rw [Finset.prod_union hST]
+  congr 1
+  · refine' Finset.prod_congr rfl fun i hi => _
+    rw [h_sets_t'_univ hi, Set.inter_univ]
+  · refine' Finset.prod_congr rfl fun i hi => _
+    rw [h_sets_s'_univ hi, Set.univ_inter]
 #align probability_theory.Indep_fun.indep_fun_finset ProbabilityTheory.iIndepFun.indepFun_finset
 -/
 
@@ -828,7 +1021,38 @@ theorem iIndepFun.indepFun_finset [IsProbabilityMeasure μ] {ι : Type _} {β :
 theorem iIndepFun.indepFun_prod_mk [IsProbabilityMeasure μ] {ι : Type _} {β : ι → Type _}
     {m : ∀ i, MeasurableSpace (β i)} {f : ∀ i, Ω → β i} (hf_Indep : iIndepFun m f μ)
     (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
-    IndepFun (fun a => (f i a, f j a)) (f k) μ := by classical
+    IndepFun (fun a => (f i a, f j a)) (f k) μ := by
+  classical
+  have h_right :
+    f k =
+      (fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩) ∘
+        fun a (j : ({k} : Finset ι)) => f j a :=
+    rfl
+  have h_meas_right :
+    Measurable fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩ :=
+    measurable_pi_apply ⟨k, Finset.mem_singleton_self k⟩
+  let s : Finset ι := {i, j}
+  have h_left :
+    (fun ω => (f i ω, f j ω)) =
+      (fun p : ∀ l : s, β l =>
+          (p ⟨i, Finset.mem_insert_self i _⟩,
+            p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩)) ∘
+        fun a (j : s) => f j a :=
+    by
+    ext1 a
+    simp only [Prod.mk.inj_iff]
+    constructor <;> rfl
+  have h_meas_left :
+    Measurable fun p : ∀ l : s, β l =>
+      (p ⟨i, Finset.mem_insert_self i _⟩,
+        p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩) :=
+    Measurable.prod (measurable_pi_apply ⟨i, Finset.mem_insert_self i {j}⟩)
+      (measurable_pi_apply ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self j)⟩)
+  rw [h_left, h_right]
+  refine' (hf_Indep.indep_fun_finset s {k} _ hf_meas).comp h_meas_left h_meas_right
+  rw [Finset.disjoint_singleton_right]
+  simp only [Finset.mem_insert, Finset.mem_singleton, not_or]
+  exact ⟨hik.symm, hjk.symm⟩
 #align probability_theory.Indep_fun.indep_fun_prod ProbabilityTheory.iIndepFun.indepFun_prod_mk
 -/
 
@@ -852,7 +1076,28 @@ theorem iIndepFun.indepFun_mul_left [IsProbabilityMeasure μ] {ι : Type _} {β
 theorem iIndepFun.indepFun_finset_prod_of_not_mem [IsProbabilityMeasure μ] {ι : Type _} {β : Type _}
     {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂ β] {f : ι → Ω → β}
     (hf_Indep : iIndepFun (fun _ => m) f μ) (hf_meas : ∀ i, Measurable (f i)) {s : Finset ι} {i : ι}
-    (hi : i ∉ s) : IndepFun (∏ j in s, f j) (f i) μ := by classical
+    (hi : i ∉ s) : IndepFun (∏ j in s, f j) (f i) μ := by
+  classical
+  have h_right :
+    f i =
+      (fun p : ∀ j : ({i} : Finset ι), β => p ⟨i, Finset.mem_singleton_self i⟩) ∘
+        fun a (j : ({i} : Finset ι)) => f j a :=
+    rfl
+  have h_meas_right :
+    Measurable fun p : ∀ j : ({i} : Finset ι), β => p ⟨i, Finset.mem_singleton_self i⟩ :=
+    measurable_pi_apply ⟨i, Finset.mem_singleton_self i⟩
+  have h_left : ∏ j in s, f j = (fun p : ∀ j : s, β => ∏ j, p j) ∘ fun a (j : s) => f j a :=
+    by
+    ext1 a
+    simp only [Function.comp_apply]
+    have : ∏ j : ↥s, f (↑j) a = (∏ j : ↥s, f ↑j) a := by rw [Finset.prod_apply]
+    rw [this, Finset.prod_coe_sort]
+  have h_meas_left : Measurable fun p : ∀ j : s, β => ∏ j, p j :=
+    finset.univ.measurable_prod fun (j : ↥s) (H : j ∈ Finset.univ) => measurable_pi_apply j
+  rw [h_left, h_right]
+  exact
+    (hf_Indep.indep_fun_finset s {i} (finset.disjoint_singleton_left.mpr hi).symm hf_meas).comp
+      h_meas_left h_meas_right
 #align probability_theory.Indep_fun.indep_fun_finset_prod_of_not_mem ProbabilityTheory.iIndepFun.indepFun_finset_prod_of_not_mem
 #align probability_theory.Indep_fun.indep_fun_finset_sum_of_not_mem ProbabilityTheory.iIndepFun.indepFun_finset_sum_of_not_mem
 -/
@@ -872,6 +1117,17 @@ theorem iIndepFun.indepFun_prod_range_succ [IsProbabilityMeasure μ] {β : Type
 theorem iIndepSet.iIndepFun_indicator [Zero β] [One β] {m : MeasurableSpace β} {s : ι → Set Ω}
     (hs : iIndepSet s μ) : iIndepFun (fun n => m) (fun n => (s n).indicator fun ω => 1) μ := by
   classical
+  rw [Indep_fun_iff_measure_inter_preimage_eq_mul]
+  rintro S π hπ
+  simp_rw [Set.indicator_const_preimage_eq_union]
+  refine' @hs S (fun i => ite (1 ∈ π i) (s i) ∅ ∪ ite ((0 : β) ∈ π i) (s iᶜ) ∅) fun i hi => _
+  have hsi : measurable_set[generate_from {s i}] (s i) :=
+    measurable_set_generate_from (Set.mem_singleton _)
+  refine'
+    MeasurableSet.union (MeasurableSet.ite' (fun _ => hsi) fun _ => _)
+      (MeasurableSet.ite' (fun _ => hsi.compl) fun _ => _)
+  · exact @MeasurableSet.empty _ (generate_from {s i})
+  · exact @MeasurableSet.empty _ (generate_from {s i})
 #align probability_theory.Indep_set.Indep_fun_indicator ProbabilityTheory.iIndepSet.iIndepFun_indicator
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -325,30 +325,7 @@ section FromIndepToIndep
 
 #print ProbabilityTheory.iIndepSets.indepSets /-
 theorem iIndepSets.indepSets {s : ι → Set (Set Ω)} [MeasurableSpace Ω] {μ : Measure Ω}
-    (h_indep : iIndepSets s μ) {i j : ι} (hij : i ≠ j) : IndepSets (s i) (s j) μ := by
-  classical
-  intro t₁ t₂ ht₁ ht₂
-  have hf_m : ∀ x : ι, x ∈ {i, j} → ite (x = i) t₁ t₂ ∈ s x :=
-    by
-    intro x hx
-    cases' finset.mem_insert.mp hx with hx hx
-    · simp [hx, ht₁]
-    · simp [finset.mem_singleton.mp hx, hij.symm, ht₂]
-  have h1 : t₁ = ite (i = i) t₁ t₂ := by simp only [if_true, eq_self_iff_true]
-  have h2 : t₂ = ite (j = i) t₁ t₂ := by simp only [hij.symm, if_false]
-  have h_inter :
-    (⋂ (t : ι) (H : t ∈ ({i, j} : Finset ι)), ite (t = i) t₁ t₂) =
-      ite (i = i) t₁ t₂ ∩ ite (j = i) t₁ t₂ :=
-    by simp only [Finset.set_biInter_singleton, Finset.set_biInter_insert]
-  have h_prod :
-    ∏ t : ι in ({i, j} : Finset ι), μ (ite (t = i) t₁ t₂) =
-      μ (ite (i = i) t₁ t₂) * μ (ite (j = i) t₁ t₂) :=
-    by
-    simp only [hij, Finset.prod_singleton, Finset.prod_insert, not_false_iff, Finset.mem_singleton]
-  rw [h1]
-  nth_rw 2 [h2]
-  nth_rw 4 [h2]
-  rw [← h_inter, ← h_prod, h_indep {i, j} hf_m]
+    (h_indep : iIndepSets s μ) {i j : ι} (hij : i ≠ j) : IndepSets (s i) (s j) μ := by classical
 #align probability_theory.Indep_sets.indep_sets ProbabilityTheory.iIndepSets.indepSets
 -/
 
@@ -467,41 +444,6 @@ theorem indepSets_piiUnionInter_of_disjoint [IsProbabilityMeasure μ] {s : ι 
   by
   rintro t1 t2 ⟨p1, hp1, f1, ht1_m, ht1_eq⟩ ⟨p2, hp2, f2, ht2_m, ht2_eq⟩
   classical
-  let g i := ite (i ∈ p1) (f1 i) Set.univ ∩ ite (i ∈ p2) (f2 i) Set.univ
-  have h_P_inter : μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, μ (g n) :=
-    by
-    have hgm : ∀ i ∈ p1 ∪ p2, g i ∈ s i :=
-      by
-      intro i hi_mem_union
-      rw [Finset.mem_union] at hi_mem_union 
-      cases' hi_mem_union with hi1 hi2
-      · have hi2 : i ∉ p2 := fun hip2 => set.disjoint_left.mp hST (hp1 hi1) (hp2 hip2)
-        simp_rw [g, if_pos hi1, if_neg hi2, Set.inter_univ]
-        exact ht1_m i hi1
-      · have hi1 : i ∉ p1 := fun hip1 => set.disjoint_right.mp hST (hp2 hi2) (hp1 hip1)
-        simp_rw [g, if_neg hi1, if_pos hi2, Set.univ_inter]
-        exact ht2_m i hi2
-    have h_p1_inter_p2 :
-      ((⋂ x ∈ p1, f1 x) ∩ ⋂ x ∈ p2, f2 x) =
-        ⋂ i ∈ p1 ∪ p2, ite (i ∈ p1) (f1 i) Set.univ ∩ ite (i ∈ p2) (f2 i) Set.univ :=
-      by
-      ext1 x
-      simp only [Set.mem_ite_univ_right, Set.mem_inter_iff, Set.mem_iInter, Finset.mem_union]
-      exact
-        ⟨fun h i _ => ⟨h.1 i, h.2 i⟩, fun h =>
-          ⟨fun i hi => (h i (Or.inl hi)).1 hi, fun i hi => (h i (Or.inr hi)).2 hi⟩⟩
-    rw [ht1_eq, ht2_eq, h_p1_inter_p2, ← h_indep _ hgm]
-  have h_μg : ∀ n, μ (g n) = ite (n ∈ p1) (μ (f1 n)) 1 * ite (n ∈ p2) (μ (f2 n)) 1 :=
-    by
-    intro n
-    simp_rw [g]
-    split_ifs
-    · exact absurd rfl (set.disjoint_iff_forall_ne.mp hST _ (hp1 h) _ (hp2 h_1))
-    all_goals simp only [measure_univ, one_mul, mul_one, Set.inter_univ, Set.univ_inter]
-  simp_rw [h_P_inter, h_μg, Finset.prod_mul_distrib,
-    Finset.prod_ite_mem (p1 ∪ p2) p1 fun x => μ (f1 x), Finset.union_inter_cancel_left,
-    Finset.prod_ite_mem (p1 ∪ p2) p2 fun x => μ (f2 x), Finset.union_inter_cancel_right, ht1_eq, ←
-    h_indep p1 ht1_m, ht2_eq, ← h_indep p2 ht2_m]
 #align probability_theory.indep_sets_pi_Union_Inter_of_disjoint ProbabilityTheory.indepSets_piiUnionInter_of_disjoint
 -/
 
@@ -519,7 +461,7 @@ theorem iIndepSet.indep_generateFrom_of_disjoint [IsProbabilityMeasure μ] {s :
   · exact fun k => generate_from_le fun t ht => (Set.mem_singleton_iff.1 ht).symm ▸ hsm k
   · exact isPiSystem_piiUnionInter _ (fun k => IsPiSystem.singleton _) _
   · exact isPiSystem_piiUnionInter _ (fun k => IsPiSystem.singleton _) _
-  · classical exact indep_sets_pi_Union_Inter_of_disjoint (Indep.Indep_sets (fun n => rfl) hs) hST
+  · classical
 #align probability_theory.Indep_set.indep_generate_from_of_disjoint ProbabilityTheory.iIndepSet.indep_generateFrom_of_disjoint
 -/
 
@@ -534,7 +476,7 @@ theorem indep_iSup_of_disjoint [IsProbabilityMeasure μ] {m : ι → MeasurableS
       (generateFrom_piiUnionInter_measurableSet m T).symm _
   · exact isPiSystem_piiUnionInter _ (fun n => @is_pi_system_measurable_set Ω (m n)) _
   · exact isPiSystem_piiUnionInter _ (fun n => @is_pi_system_measurable_set Ω (m n)) _
-  · classical exact indep_sets_pi_Union_Inter_of_disjoint h_indep hST
+  · classical
 #align probability_theory.indep_supr_of_disjoint ProbabilityTheory.indep_iSup_of_disjoint
 -/
 
@@ -620,35 +562,6 @@ theorem iIndepSets.piiUnionInter_of_not_mem {π : ι → Set (Set Ω)} {a : ι}
   rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
   rw [Finset.coe_subset] at hs_mem 
   classical
-  let f n := ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
-  have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n :=
-    by
-    intro n hn_mem_insert
-    simp_rw [f]
-    cases' finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
-    · simp [hn_mem, ht2_mem_pia]
-    · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
-      simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
-  have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
-  have h_t1 : t1 = ⋂ n ∈ s, f n :=
-    by
-    suffices h_forall : ∀ n ∈ s, f n = ft1 n
-    · rw [ht1_eq]
-      congr with (n x)
-      congr with (hns y)
-      simp only [(h_forall n hns).symm]
-    intro n hnS
-    have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hnS)
-    simp_rw [f, if_pos hnS, if_neg hn_ne_a]
-  have h_μ_t1 : μ t1 = ∏ n in s, μ (f n) := by rw [h_t1, ← hp_ind s h_f_mem_pi]
-  have h_t2 : t2 = f a := by simp_rw [f]; simp
-  have h_μ_inter : μ (t1 ∩ t2) = ∏ n in insert a s, μ (f n) :=
-    by
-    have h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n := by
-      rw [h_t1, h_t2, Finset.set_biInter_insert, Set.inter_comm]
-    rw [h_t1_inter_t2, ← hp_ind (insert a s) h_f_mem]
-  have has : a ∉ s := fun has_mem => haS (hs_mem Membership)
-  rw [h_μ_inter, Finset.prod_insert has, h_t2, mul_comm, h_μ_t1]
 #align probability_theory.Indep_sets.pi_Union_Inter_of_not_mem ProbabilityTheory.iIndepSets.piiUnionInter_of_not_mem
 -/
 
@@ -658,30 +571,6 @@ theorem iIndepSets.iIndep [IsProbabilityMeasure μ] (m : ι → MeasurableSpace
     (h_le : ∀ i, m i ≤ m0) (π : ι → Set (Set Ω)) (h_pi : ∀ n, IsPiSystem (π n))
     (h_generate : ∀ i, m i = generateFrom (π i)) (h_ind : iIndepSets π μ) : iIndep m μ := by
   classical
-  refine' Finset.induction _ _
-  ·
-    simp only [measure_univ, imp_true_iff, Set.iInter_false, Set.iInter_univ, Finset.prod_empty,
-      eq_self_iff_true]
-  intro a S ha_notin_S h_rec f hf_m
-  have hf_m_S : ∀ x ∈ S, measurable_set[m x] (f x) := fun x hx => hf_m x (by simp [hx])
-  rw [Finset.set_biInter_insert, Finset.prod_insert ha_notin_S, ← h_rec hf_m_S]
-  let p := piiUnionInter π S
-  set m_p := generate_from p with hS_eq_generate
-  have h_indep : indep m_p (m a) μ :=
-    by
-    have hp : IsPiSystem p := isPiSystem_piiUnionInter π h_pi S
-    have h_le' : ∀ i, generate_from (π i) ≤ m0 := fun i => (h_generate i).symm.trans_le (h_le i)
-    have hm_p : m_p ≤ m0 := generateFrom_piiUnionInter_le π h_le' S
-    exact
-      indep_sets.indep hm_p (h_le a) hp (h_pi a) hS_eq_generate (h_generate a)
-        (h_ind.pi_Union_Inter_of_not_mem ha_notin_S)
-  refine' h_indep.symm (f a) (⋂ n ∈ S, f n) (hf_m a (Finset.mem_insert_self a S)) _
-  have h_le_p : ∀ i ∈ S, m i ≤ m_p := by
-    intro n hn
-    rw [hS_eq_generate, h_generate n]
-    exact le_generateFrom_piiUnionInter S hn
-  have h_S_f : ∀ i ∈ S, measurable_set[m_p] (f i) := fun i hi => (h_le_p i hi) (f i) (hf_m_S i hi)
-  exact S.measurable_set_bInter h_S_f
 #align probability_theory.Indep_sets.Indep ProbabilityTheory.iIndepSets.iIndep
 -/
 
@@ -844,31 +733,6 @@ theorem iIndepFun_iff_measure_inter_preimage_eq_mul {ι : Type _} {β : ι → T
   refine' ⟨fun h S sets h_meas => h _ fun i hi_mem => ⟨sets i, h_meas i hi_mem, rfl⟩, _⟩
   intro h S setsΩ h_meas
   classical
-  let setsβ : ∀ i : ι, Set (β i) := fun i =>
-    dite (i ∈ S) (fun hi_mem => (h_meas i hi_mem).some) fun _ => Set.univ
-  have h_measβ : ∀ i ∈ S, measurable_set[m i] (setsβ i) :=
-    by
-    intro i hi_mem
-    simp_rw [setsβ, dif_pos hi_mem]
-    exact (h_meas i hi_mem).choose_spec.1
-  have h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i :=
-    by
-    intro i hi_mem
-    simp_rw [setsβ, dif_pos hi_mem]
-    exact (h_meas i hi_mem).choose_spec.2.symm
-  have h_left_eq : μ (⋂ i ∈ S, setsΩ i) = μ (⋂ i ∈ S, f i ⁻¹' setsβ i) :=
-    by
-    congr with (i x)
-    simp only [Set.mem_iInter]
-    constructor <;> intro h hi_mem <;> specialize h hi_mem
-    · rwa [h_preim i hi_mem] at h 
-    · rwa [h_preim i hi_mem]
-  have h_right_eq : ∏ i in S, μ (setsΩ i) = ∏ i in S, μ (f i ⁻¹' setsβ i) :=
-    by
-    refine' Finset.prod_congr rfl fun i hi_mem => _
-    rw [h_preim i hi_mem]
-  rw [h_left_eq, h_right_eq]
-  exact h S h_measβ
 #align probability_theory.Indep_fun_iff_measure_inter_preimage_eq_mul ProbabilityTheory.iIndepFun_iff_measure_inter_preimage_eq_mul
 -/
 
@@ -957,63 +821,6 @@ theorem iIndepFun.indepFun_finset [IsProbabilityMeasure μ] {ι : Type _} {β :
   simp only [Set.mem_univ_pi, Set.mem_setOf_eq] at hs1 ht1 
   rw [← hs2, ← ht2]
   classical
-  let sets_s' : ∀ i : ι, Set (β i) := fun i =>
-    dite (i ∈ S) (fun hi => sets_s ⟨i, hi⟩) fun _ => Set.univ
-  have h_sets_s'_eq : ∀ {i} (hi : i ∈ S), sets_s' i = sets_s ⟨i, hi⟩ := by intro i hi;
-    simp_rw [sets_s', dif_pos hi]
-  have h_sets_s'_univ : ∀ {i} (hi : i ∈ T), sets_s' i = Set.univ := by intro i hi;
-    simp_rw [sets_s', dif_neg (finset.disjoint_right.mp hST hi)]
-  let sets_t' : ∀ i : ι, Set (β i) := fun i =>
-    dite (i ∈ T) (fun hi => sets_t ⟨i, hi⟩) fun _ => Set.univ
-  have h_sets_t'_univ : ∀ {i} (hi : i ∈ S), sets_t' i = Set.univ := by intro i hi;
-    simp_rw [sets_t', dif_neg (finset.disjoint_left.mp hST hi)]
-  have h_meas_s' : ∀ i ∈ S, MeasurableSet (sets_s' i) := by intro i hi; rw [h_sets_s'_eq hi];
-    exact hs1 _
-  have h_meas_t' : ∀ i ∈ T, MeasurableSet (sets_t' i) := by intro i hi;
-    simp_rw [sets_t', dif_pos hi]; exact ht1 _
-  have h_eq_inter_S :
-    (fun (ω : Ω) (i : ↥S) => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i :=
-    by
-    ext1 x
-    simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
-    constructor <;> intro h
-    · intro i hi; rw [h_sets_s'_eq hi]; exact h ⟨i, hi⟩
-    · rintro ⟨i, hi⟩; specialize h i hi; rw [h_sets_s'_eq hi] at h ; exact h
-  have h_eq_inter_T :
-    (fun (ω : Ω) (i : ↥T) => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i :=
-    by
-    ext1 x
-    simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
-    constructor <;> intro h
-    · intro i hi; simp_rw [sets_t', dif_pos hi]; exact h ⟨i, hi⟩
-    · rintro ⟨i, hi⟩; specialize h i hi; simp_rw [sets_t', dif_pos hi] at h ; exact h
-  rw [Indep_fun_iff_measure_inter_preimage_eq_mul] at hf_Indep 
-  rw [h_eq_inter_S, h_eq_inter_T, hf_Indep S h_meas_s', hf_Indep T h_meas_t']
-  have h_Inter_inter :
-    ((⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i) =
-      ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) :=
-    by
-    ext1 x
-    simp only [Set.mem_inter_iff, Set.mem_iInter, Set.mem_preimage, Finset.mem_union]
-    constructor <;> intro h
-    · intro i hi
-      cases hi
-      · rw [h_sets_t'_univ hi]; exact ⟨h.1 i hi, Set.mem_univ _⟩
-      · rw [h_sets_s'_univ hi]; exact ⟨Set.mem_univ _, h.2 i hi⟩
-    · exact ⟨fun i hi => (h i (Or.inl hi)).1, fun i hi => (h i (Or.inr hi)).2⟩
-  rw [h_Inter_inter, hf_Indep (S ∪ T)]
-  swap
-  · intro i hi_mem
-    rw [Finset.mem_union] at hi_mem 
-    cases hi_mem
-    · rw [h_sets_t'_univ hi_mem, Set.inter_univ]; exact h_meas_s' i hi_mem
-    · rw [h_sets_s'_univ hi_mem, Set.univ_inter]; exact h_meas_t' i hi_mem
-  rw [Finset.prod_union hST]
-  congr 1
-  · refine' Finset.prod_congr rfl fun i hi => _
-    rw [h_sets_t'_univ hi, Set.inter_univ]
-  · refine' Finset.prod_congr rfl fun i hi => _
-    rw [h_sets_s'_univ hi, Set.univ_inter]
 #align probability_theory.Indep_fun.indep_fun_finset ProbabilityTheory.iIndepFun.indepFun_finset
 -/
 
@@ -1021,38 +828,7 @@ theorem iIndepFun.indepFun_finset [IsProbabilityMeasure μ] {ι : Type _} {β :
 theorem iIndepFun.indepFun_prod_mk [IsProbabilityMeasure μ] {ι : Type _} {β : ι → Type _}
     {m : ∀ i, MeasurableSpace (β i)} {f : ∀ i, Ω → β i} (hf_Indep : iIndepFun m f μ)
     (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
-    IndepFun (fun a => (f i a, f j a)) (f k) μ := by
-  classical
-  have h_right :
-    f k =
-      (fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩) ∘
-        fun a (j : ({k} : Finset ι)) => f j a :=
-    rfl
-  have h_meas_right :
-    Measurable fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩ :=
-    measurable_pi_apply ⟨k, Finset.mem_singleton_self k⟩
-  let s : Finset ι := {i, j}
-  have h_left :
-    (fun ω => (f i ω, f j ω)) =
-      (fun p : ∀ l : s, β l =>
-          (p ⟨i, Finset.mem_insert_self i _⟩,
-            p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩)) ∘
-        fun a (j : s) => f j a :=
-    by
-    ext1 a
-    simp only [Prod.mk.inj_iff]
-    constructor <;> rfl
-  have h_meas_left :
-    Measurable fun p : ∀ l : s, β l =>
-      (p ⟨i, Finset.mem_insert_self i _⟩,
-        p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩) :=
-    Measurable.prod (measurable_pi_apply ⟨i, Finset.mem_insert_self i {j}⟩)
-      (measurable_pi_apply ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self j)⟩)
-  rw [h_left, h_right]
-  refine' (hf_Indep.indep_fun_finset s {k} _ hf_meas).comp h_meas_left h_meas_right
-  rw [Finset.disjoint_singleton_right]
-  simp only [Finset.mem_insert, Finset.mem_singleton, not_or]
-  exact ⟨hik.symm, hjk.symm⟩
+    IndepFun (fun a => (f i a, f j a)) (f k) μ := by classical
 #align probability_theory.Indep_fun.indep_fun_prod ProbabilityTheory.iIndepFun.indepFun_prod_mk
 -/
 
@@ -1076,28 +852,7 @@ theorem iIndepFun.indepFun_mul_left [IsProbabilityMeasure μ] {ι : Type _} {β
 theorem iIndepFun.indepFun_finset_prod_of_not_mem [IsProbabilityMeasure μ] {ι : Type _} {β : Type _}
     {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂ β] {f : ι → Ω → β}
     (hf_Indep : iIndepFun (fun _ => m) f μ) (hf_meas : ∀ i, Measurable (f i)) {s : Finset ι} {i : ι}
-    (hi : i ∉ s) : IndepFun (∏ j in s, f j) (f i) μ := by
-  classical
-  have h_right :
-    f i =
-      (fun p : ∀ j : ({i} : Finset ι), β => p ⟨i, Finset.mem_singleton_self i⟩) ∘
-        fun a (j : ({i} : Finset ι)) => f j a :=
-    rfl
-  have h_meas_right :
-    Measurable fun p : ∀ j : ({i} : Finset ι), β => p ⟨i, Finset.mem_singleton_self i⟩ :=
-    measurable_pi_apply ⟨i, Finset.mem_singleton_self i⟩
-  have h_left : ∏ j in s, f j = (fun p : ∀ j : s, β => ∏ j, p j) ∘ fun a (j : s) => f j a :=
-    by
-    ext1 a
-    simp only [Function.comp_apply]
-    have : ∏ j : ↥s, f (↑j) a = (∏ j : ↥s, f ↑j) a := by rw [Finset.prod_apply]
-    rw [this, Finset.prod_coe_sort]
-  have h_meas_left : Measurable fun p : ∀ j : s, β => ∏ j, p j :=
-    finset.univ.measurable_prod fun (j : ↥s) (H : j ∈ Finset.univ) => measurable_pi_apply j
-  rw [h_left, h_right]
-  exact
-    (hf_Indep.indep_fun_finset s {i} (finset.disjoint_singleton_left.mpr hi).symm hf_meas).comp
-      h_meas_left h_meas_right
+    (hi : i ∉ s) : IndepFun (∏ j in s, f j) (f i) μ := by classical
 #align probability_theory.Indep_fun.indep_fun_finset_prod_of_not_mem ProbabilityTheory.iIndepFun.indepFun_finset_prod_of_not_mem
 #align probability_theory.Indep_fun.indep_fun_finset_sum_of_not_mem ProbabilityTheory.iIndepFun.indepFun_finset_sum_of_not_mem
 -/
@@ -1117,17 +872,6 @@ theorem iIndepFun.indepFun_prod_range_succ [IsProbabilityMeasure μ] {β : Type
 theorem iIndepSet.iIndepFun_indicator [Zero β] [One β] {m : MeasurableSpace β} {s : ι → Set Ω}
     (hs : iIndepSet s μ) : iIndepFun (fun n => m) (fun n => (s n).indicator fun ω => 1) μ := by
   classical
-  rw [Indep_fun_iff_measure_inter_preimage_eq_mul]
-  rintro S π hπ
-  simp_rw [Set.indicator_const_preimage_eq_union]
-  refine' @hs S (fun i => ite (1 ∈ π i) (s i) ∅ ∪ ite ((0 : β) ∈ π i) (s iᶜ) ∅) fun i hi => _
-  have hsi : measurable_set[generate_from {s i}] (s i) :=
-    measurable_set_generate_from (Set.mem_singleton _)
-  refine'
-    MeasurableSet.union (MeasurableSet.ite' (fun _ => hsi) fun _ => _)
-      (MeasurableSet.ite' (fun _ => hsi.compl) fun _ => _)
-  · exact @MeasurableSet.empty _ (generate_from {s i})
-  · exact @MeasurableSet.empty _ (generate_from {s i})
 #align probability_theory.Indep_set.Indep_fun_indicator ProbabilityTheory.iIndepSet.iIndepFun_indicator
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -1017,8 +1017,8 @@ theorem iIndepFun.indepFun_finset [IsProbabilityMeasure μ] {ι : Type _} {β :
 #align probability_theory.Indep_fun.indep_fun_finset ProbabilityTheory.iIndepFun.indepFun_finset
 -/
 
-#print ProbabilityTheory.iIndepFun.indepFun_prod /-
-theorem iIndepFun.indepFun_prod [IsProbabilityMeasure μ] {ι : Type _} {β : ι → Type _}
+#print ProbabilityTheory.iIndepFun.indepFun_prod_mk /-
+theorem iIndepFun.indepFun_prod_mk [IsProbabilityMeasure μ] {ι : Type _} {β : ι → Type _}
     {m : ∀ i, MeasurableSpace (β i)} {f : ∀ i, Ω → β i} (hf_Indep : iIndepFun m f μ)
     (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
     IndepFun (fun a => (f i a, f j a)) (f k) μ := by
@@ -1053,22 +1053,22 @@ theorem iIndepFun.indepFun_prod [IsProbabilityMeasure μ] {ι : Type _} {β : ι
   rw [Finset.disjoint_singleton_right]
   simp only [Finset.mem_insert, Finset.mem_singleton, not_or]
   exact ⟨hik.symm, hjk.symm⟩
-#align probability_theory.Indep_fun.indep_fun_prod ProbabilityTheory.iIndepFun.indepFun_prod
+#align probability_theory.Indep_fun.indep_fun_prod ProbabilityTheory.iIndepFun.indepFun_prod_mk
 -/
 
-#print ProbabilityTheory.iIndepFun.mul /-
+#print ProbabilityTheory.iIndepFun.indepFun_mul_left /-
 @[to_additive]
-theorem iIndepFun.mul [IsProbabilityMeasure μ] {ι : Type _} {β : Type _} {m : MeasurableSpace β}
-    [Mul β] [MeasurableMul₂ β] {f : ι → Ω → β} (hf_Indep : iIndepFun (fun _ => m) f μ)
-    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
-    IndepFun (f i * f j) (f k) μ :=
+theorem iIndepFun.indepFun_mul_left [IsProbabilityMeasure μ] {ι : Type _} {β : Type _}
+    {m : MeasurableSpace β} [Mul β] [MeasurableMul₂ β] {f : ι → Ω → β}
+    (hf_Indep : iIndepFun (fun _ => m) f μ) (hf_meas : ∀ i, Measurable (f i)) (i j k : ι)
+    (hik : i ≠ k) (hjk : j ≠ k) : IndepFun (f i * f j) (f k) μ :=
   by
   have : indep_fun (fun ω => (f i ω, f j ω)) (f k) μ :=
     hf_Indep.indep_fun_prod hf_meas i j k hik hjk
   change indep_fun ((fun p : β × β => p.fst * p.snd) ∘ fun ω => (f i ω, f j ω)) (id ∘ f k) μ
   exact indep_fun.comp this (measurable_fst.mul measurable_snd) measurable_id
-#align probability_theory.Indep_fun.mul ProbabilityTheory.iIndepFun.mul
-#align probability_theory.Indep_fun.add ProbabilityTheory.iIndepFun.add
+#align probability_theory.Indep_fun.mul ProbabilityTheory.iIndepFun.indepFun_mul_left
+#align probability_theory.Indep_fun.add ProbabilityTheory.iIndepFun.indepFun_add_left
 -/
 
 #print ProbabilityTheory.iIndepFun.indepFun_finset_prod_of_not_mem /-

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -652,12 +652,10 @@ theorem iIndepSets.piiUnionInter_of_not_mem {π : ι → Set (Set Ω)} {a : ι}
 #align probability_theory.Indep_sets.pi_Union_Inter_of_not_mem ProbabilityTheory.iIndepSets.piiUnionInter_of_not_mem
 -/
 
-/- warning: probability_theory.Indep_sets.Indep clashes with probability_theory.indep_sets.indep -> ProbabilityTheory.IndepSets.indep
-Case conversion may be inaccurate. Consider using '#align probability_theory.Indep_sets.Indep ProbabilityTheory.IndepSets.indepₓ'. -/
-#print ProbabilityTheory.IndepSets.indep /-
+#print ProbabilityTheory.iIndepSets.iIndep /-
 /-- The measurable space structures generated by independent pi-systems are independent. -/
-theorem IndepSets.indep [IsProbabilityMeasure μ] (m : ι → MeasurableSpace Ω) (h_le : ∀ i, m i ≤ m0)
-    (π : ι → Set (Set Ω)) (h_pi : ∀ n, IsPiSystem (π n))
+theorem iIndepSets.iIndep [IsProbabilityMeasure μ] (m : ι → MeasurableSpace Ω)
+    (h_le : ∀ i, m i ≤ m0) (π : ι → Set (Set Ω)) (h_pi : ∀ n, IsPiSystem (π n))
     (h_generate : ∀ i, m i = generateFrom (π i)) (h_ind : iIndepSets π μ) : iIndep m μ := by
   classical
   refine' Finset.induction _ _
@@ -684,7 +682,7 @@ theorem IndepSets.indep [IsProbabilityMeasure μ] (m : ι → MeasurableSpace Ω
     exact le_generateFrom_piiUnionInter S hn
   have h_S_f : ∀ i ∈ S, measurable_set[m_p] (f i) := fun i hi => (h_le_p i hi) (f i) (hf_m_S i hi)
   exact S.measurable_set_bInter h_S_f
-#align probability_theory.Indep_sets.Indep ProbabilityTheory.IndepSets.indep
+#align probability_theory.Indep_sets.Indep ProbabilityTheory.iIndepSets.iIndep
 -/
 
 end FromPiSystemsToMeasurableSpaces
@@ -759,18 +757,23 @@ theorem indep_iff_forall_indepSet (m₁ m₂ : MeasurableSpace Ω) {m0 : Measura
 #align probability_theory.indep_iff_forall_indep_set ProbabilityTheory.indep_iff_forall_indepSet
 -/
 
+#print ProbabilityTheory.iIndepSets.meas_iInter /-
 theorem iIndepSets.meas_iInter [Fintype ι] (h : iIndepSets π μ) (hf : ∀ i, f i ∈ π i) :
     μ (⋂ i, f i) = ∏ i, μ (f i) := by simp [← h _ fun i _ => hf _]
 #align probability_theory.Indep_sets.meas_Inter ProbabilityTheory.iIndepSets.meas_iInter
+-/
 
+#print ProbabilityTheory.iIndep_comap_mem_iff /-
 theorem iIndep_comap_mem_iff :
     iIndep (fun i => MeasurableSpace.comap (· ∈ f i) ⊤) μ ↔ iIndepSet f μ := by
   simp_rw [← generate_from_singleton]; rfl
 #align probability_theory.Indep_comap_mem_iff ProbabilityTheory.iIndep_comap_mem_iff
+-/
 
 alias ⟨_, Indep_set.Indep_comap_mem⟩ := Indep_comap_mem_iff
 #align probability_theory.Indep_set.Indep_comap_mem ProbabilityTheory.iIndepSet.iIndep_comap_mem
 
+#print ProbabilityTheory.iIndepSets_singleton_iff /-
 theorem iIndepSets_singleton_iff :
     iIndepSets (fun i => {f i}) μ ↔ ∀ t, μ (⋂ i ∈ t, f i) = ∏ i in t, μ (f i) :=
   forall_congr' fun t =>
@@ -780,25 +783,32 @@ theorem iIndepSets_singleton_iff :
         Eq.trans _ (h.trans <| Finset.prod_congr rfl fun i hi => congr_arg _ <| (hf i hi).symm)
       rw [Inter₂_congr hf]⟩
 #align probability_theory.Indep_sets_singleton_iff ProbabilityTheory.iIndepSets_singleton_iff
+-/
 
 variable [IsProbabilityMeasure μ]
 
+#print ProbabilityTheory.iIndepSet_iff_iIndepSets_singleton /-
 theorem iIndepSet_iff_iIndepSets_singleton (hf : ∀ i, MeasurableSet (f i)) :
     iIndepSet f μ ↔ iIndepSets (fun i => {f i}) μ :=
   ⟨iIndep.iIndepSets fun _ => rfl,
-    IndepSets.indep _ (fun i => generateFrom_le <| by rintro t (rfl : t = _); exact hf _) _
+    iIndepSets.iIndep _ (fun i => generateFrom_le <| by rintro t (rfl : t = _); exact hf _) _
       (fun _ => IsPiSystem.singleton _) fun _ => rfl⟩
 #align probability_theory.Indep_set_iff_Indep_sets_singleton ProbabilityTheory.iIndepSet_iff_iIndepSets_singleton
+-/
 
-theorem iIndepSet_iff_measure_iInter_eq_prod (hf : ∀ i, MeasurableSet (f i)) :
+#print ProbabilityTheory.iIndepSet_iff_meas_biInter /-
+theorem iIndepSet_iff_meas_biInter (hf : ∀ i, MeasurableSet (f i)) :
     iIndepSet f μ ↔ ∀ s, μ (⋂ i ∈ s, f i) = ∏ i in s, μ (f i) :=
   (iIndepSet_iff_iIndepSets_singleton hf).trans iIndepSets_singleton_iff
-#align probability_theory.Indep_set_iff_measure_Inter_eq_prod ProbabilityTheory.iIndepSet_iff_measure_iInter_eq_prod
+#align probability_theory.Indep_set_iff_measure_Inter_eq_prod ProbabilityTheory.iIndepSet_iff_meas_biInter
+-/
 
+#print ProbabilityTheory.iIndepSets.iIndepSet_of_mem /-
 theorem iIndepSets.iIndepSet_of_mem (hfπ : ∀ i, f i ∈ π i) (hf : ∀ i, MeasurableSet (f i))
     (hπ : iIndepSets π μ) : iIndepSet f μ :=
-  (iIndepSet_iff_measure_iInter_eq_prod hf).2 fun t => hπ _ fun i _ => hfπ _
+  (iIndepSet_iff_meas_biInter hf).2 fun t => hπ _ fun i _ => hfπ _
 #align probability_theory.Indep_sets.Indep_set_of_mem ProbabilityTheory.iIndepSets.iIndepSet_of_mem
+-/
 
 end IndepSet
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe

@@ -609,7 +609,7 @@ theorem indep_iSup_of_antitone [SemilatticeInf ι] {Ω} {m : ι → MeasurableSp
     {m' m0 : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
     (h_indep : ∀ i, Indep (m i) m' μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0) (hm : Antitone m) :
     Indep (⨆ i, m i) m' μ :=
-  indep_iSup_of_directed_le h_indep h_le h_le' (directed_of_inf hm)
+  indep_iSup_of_directed_le h_indep h_le h_le' (directed_of_isDirected_ge hm)
 #align probability_theory.indep_supr_of_antitone ProbabilityTheory.indep_iSup_of_antitone
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,7 +3,7 @@ Copyright (c) 2021 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 -/
-import Mathbin.MeasureTheory.Constructions.Pi
+import MeasureTheory.Constructions.Pi
 
 #align_import probability.independence.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/001ffdc42920050657fd45bd2b8bfbec8eaaeb29

@@ -5,7 +5,7 @@ Authors: Rémy Degenne
 -/
 import Mathbin.MeasureTheory.Constructions.Pi
 
-#align_import probability.independence.basic from "leanprover-community/mathlib"@"af471b9e3ce868f296626d33189b4ce730fa4c00"
+#align_import probability.independence.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
 
 /-!
 # Independence of sets of sets and measure spaces (σ-algebras)
@@ -67,7 +67,7 @@ Part A, Chapter 4.
 -/
 
 
-open MeasureTheory MeasurableSpace
+open MeasureTheory MeasurableSpace Set
 
 open scoped BigOperators MeasureTheory ENNReal
 
@@ -699,7 +699,8 @@ We prove the following equivalences on `indep_set`, for measurable sets `s, t`.
 -/
 
 
-variable {s t : Set Ω} (S T : Set (Set Ω))
+variable {s t : Set Ω} (S T : Set (Set Ω)) {π : ι → Set (Set Ω)} {f : ι → Set Ω}
+  {m0 : MeasurableSpace Ω} {μ : Measure Ω}
 
 #print ProbabilityTheory.indepSet_iff_indepSets_singleton /-
 theorem indepSet_iff_indepSets_singleton {m0 : MeasurableSpace Ω} (hs_meas : MeasurableSet s)
@@ -758,6 +759,47 @@ theorem indep_iff_forall_indepSet (m₁ m₂ : MeasurableSpace Ω) {m0 : Measura
 #align probability_theory.indep_iff_forall_indep_set ProbabilityTheory.indep_iff_forall_indepSet
 -/
 
+theorem iIndepSets.meas_iInter [Fintype ι] (h : iIndepSets π μ) (hf : ∀ i, f i ∈ π i) :
+    μ (⋂ i, f i) = ∏ i, μ (f i) := by simp [← h _ fun i _ => hf _]
+#align probability_theory.Indep_sets.meas_Inter ProbabilityTheory.iIndepSets.meas_iInter
+
+theorem iIndep_comap_mem_iff :
+    iIndep (fun i => MeasurableSpace.comap (· ∈ f i) ⊤) μ ↔ iIndepSet f μ := by
+  simp_rw [← generate_from_singleton]; rfl
+#align probability_theory.Indep_comap_mem_iff ProbabilityTheory.iIndep_comap_mem_iff
+
+alias ⟨_, Indep_set.Indep_comap_mem⟩ := Indep_comap_mem_iff
+#align probability_theory.Indep_set.Indep_comap_mem ProbabilityTheory.iIndepSet.iIndep_comap_mem
+
+theorem iIndepSets_singleton_iff :
+    iIndepSets (fun i => {f i}) μ ↔ ∀ t, μ (⋂ i ∈ t, f i) = ∏ i in t, μ (f i) :=
+  forall_congr' fun t =>
+    ⟨fun h => h fun _ _ => mem_singleton _, fun h f hf =>
+      by
+      refine'
+        Eq.trans _ (h.trans <| Finset.prod_congr rfl fun i hi => congr_arg _ <| (hf i hi).symm)
+      rw [Inter₂_congr hf]⟩
+#align probability_theory.Indep_sets_singleton_iff ProbabilityTheory.iIndepSets_singleton_iff
+
+variable [IsProbabilityMeasure μ]
+
+theorem iIndepSet_iff_iIndepSets_singleton (hf : ∀ i, MeasurableSet (f i)) :
+    iIndepSet f μ ↔ iIndepSets (fun i => {f i}) μ :=
+  ⟨iIndep.iIndepSets fun _ => rfl,
+    IndepSets.indep _ (fun i => generateFrom_le <| by rintro t (rfl : t = _); exact hf _) _
+      (fun _ => IsPiSystem.singleton _) fun _ => rfl⟩
+#align probability_theory.Indep_set_iff_Indep_sets_singleton ProbabilityTheory.iIndepSet_iff_iIndepSets_singleton
+
+theorem iIndepSet_iff_measure_iInter_eq_prod (hf : ∀ i, MeasurableSet (f i)) :
+    iIndepSet f μ ↔ ∀ s, μ (⋂ i ∈ s, f i) = ∏ i in s, μ (f i) :=
+  (iIndepSet_iff_iIndepSets_singleton hf).trans iIndepSets_singleton_iff
+#align probability_theory.Indep_set_iff_measure_Inter_eq_prod ProbabilityTheory.iIndepSet_iff_measure_iInter_eq_prod
+
+theorem iIndepSets.iIndepSet_of_mem (hfπ : ∀ i, f i ∈ π i) (hf : ∀ i, MeasurableSet (f i))
+    (hπ : iIndepSets π μ) : iIndepSet f μ :=
+  (iIndepSet_iff_measure_iInter_eq_prod hf).2 fun t => hπ _ fun i _ => hfπ _
+#align probability_theory.Indep_sets.Indep_set_of_mem ProbabilityTheory.iIndepSets.iIndepSet_of_mem
+
 end IndepSet
 
 section IndepFun

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,14 +2,11 @@
 Copyright (c) 2021 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
-
-! This file was ported from Lean 3 source module probability.independence.basic
-! leanprover-community/mathlib commit af471b9e3ce868f296626d33189b4ce730fa4c00
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.MeasureTheory.Constructions.Pi
 
+#align_import probability.independence.basic from "leanprover-community/mathlib"@"af471b9e3ce868f296626d33189b4ce730fa4c00"
+
 /-!
 # Independence of sets of sets and measure spaces (σ-algebras)
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -182,6 +182,7 @@ theorem Indep.symm {m₁ m₂ : MeasurableSpace Ω} [MeasurableSpace Ω] {μ : M
 #align probability_theory.indep.symm ProbabilityTheory.Indep.symm
 -/
 
+#print ProbabilityTheory.indep_bot_right /-
 theorem indep_bot_right (m' : MeasurableSpace Ω) {m : MeasurableSpace Ω} {μ : Measure Ω}
     [IsProbabilityMeasure μ] : Indep m' ⊥ μ :=
   by
@@ -191,11 +192,14 @@ theorem indep_bot_right (m' : MeasurableSpace Ω) {m : MeasurableSpace Ω} {μ :
   · rw [ht, Set.inter_empty, measure_empty, MulZeroClass.mul_zero]
   · rw [ht, Set.inter_univ, measure_univ, mul_one]
 #align probability_theory.indep_bot_right ProbabilityTheory.indep_bot_right
+-/
 
+#print ProbabilityTheory.indep_bot_left /-
 theorem indep_bot_left (m' : MeasurableSpace Ω) {m : MeasurableSpace Ω} {μ : Measure Ω}
     [IsProbabilityMeasure μ] : Indep ⊥ m' μ :=
   (indep_bot_right m').symm
 #align probability_theory.indep_bot_left ProbabilityTheory.indep_bot_left
+-/
 
 #print ProbabilityTheory.indepSet_empty_right /-
 theorem indepSet_empty_right {m : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
@@ -239,6 +243,7 @@ theorem indep_of_indep_of_le_right {m₁ m₂ m₃ : MeasurableSpace Ω} [Measur
 #align probability_theory.indep_of_indep_of_le_right ProbabilityTheory.indep_of_indep_of_le_right
 -/
 
+#print ProbabilityTheory.IndepSets.union /-
 theorem IndepSets.union [MeasurableSpace Ω] {s₁ s₂ s' : Set (Set Ω)} {μ : Measure Ω}
     (h₁ : IndepSets s₁ s' μ) (h₂ : IndepSets s₂ s' μ) : IndepSets (s₁ ∪ s₂) s' μ :=
   by
@@ -247,7 +252,9 @@ theorem IndepSets.union [MeasurableSpace Ω] {s₁ s₂ s' : Set (Set Ω)} {μ :
   · exact h₁ t1 t2 ht1₁ ht2
   · exact h₂ t1 t2 ht1₂ ht2
 #align probability_theory.indep_sets.union ProbabilityTheory.IndepSets.union
+-/
 
+#print ProbabilityTheory.IndepSets.union_iff /-
 @[simp]
 theorem IndepSets.union_iff [MeasurableSpace Ω] {s₁ s₂ s' : Set (Set Ω)} {μ : Measure Ω} :
     IndepSets (s₁ ∪ s₂) s' μ ↔ IndepSets s₁ s' μ ∧ IndepSets s₂ s' μ :=
@@ -256,7 +263,9 @@ theorem IndepSets.union_iff [MeasurableSpace Ω] {s₁ s₂ s' : Set (Set Ω)} {
       indepSets_of_indepSets_of_le_left h (Set.subset_union_right s₁ s₂)⟩,
     fun h => IndepSets.union h.left h.right⟩
 #align probability_theory.indep_sets.union_iff ProbabilityTheory.IndepSets.union_iff
+-/
 
+#print ProbabilityTheory.IndepSets.iUnion /-
 theorem IndepSets.iUnion [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
     {μ : Measure Ω} (hyp : ∀ n, IndepSets (s n) s' μ) : IndepSets (⋃ n, s n) s' μ :=
   by
@@ -265,7 +274,9 @@ theorem IndepSets.iUnion [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Se
   cases' ht1 with n ht1
   exact hyp n t1 t2 ht1 ht2
 #align probability_theory.indep_sets.Union ProbabilityTheory.IndepSets.iUnion
+-/
 
+#print ProbabilityTheory.IndepSets.bUnion /-
 theorem IndepSets.bUnion [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
     {μ : Measure Ω} {u : Set ι} (hyp : ∀ n ∈ u, IndepSets (s n) s' μ) :
     IndepSets (⋃ n ∈ u, s n) s' μ := by
@@ -274,17 +285,23 @@ theorem IndepSets.bUnion [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Se
   rcases ht1 with ⟨n, hpn, ht1⟩
   exact hyp n hpn t1 t2 ht1 ht2
 #align probability_theory.indep_sets.bUnion ProbabilityTheory.IndepSets.bUnion
+-/
 
+#print ProbabilityTheory.IndepSets.inter /-
 theorem IndepSets.inter [MeasurableSpace Ω] {s₁ s' : Set (Set Ω)} (s₂ : Set (Set Ω)) {μ : Measure Ω}
     (h₁ : IndepSets s₁ s' μ) : IndepSets (s₁ ∩ s₂) s' μ := fun t1 t2 ht1 ht2 =>
   h₁ t1 t2 ((Set.mem_inter_iff _ _ _).mp ht1).left ht2
 #align probability_theory.indep_sets.inter ProbabilityTheory.IndepSets.inter
+-/
 
+#print ProbabilityTheory.IndepSets.iInter /-
 theorem IndepSets.iInter [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
     {μ : Measure Ω} (h : ∃ n, IndepSets (s n) s' μ) : IndepSets (⋂ n, s n) s' μ := by
   intro t1 t2 ht1 ht2; cases' h with n h; exact h t1 t2 (set.mem_Inter.mp ht1 n) ht2
 #align probability_theory.indep_sets.Inter ProbabilityTheory.IndepSets.iInter
+-/
 
+#print ProbabilityTheory.IndepSets.bInter /-
 theorem IndepSets.bInter [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
     {μ : Measure Ω} {u : Set ι} (h : ∃ n ∈ u, IndepSets (s n) s' μ) :
     IndepSets (⋂ n ∈ u, s n) s' μ := by
@@ -292,12 +309,15 @@ theorem IndepSets.bInter [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Se
   rcases h with ⟨n, hn, h⟩
   exact h t1 t2 (Set.biInter_subset_of_mem hn ht1) ht2
 #align probability_theory.indep_sets.bInter ProbabilityTheory.IndepSets.bInter
+-/
 
+#print ProbabilityTheory.indepSets_singleton_iff /-
 theorem indepSets_singleton_iff [MeasurableSpace Ω] {s t : Set Ω} {μ : Measure Ω} :
     IndepSets {s} {t} μ ↔ μ (s ∩ t) = μ s * μ t :=
   ⟨fun h => h s t rfl rfl, fun h s1 t1 hs1 ht1 => by
     rwa [set.mem_singleton_iff.mp hs1, set.mem_singleton_iff.mp ht1]⟩
 #align probability_theory.indep_sets_singleton_iff ProbabilityTheory.indepSets_singleton_iff
+-/
 
 end Indep
 
@@ -306,6 +326,7 @@ end Indep
 
 section FromIndepToIndep
 
+#print ProbabilityTheory.iIndepSets.indepSets /-
 theorem iIndepSets.indepSets {s : ι → Set (Set Ω)} [MeasurableSpace Ω] {μ : Measure Ω}
     (h_indep : iIndepSets s μ) {i j : ι} (hij : i ≠ j) : IndepSets (s i) (s j) μ := by
   classical
@@ -332,19 +353,24 @@ theorem iIndepSets.indepSets {s : ι → Set (Set Ω)} [MeasurableSpace Ω] {μ
   nth_rw 4 [h2]
   rw [← h_inter, ← h_prod, h_indep {i, j} hf_m]
 #align probability_theory.Indep_sets.indep_sets ProbabilityTheory.iIndepSets.indepSets
+-/
 
+#print ProbabilityTheory.iIndep.indep /-
 theorem iIndep.indep {m : ι → MeasurableSpace Ω} [MeasurableSpace Ω] {μ : Measure Ω}
     (h_indep : iIndep m μ) {i j : ι} (hij : i ≠ j) : Indep (m i) (m j) μ :=
   by
   change indep_sets ((fun x => measurable_set[m x]) i) ((fun x => measurable_set[m x]) j) μ
   exact Indep_sets.indep_sets h_indep hij
 #align probability_theory.Indep.indep ProbabilityTheory.iIndep.indep
+-/
 
+#print ProbabilityTheory.iIndepFun.indepFun /-
 theorem iIndepFun.indepFun {m₀ : MeasurableSpace Ω} {μ : Measure Ω} {β : ι → Type _}
     {m : ∀ x, MeasurableSpace (β x)} {f : ∀ i, Ω → β i} (hf_Indep : iIndepFun m f μ) {i j : ι}
     (hij : i ≠ j) : IndepFun (f i) (f j) μ :=
   hf_Indep.indep hij
 #align probability_theory.Indep_fun.indep_fun ProbabilityTheory.iIndepFun.indepFun
+-/
 
 end FromIndepToIndep
 
@@ -360,12 +386,14 @@ section FromMeasurableSpacesToSetsOfSets
 /-! ### Independence of measurable space structures implies independence of generating π-systems -/
 
 
+#print ProbabilityTheory.iIndep.iIndepSets /-
 theorem iIndep.iIndepSets [MeasurableSpace Ω] {μ : Measure Ω} {m : ι → MeasurableSpace Ω}
     {s : ι → Set (Set Ω)} (hms : ∀ n, m n = generateFrom (s n)) (h_indep : iIndep m μ) :
     iIndepSets s μ := fun S f hfs =>
   h_indep S fun x hxS =>
     ((hms x).symm ▸ measurableSet_generateFrom (hfs x hxS) : measurable_set[m x] (f x))
 #align probability_theory.Indep.Indep_sets ProbabilityTheory.iIndep.iIndepSets
+-/
 
 #print ProbabilityTheory.Indep.indepSets /-
 theorem Indep.indepSets [MeasurableSpace Ω] {μ : Measure Ω} {s1 s2 : Set (Set Ω)}
@@ -435,6 +463,7 @@ theorem IndepSets.indep' {m : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabili
 
 variable {m0 : MeasurableSpace Ω} {μ : Measure Ω}
 
+#print ProbabilityTheory.indepSets_piiUnionInter_of_disjoint /-
 theorem indepSets_piiUnionInter_of_disjoint [IsProbabilityMeasure μ] {s : ι → Set (Set Ω)}
     {S T : Set ι} (h_indep : iIndepSets s μ) (hST : Disjoint S T) :
     IndepSets (piiUnionInter s S) (piiUnionInter s T) μ :=
@@ -477,7 +506,9 @@ theorem indepSets_piiUnionInter_of_disjoint [IsProbabilityMeasure μ] {s : ι 
     Finset.prod_ite_mem (p1 ∪ p2) p2 fun x => μ (f2 x), Finset.union_inter_cancel_right, ht1_eq, ←
     h_indep p1 ht1_m, ht2_eq, ← h_indep p2 ht2_m]
 #align probability_theory.indep_sets_pi_Union_Inter_of_disjoint ProbabilityTheory.indepSets_piiUnionInter_of_disjoint
+-/
 
+#print ProbabilityTheory.iIndepSet.indep_generateFrom_of_disjoint /-
 theorem iIndepSet.indep_generateFrom_of_disjoint [IsProbabilityMeasure μ] {s : ι → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (S T : Set ι) (hST : Disjoint S T) :
     Indep (generateFrom {t | ∃ n ∈ S, s n = t}) (generateFrom {t | ∃ k ∈ T, s k = t}) μ :=
@@ -493,7 +524,9 @@ theorem iIndepSet.indep_generateFrom_of_disjoint [IsProbabilityMeasure μ] {s :
   · exact isPiSystem_piiUnionInter _ (fun k => IsPiSystem.singleton _) _
   · classical exact indep_sets_pi_Union_Inter_of_disjoint (Indep.Indep_sets (fun n => rfl) hs) hST
 #align probability_theory.Indep_set.indep_generate_from_of_disjoint ProbabilityTheory.iIndepSet.indep_generateFrom_of_disjoint
+-/
 
+#print ProbabilityTheory.indep_iSup_of_disjoint /-
 theorem indep_iSup_of_disjoint [IsProbabilityMeasure μ] {m : ι → MeasurableSpace Ω}
     (h_le : ∀ i, m i ≤ m0) (h_indep : iIndep m μ) {S T : Set ι} (hST : Disjoint S T) :
     Indep (⨆ i ∈ S, m i) (⨆ i ∈ T, m i) μ :=
@@ -506,7 +539,9 @@ theorem indep_iSup_of_disjoint [IsProbabilityMeasure μ] {m : ι → MeasurableS
   · exact isPiSystem_piiUnionInter _ (fun n => @is_pi_system_measurable_set Ω (m n)) _
   · classical exact indep_sets_pi_Union_Inter_of_disjoint h_indep hST
 #align probability_theory.indep_supr_of_disjoint ProbabilityTheory.indep_iSup_of_disjoint
+-/
 
+#print ProbabilityTheory.indep_iSup_of_directed_le /-
 theorem indep_iSup_of_directed_le {Ω} {m : ι → MeasurableSpace Ω} {m' m0 : MeasurableSpace Ω}
     {μ : Measure Ω} [IsProbabilityMeasure μ] (h_indep : ∀ i, Indep (m i) m' μ)
     (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0) (hm : Directed (· ≤ ·) m) : Indep (⨆ i, m i) m' μ :=
@@ -529,7 +564,9 @@ theorem indep_iSup_of_directed_le {Ω} {m : ι → MeasurableSpace Ω} {m' m0 :
   refine' indep_sets.indep (iSup_le h_le) h_le' hp_supr_pi hp'_pi _ h_gen' h_pi_system_indep
   exact (generate_from_Union_measurable_set _).symm
 #align probability_theory.indep_supr_of_directed_le ProbabilityTheory.indep_iSup_of_directed_le
+-/
 
+#print ProbabilityTheory.iIndepSet.indep_generateFrom_lt /-
 theorem iIndepSet.indep_generateFrom_lt [Preorder ι] [IsProbabilityMeasure μ] {s : ι → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (i : ι) :
     Indep (generateFrom {s i}) (generateFrom {t | ∃ j < i, s j = t}) μ :=
@@ -539,7 +576,9 @@ theorem iIndepSet.indep_generateFrom_lt [Preorder ι] [IsProbabilityMeasure μ]
       (set.disjoint_singleton_left.mpr (lt_irrefl _))
   simp only [Set.mem_singleton_iff, exists_prop, exists_eq_left, Set.setOf_eq_eq_singleton']
 #align probability_theory.Indep_set.indep_generate_from_lt ProbabilityTheory.iIndepSet.indep_generateFrom_lt
+-/
 
+#print ProbabilityTheory.iIndepSet.indep_generateFrom_le /-
 theorem iIndepSet.indep_generateFrom_le [LinearOrder ι] [IsProbabilityMeasure μ] {s : ι → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (i : ι) {k : ι} (hk : i < k) :
     Indep (generateFrom {s k}) (generateFrom {t | ∃ j ≤ i, s j = t}) μ :=
@@ -549,27 +588,35 @@ theorem iIndepSet.indep_generateFrom_le [LinearOrder ι] [IsProbabilityMeasure 
       (set.disjoint_singleton_left.mpr hk.not_le)
   simp only [Set.mem_singleton_iff, exists_prop, exists_eq_left, Set.setOf_eq_eq_singleton']
 #align probability_theory.Indep_set.indep_generate_from_le ProbabilityTheory.iIndepSet.indep_generateFrom_le
+-/
 
+#print ProbabilityTheory.iIndepSet.indep_generateFrom_le_nat /-
 theorem iIndepSet.indep_generateFrom_le_nat [IsProbabilityMeasure μ] {s : ℕ → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (n : ℕ) :
     Indep (generateFrom {s (n + 1)}) (generateFrom {t | ∃ k ≤ n, s k = t}) μ :=
   hs.indep_generateFrom_le hsm _ n.lt_succ_self
 #align probability_theory.Indep_set.indep_generate_from_le_nat ProbabilityTheory.iIndepSet.indep_generateFrom_le_nat
+-/
 
+#print ProbabilityTheory.indep_iSup_of_monotone /-
 theorem indep_iSup_of_monotone [SemilatticeSup ι] {Ω} {m : ι → MeasurableSpace Ω}
     {m' m0 : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
     (h_indep : ∀ i, Indep (m i) m' μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0) (hm : Monotone m) :
     Indep (⨆ i, m i) m' μ :=
   indep_iSup_of_directed_le h_indep h_le h_le' (Monotone.directed_le hm)
 #align probability_theory.indep_supr_of_monotone ProbabilityTheory.indep_iSup_of_monotone
+-/
 
+#print ProbabilityTheory.indep_iSup_of_antitone /-
 theorem indep_iSup_of_antitone [SemilatticeInf ι] {Ω} {m : ι → MeasurableSpace Ω}
     {m' m0 : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
     (h_indep : ∀ i, Indep (m i) m' μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0) (hm : Antitone m) :
     Indep (⨆ i, m i) m' μ :=
   indep_iSup_of_directed_le h_indep h_le h_le' (directed_of_inf hm)
 #align probability_theory.indep_supr_of_antitone ProbabilityTheory.indep_iSup_of_antitone
+-/
 
+#print ProbabilityTheory.iIndepSets.piiUnionInter_of_not_mem /-
 theorem iIndepSets.piiUnionInter_of_not_mem {π : ι → Set (Set Ω)} {a : ι} {S : Finset ι}
     (hp_ind : iIndepSets π μ) (haS : a ∉ S) : IndepSets (piiUnionInter π S) (π a) μ :=
   by
@@ -606,9 +653,11 @@ theorem iIndepSets.piiUnionInter_of_not_mem {π : ι → Set (Set Ω)} {a : ι}
   have has : a ∉ s := fun has_mem => haS (hs_mem Membership)
   rw [h_μ_inter, Finset.prod_insert has, h_t2, mul_comm, h_μ_t1]
 #align probability_theory.Indep_sets.pi_Union_Inter_of_not_mem ProbabilityTheory.iIndepSets.piiUnionInter_of_not_mem
+-/
 
 /- warning: probability_theory.Indep_sets.Indep clashes with probability_theory.indep_sets.indep -> ProbabilityTheory.IndepSets.indep
 Case conversion may be inaccurate. Consider using '#align probability_theory.Indep_sets.Indep ProbabilityTheory.IndepSets.indepₓ'. -/
+#print ProbabilityTheory.IndepSets.indep /-
 /-- The measurable space structures generated by independent pi-systems are independent. -/
 theorem IndepSets.indep [IsProbabilityMeasure μ] (m : ι → MeasurableSpace Ω) (h_le : ∀ i, m i ≤ m0)
     (π : ι → Set (Set Ω)) (h_pi : ∀ n, IsPiSystem (π n))
@@ -639,6 +688,7 @@ theorem IndepSets.indep [IsProbabilityMeasure μ] (m : ι → MeasurableSpace Ω
   have h_S_f : ∀ i ∈ S, measurable_set[m_p] (f i) := fun i hi => (h_le_p i hi) (f i) (hf_m_S i hi)
   exact S.measurable_set_bInter h_S_f
 #align probability_theory.Indep_sets.Indep ProbabilityTheory.IndepSets.indep
+-/
 
 end FromPiSystemsToMeasurableSpaces
 
@@ -665,11 +715,13 @@ theorem indepSet_iff_indepSets_singleton {m0 : MeasurableSpace Ω} (hs_meas : Me
 #align probability_theory.indep_set_iff_indep_sets_singleton ProbabilityTheory.indepSet_iff_indepSets_singleton
 -/
 
+#print ProbabilityTheory.indepSet_iff_measure_inter_eq_mul /-
 theorem indepSet_iff_measure_inter_eq_mul {m0 : MeasurableSpace Ω} (hs_meas : MeasurableSet s)
     (ht_meas : MeasurableSet t) (μ : Measure Ω := by exact MeasureTheory.MeasureSpace.volume)
     [IsProbabilityMeasure μ] : IndepSet s t μ ↔ μ (s ∩ t) = μ s * μ t :=
   (indepSet_iff_indepSets_singleton hs_meas ht_meas μ).trans indepSets_singleton_iff
 #align probability_theory.indep_set_iff_measure_inter_eq_mul ProbabilityTheory.indepSet_iff_measure_inter_eq_mul
+-/
 
 #print ProbabilityTheory.IndepSets.indepSet_of_mem /-
 theorem IndepSets.indepSet_of_mem {m0 : MeasurableSpace Ω} (hs : s ∈ S) (ht : t ∈ T)
@@ -720,6 +772,7 @@ section IndepFun
 
 variable {β β' γ γ' : Type _} {mΩ : MeasurableSpace Ω} {μ : Measure Ω} {f : Ω → β} {g : Ω → β'}
 
+#print ProbabilityTheory.indepFun_iff_measure_inter_preimage_eq_mul /-
 theorem indepFun_iff_measure_inter_preimage_eq_mul {mβ : MeasurableSpace β}
     {mβ' : MeasurableSpace β'} :
     IndepFun f g μ ↔
@@ -730,7 +783,9 @@ theorem indepFun_iff_measure_inter_preimage_eq_mul {mβ : MeasurableSpace β}
   · refine' fun s t hs ht => h (f ⁻¹' s) (g ⁻¹' t) ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩
   · rintro _ _ ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩; exact h s t hs ht
 #align probability_theory.indep_fun_iff_measure_inter_preimage_eq_mul ProbabilityTheory.indepFun_iff_measure_inter_preimage_eq_mul
+-/
 
+#print ProbabilityTheory.iIndepFun_iff_measure_inter_preimage_eq_mul /-
 theorem iIndepFun_iff_measure_inter_preimage_eq_mul {ι : Type _} {β : ι → Type _}
     (m : ∀ x, MeasurableSpace (β x)) (f : ∀ i, Ω → β i) :
     iIndepFun m f μ ↔
@@ -766,7 +821,9 @@ theorem iIndepFun_iff_measure_inter_preimage_eq_mul {ι : Type _} {β : ι → T
   rw [h_left_eq, h_right_eq]
   exact h S h_measβ
 #align probability_theory.Indep_fun_iff_measure_inter_preimage_eq_mul ProbabilityTheory.iIndepFun_iff_measure_inter_preimage_eq_mul
+-/
 
+#print ProbabilityTheory.indepFun_iff_indepSet_preimage /-
 theorem indepFun_iff_indepSet_preimage {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
     [IsProbabilityMeasure μ] (hf : Measurable f) (hg : Measurable g) :
     IndepFun f g μ ↔ ∀ s t, MeasurableSet s → MeasurableSet t → IndepSet (f ⁻¹' s) (g ⁻¹' t) μ :=
@@ -776,6 +833,7 @@ theorem indepFun_iff_indepSet_preimage {mβ : MeasurableSpace β} {mβ' : Measur
   · rwa [indep_set_iff_measure_inter_eq_mul (hf hs) (hg ht) μ]
   · rwa [← indep_set_iff_measure_inter_eq_mul (hf hs) (hg ht) μ]
 #align probability_theory.indep_fun_iff_indep_set_preimage ProbabilityTheory.indepFun_iff_indepSet_preimage
+-/
 
 #print ProbabilityTheory.IndepFun.symm /-
 @[symm]
@@ -797,6 +855,7 @@ theorem IndepFun.ae_eq {mβ : MeasurableSpace β} {f g f' g' : Ω → β} (hfg :
 #align probability_theory.indep_fun.ae_eq ProbabilityTheory.IndepFun.ae_eq
 -/
 
+#print ProbabilityTheory.IndepFun.comp /-
 theorem IndepFun.comp {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'} {mγ : MeasurableSpace γ}
     {mγ' : MeasurableSpace γ'} {φ : β → γ} {ψ : β' → γ'} (hfg : IndepFun f g μ) (hφ : Measurable φ)
     (hψ : Measurable ψ) : IndepFun (φ ∘ f) (ψ ∘ g) μ :=
@@ -806,7 +865,9 @@ theorem IndepFun.comp {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'} {m
   · exact ⟨φ ⁻¹' A, hφ hA, set.preimage_comp.symm⟩
   · exact ⟨ψ ⁻¹' B, hψ hB, set.preimage_comp.symm⟩
 #align probability_theory.indep_fun.comp ProbabilityTheory.IndepFun.comp
+-/
 
+#print ProbabilityTheory.iIndepFun.indepFun_finset /-
 /-- If `f` is a family of mutually independent random variables (`Indep_fun m f μ`) and `S, T` are
 two disjoint finite index sets, then the tuple formed by `f i` for `i ∈ S` is independent of the
 tuple `(f i)_i` for `i ∈ T`. -/
@@ -905,7 +966,9 @@ theorem iIndepFun.indepFun_finset [IsProbabilityMeasure μ] {ι : Type _} {β :
   · refine' Finset.prod_congr rfl fun i hi => _
     rw [h_sets_s'_univ hi, Set.univ_inter]
 #align probability_theory.Indep_fun.indep_fun_finset ProbabilityTheory.iIndepFun.indepFun_finset
+-/
 
+#print ProbabilityTheory.iIndepFun.indepFun_prod /-
 theorem iIndepFun.indepFun_prod [IsProbabilityMeasure μ] {ι : Type _} {β : ι → Type _}
     {m : ∀ i, MeasurableSpace (β i)} {f : ∀ i, Ω → β i} (hf_Indep : iIndepFun m f μ)
     (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
@@ -942,7 +1005,9 @@ theorem iIndepFun.indepFun_prod [IsProbabilityMeasure μ] {ι : Type _} {β : ι
   simp only [Finset.mem_insert, Finset.mem_singleton, not_or]
   exact ⟨hik.symm, hjk.symm⟩
 #align probability_theory.Indep_fun.indep_fun_prod ProbabilityTheory.iIndepFun.indepFun_prod
+-/
 
+#print ProbabilityTheory.iIndepFun.mul /-
 @[to_additive]
 theorem iIndepFun.mul [IsProbabilityMeasure μ] {ι : Type _} {β : Type _} {m : MeasurableSpace β}
     [Mul β] [MeasurableMul₂ β] {f : ι → Ω → β} (hf_Indep : iIndepFun (fun _ => m) f μ)
@@ -955,7 +1020,9 @@ theorem iIndepFun.mul [IsProbabilityMeasure μ] {ι : Type _} {β : Type _} {m :
   exact indep_fun.comp this (measurable_fst.mul measurable_snd) measurable_id
 #align probability_theory.Indep_fun.mul ProbabilityTheory.iIndepFun.mul
 #align probability_theory.Indep_fun.add ProbabilityTheory.iIndepFun.add
+-/
 
+#print ProbabilityTheory.iIndepFun.indepFun_finset_prod_of_not_mem /-
 @[to_additive]
 theorem iIndepFun.indepFun_finset_prod_of_not_mem [IsProbabilityMeasure μ] {ι : Type _} {β : Type _}
     {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂ β] {f : ι → Ω → β}
@@ -984,7 +1051,9 @@ theorem iIndepFun.indepFun_finset_prod_of_not_mem [IsProbabilityMeasure μ] {ι
       h_meas_left h_meas_right
 #align probability_theory.Indep_fun.indep_fun_finset_prod_of_not_mem ProbabilityTheory.iIndepFun.indepFun_finset_prod_of_not_mem
 #align probability_theory.Indep_fun.indep_fun_finset_sum_of_not_mem ProbabilityTheory.iIndepFun.indepFun_finset_sum_of_not_mem
+-/
 
+#print ProbabilityTheory.iIndepFun.indepFun_prod_range_succ /-
 @[to_additive]
 theorem iIndepFun.indepFun_prod_range_succ [IsProbabilityMeasure μ] {β : Type _}
     {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂ β] {f : ℕ → Ω → β}
@@ -993,7 +1062,9 @@ theorem iIndepFun.indepFun_prod_range_succ [IsProbabilityMeasure μ] {β : Type
   hf_Indep.indepFun_finset_prod_of_not_mem hf_meas Finset.not_mem_range_self
 #align probability_theory.Indep_fun.indep_fun_prod_range_succ ProbabilityTheory.iIndepFun.indepFun_prod_range_succ
 #align probability_theory.Indep_fun.indep_fun_sum_range_succ ProbabilityTheory.iIndepFun.indepFun_sum_range_succ
+-/
 
+#print ProbabilityTheory.iIndepSet.iIndepFun_indicator /-
 theorem iIndepSet.iIndepFun_indicator [Zero β] [One β] {m : MeasurableSpace β} {s : ι → Set Ω}
     (hs : iIndepSet s μ) : iIndepFun (fun n => m) (fun n => (s n).indicator fun ω => 1) μ := by
   classical
@@ -1009,6 +1080,7 @@ theorem iIndepSet.iIndepFun_indicator [Zero β] [One β] {m : MeasurableSpace β
   · exact @MeasurableSet.empty _ (generate_from {s i})
   · exact @MeasurableSet.empty _ (generate_from {s i})
 #align probability_theory.Indep_set.Indep_fun_indicator ProbabilityTheory.iIndepSet.iIndepFun_indicator
+-/
 
 end IndepFun
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -323,7 +323,7 @@ theorem iIndepSets.indepSets {s : ι → Set (Set Ω)} [MeasurableSpace Ω] {μ
       ite (i = i) t₁ t₂ ∩ ite (j = i) t₁ t₂ :=
     by simp only [Finset.set_biInter_singleton, Finset.set_biInter_insert]
   have h_prod :
-    (∏ t : ι in ({i, j} : Finset ι), μ (ite (t = i) t₁ t₂)) =
+    ∏ t : ι in ({i, j} : Finset ι), μ (ite (t = i) t₁ t₂) =
       μ (ite (i = i) t₁ t₂) * μ (ite (j = i) t₁ t₂) :=
     by
     simp only [hij, Finset.prod_singleton, Finset.prod_insert, not_false_iff, Finset.mem_singleton]
@@ -759,7 +759,7 @@ theorem iIndepFun_iff_measure_inter_preimage_eq_mul {ι : Type _} {β : ι → T
     constructor <;> intro h hi_mem <;> specialize h hi_mem
     · rwa [h_preim i hi_mem] at h 
     · rwa [h_preim i hi_mem]
-  have h_right_eq : (∏ i in S, μ (setsΩ i)) = ∏ i in S, μ (f i ⁻¹' setsβ i) :=
+  have h_right_eq : ∏ i in S, μ (setsΩ i) = ∏ i in S, μ (f i ⁻¹' setsβ i) :=
     by
     refine' Finset.prod_congr rfl fun i hi_mem => _
     rw [h_preim i hi_mem]
@@ -970,11 +970,11 @@ theorem iIndepFun.indepFun_finset_prod_of_not_mem [IsProbabilityMeasure μ] {ι
   have h_meas_right :
     Measurable fun p : ∀ j : ({i} : Finset ι), β => p ⟨i, Finset.mem_singleton_self i⟩ :=
     measurable_pi_apply ⟨i, Finset.mem_singleton_self i⟩
-  have h_left : (∏ j in s, f j) = (fun p : ∀ j : s, β => ∏ j, p j) ∘ fun a (j : s) => f j a :=
+  have h_left : ∏ j in s, f j = (fun p : ∀ j : s, β => ∏ j, p j) ∘ fun a (j : s) => f j a :=
     by
     ext1 a
     simp only [Function.comp_apply]
-    have : (∏ j : ↥s, f (↑j) a) = (∏ j : ↥s, f ↑j) a := by rw [Finset.prod_apply]
+    have : ∏ j : ↥s, f (↑j) a = (∏ j : ↥s, f ↑j) a := by rw [Finset.prod_apply]
     rw [this, Finset.prod_coe_sort]
   have h_meas_left : Measurable fun p : ∀ j : s, β => ∏ j, p j :=
     finset.univ.measurable_prod fun (j : ↥s) (H : j ∈ Finset.univ) => measurable_pi_apply j

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 
 ! This file was ported from Lean 3 source module probability.independence.basic
-! leanprover-community/mathlib commit 2f8347015b12b0864dfaf366ec4909eb70c78740
+! leanprover-community/mathlib commit af471b9e3ce868f296626d33189b4ce730fa4c00
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -13,6 +13,9 @@ import Mathbin.MeasureTheory.Constructions.Pi
 /-!
 # Independence of sets of sets and measure spaces (σ-algebras)
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 * A family of sets of sets `π : ι → set (set Ω)` is independent with respect to a measure `μ` if for
   any finite set of indices `s = {i_1, ..., i_n}`, for any sets `f i_1 ∈ π i_1, ..., f i_n ∈ π i_n`,
   `μ (⋂ i in s, f i) = ∏ i in s, μ (f i) `. It will be used for families of π-systems.
@@ -77,192 +80,224 @@ variable {Ω ι : Type _}
 
 section Definitions
 
+#print ProbabilityTheory.iIndepSets /-
 /-- A family of sets of sets `π : ι → set (set Ω)` is independent with respect to a measure `μ` if
 for any finite set of indices `s = {i_1, ..., i_n}`, for any sets
 `f i_1 ∈ π i_1, ..., f i_n ∈ π i_n`, then `μ (⋂ i in s, f i) = ∏ i in s, μ (f i) `.
 It will be used for families of pi_systems. -/
-def IndepSets [MeasurableSpace Ω] (π : ι → Set (Set Ω))
+def iIndepSets [MeasurableSpace Ω] (π : ι → Set (Set Ω))
     (μ : Measure Ω := by exact MeasureTheory.MeasureSpace.volume) : Prop :=
   ∀ (s : Finset ι) {f : ι → Set Ω} (H : ∀ i, i ∈ s → f i ∈ π i),
     μ (⋂ i ∈ s, f i) = ∏ i in s, μ (f i)
-#align probability_theory.Indep_sets ProbabilityTheory.IndepSets
+#align probability_theory.Indep_sets ProbabilityTheory.iIndepSets
+-/
 
+#print ProbabilityTheory.IndepSets /-
 /-- Two sets of sets `s₁, s₂` are independent with respect to a measure `μ` if for any sets
 `t₁ ∈ p₁, t₂ ∈ s₂`, then `μ (t₁ ∩ t₂) = μ (t₁) * μ (t₂)` -/
-def IndepSetsCat [MeasurableSpace Ω] (s1 s2 : Set (Set Ω))
+def IndepSets [MeasurableSpace Ω] (s1 s2 : Set (Set Ω))
     (μ : Measure Ω := by exact MeasureTheory.MeasureSpace.volume) : Prop :=
   ∀ t1 t2 : Set Ω, t1 ∈ s1 → t2 ∈ s2 → μ (t1 ∩ t2) = μ t1 * μ t2
-#align probability_theory.indep_sets ProbabilityTheory.IndepSetsCat
+#align probability_theory.indep_sets ProbabilityTheory.IndepSets
+-/
 
+#print ProbabilityTheory.iIndep /-
 /-- A family of measurable space structures (i.e. of σ-algebras) is independent with respect to a
 measure `μ` (typically defined on a finer σ-algebra) if the family of sets of measurable sets they
 define is independent. `m : ι → measurable_space Ω` is independent with respect to measure `μ` if
 for any finite set of indices `s = {i_1, ..., i_n}`, for any sets
 `f i_1 ∈ m i_1, ..., f i_n ∈ m i_n`, then `μ (⋂ i in s, f i) = ∏ i in s, μ (f i) `. -/
-def Indep (m : ι → MeasurableSpace Ω) [MeasurableSpace Ω]
+def iIndep (m : ι → MeasurableSpace Ω) [MeasurableSpace Ω]
     (μ : Measure Ω := by exact MeasureTheory.MeasureSpace.volume) : Prop :=
-  IndepSets (fun x => { s | measurable_set[m x] s }) μ
-#align probability_theory.Indep ProbabilityTheory.Indep
+  iIndepSets (fun x => {s | measurable_set[m x] s}) μ
+#align probability_theory.Indep ProbabilityTheory.iIndep
+-/
 
+#print ProbabilityTheory.Indep /-
 /-- Two measurable space structures (or σ-algebras) `m₁, m₂` are independent with respect to a
 measure `μ` (defined on a third σ-algebra) if for any sets `t₁ ∈ m₁, t₂ ∈ m₂`,
 `μ (t₁ ∩ t₂) = μ (t₁) * μ (t₂)` -/
-def IndepCat (m₁ m₂ : MeasurableSpace Ω) [MeasurableSpace Ω]
+def Indep (m₁ m₂ : MeasurableSpace Ω) [MeasurableSpace Ω]
     (μ : Measure Ω := by exact MeasureTheory.MeasureSpace.volume) : Prop :=
-  IndepSetsCat { s | measurable_set[m₁] s } { s | measurable_set[m₂] s } μ
-#align probability_theory.indep ProbabilityTheory.IndepCat
+  IndepSets {s | measurable_set[m₁] s} {s | measurable_set[m₂] s} μ
+#align probability_theory.indep ProbabilityTheory.Indep
+-/
 
+#print ProbabilityTheory.iIndepSet /-
 /-- A family of sets is independent if the family of measurable space structures they generate is
 independent. For a set `s`, the generated measurable space has measurable sets `∅, s, sᶜ, univ`. -/
-def IndepSet [MeasurableSpace Ω] (s : ι → Set Ω)
+def iIndepSet [MeasurableSpace Ω] (s : ι → Set Ω)
     (μ : Measure Ω := by exact MeasureTheory.MeasureSpace.volume) : Prop :=
-  Indep (fun i => generateFrom {s i}) μ
-#align probability_theory.Indep_set ProbabilityTheory.IndepSet
+  iIndep (fun i => generateFrom {s i}) μ
+#align probability_theory.Indep_set ProbabilityTheory.iIndepSet
+-/
 
+#print ProbabilityTheory.IndepSet /-
 /-- Two sets are independent if the two measurable space structures they generate are independent.
 For a set `s`, the generated measurable space structure has measurable sets `∅, s, sᶜ, univ`. -/
-def IndepSetCat [MeasurableSpace Ω] (s t : Set Ω)
+def IndepSet [MeasurableSpace Ω] (s t : Set Ω)
     (μ : Measure Ω := by exact MeasureTheory.MeasureSpace.volume) : Prop :=
-  IndepCat (generateFrom {s}) (generateFrom {t}) μ
-#align probability_theory.indep_set ProbabilityTheory.IndepSetCat
+  Indep (generateFrom {s}) (generateFrom {t}) μ
+#align probability_theory.indep_set ProbabilityTheory.IndepSet
+-/
 
+#print ProbabilityTheory.iIndepFun /-
 /-- A family of functions defined on the same space `Ω` and taking values in possibly different
 spaces, each with a measurable space structure, is independent if the family of measurable space
 structures they generate on `Ω` is independent. For a function `g` with codomain having measurable
 space structure `m`, the generated measurable space structure is `measurable_space.comap g m`. -/
-def IndepFun [MeasurableSpace Ω] {β : ι → Type _} (m : ∀ x : ι, MeasurableSpace (β x))
+def iIndepFun [MeasurableSpace Ω] {β : ι → Type _} (m : ∀ x : ι, MeasurableSpace (β x))
     (f : ∀ x : ι, Ω → β x) (μ : Measure Ω := by exact MeasureTheory.MeasureSpace.volume) : Prop :=
-  Indep (fun x => MeasurableSpace.comap (f x) (m x)) μ
-#align probability_theory.Indep_fun ProbabilityTheory.IndepFun
+  iIndep (fun x => MeasurableSpace.comap (f x) (m x)) μ
+#align probability_theory.Indep_fun ProbabilityTheory.iIndepFun
+-/
 
+#print ProbabilityTheory.IndepFun /-
 /-- Two functions are independent if the two measurable space structures they generate are
 independent. For a function `f` with codomain having measurable space structure `m`, the generated
 measurable space structure is `measurable_space.comap f m`. -/
-def IndepFunCat {β γ} [MeasurableSpace Ω] [mβ : MeasurableSpace β] [mγ : MeasurableSpace γ]
-    (f : Ω → β) (g : Ω → γ) (μ : Measure Ω := by exact MeasureTheory.MeasureSpace.volume) : Prop :=
-  IndepCat (MeasurableSpace.comap f mβ) (MeasurableSpace.comap g mγ) μ
-#align probability_theory.indep_fun ProbabilityTheory.IndepFunCat
+def IndepFun {β γ} [MeasurableSpace Ω] [mβ : MeasurableSpace β] [mγ : MeasurableSpace γ] (f : Ω → β)
+    (g : Ω → γ) (μ : Measure Ω := by exact MeasureTheory.MeasureSpace.volume) : Prop :=
+  Indep (MeasurableSpace.comap f mβ) (MeasurableSpace.comap g mγ) μ
+#align probability_theory.indep_fun ProbabilityTheory.IndepFun
+-/
 
 end Definitions
 
 section Indep
 
+#print ProbabilityTheory.IndepSets.symm /-
 @[symm]
-theorem IndepSetsCat.symm {s₁ s₂ : Set (Set Ω)} [MeasurableSpace Ω] {μ : Measure Ω}
-    (h : IndepSetsCat s₁ s₂ μ) : IndepSetsCat s₂ s₁ μ := by intro t1 t2 ht1 ht2;
+theorem IndepSets.symm {s₁ s₂ : Set (Set Ω)} [MeasurableSpace Ω] {μ : Measure Ω}
+    (h : IndepSets s₁ s₂ μ) : IndepSets s₂ s₁ μ := by intro t1 t2 ht1 ht2;
   rw [Set.inter_comm, mul_comm]; exact h t2 t1 ht2 ht1
-#align probability_theory.indep_sets.symm ProbabilityTheory.IndepSetsCat.symm
+#align probability_theory.indep_sets.symm ProbabilityTheory.IndepSets.symm
+-/
 
+#print ProbabilityTheory.Indep.symm /-
 @[symm]
-theorem IndepCat.symm {m₁ m₂ : MeasurableSpace Ω} [MeasurableSpace Ω] {μ : Measure Ω}
-    (h : IndepCat m₁ m₂ μ) : IndepCat m₂ m₁ μ :=
-  IndepSetsCat.symm h
-#align probability_theory.indep.symm ProbabilityTheory.IndepCat.symm
+theorem Indep.symm {m₁ m₂ : MeasurableSpace Ω} [MeasurableSpace Ω] {μ : Measure Ω}
+    (h : Indep m₁ m₂ μ) : Indep m₂ m₁ μ :=
+  IndepSets.symm h
+#align probability_theory.indep.symm ProbabilityTheory.Indep.symm
+-/
 
-theorem indepCat_bot_right (m' : MeasurableSpace Ω) {m : MeasurableSpace Ω} {μ : Measure Ω}
-    [ProbabilityMeasure μ] : IndepCat m' ⊥ μ :=
+theorem indep_bot_right (m' : MeasurableSpace Ω) {m : MeasurableSpace Ω} {μ : Measure Ω}
+    [IsProbabilityMeasure μ] : Indep m' ⊥ μ :=
   by
   intro s t hs ht
   rw [Set.mem_setOf_eq, MeasurableSpace.measurableSet_bot_iff] at ht 
   cases ht
   · rw [ht, Set.inter_empty, measure_empty, MulZeroClass.mul_zero]
   · rw [ht, Set.inter_univ, measure_univ, mul_one]
-#align probability_theory.indep_bot_right ProbabilityTheory.indepCat_bot_right
+#align probability_theory.indep_bot_right ProbabilityTheory.indep_bot_right
 
-theorem indepCat_bot_left (m' : MeasurableSpace Ω) {m : MeasurableSpace Ω} {μ : Measure Ω}
-    [ProbabilityMeasure μ] : IndepCat ⊥ m' μ :=
-  (indepCat_bot_right m').symm
-#align probability_theory.indep_bot_left ProbabilityTheory.indepCat_bot_left
+theorem indep_bot_left (m' : MeasurableSpace Ω) {m : MeasurableSpace Ω} {μ : Measure Ω}
+    [IsProbabilityMeasure μ] : Indep ⊥ m' μ :=
+  (indep_bot_right m').symm
+#align probability_theory.indep_bot_left ProbabilityTheory.indep_bot_left
 
-theorem indepSetCat_empty_right {m : MeasurableSpace Ω} {μ : Measure Ω} [ProbabilityMeasure μ]
-    (s : Set Ω) : IndepSetCat s ∅ μ := by simp only [indep_set, generate_from_singleton_empty];
+#print ProbabilityTheory.indepSet_empty_right /-
+theorem indepSet_empty_right {m : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
+    (s : Set Ω) : IndepSet s ∅ μ := by simp only [indep_set, generate_from_singleton_empty];
   exact indep_bot_right _
-#align probability_theory.indep_set_empty_right ProbabilityTheory.indepSetCat_empty_right
+#align probability_theory.indep_set_empty_right ProbabilityTheory.indepSet_empty_right
+-/
 
-theorem indepSetCat_empty_left {m : MeasurableSpace Ω} {μ : Measure Ω} [ProbabilityMeasure μ]
-    (s : Set Ω) : IndepSetCat ∅ s μ :=
-  (indepSetCat_empty_right s).symm
-#align probability_theory.indep_set_empty_left ProbabilityTheory.indepSetCat_empty_left
+#print ProbabilityTheory.indepSet_empty_left /-
+theorem indepSet_empty_left {m : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
+    (s : Set Ω) : IndepSet ∅ s μ :=
+  (indepSet_empty_right s).symm
+#align probability_theory.indep_set_empty_left ProbabilityTheory.indepSet_empty_left
+-/
 
-theorem indepSetsCat_of_indepSetsCat_of_le_left {s₁ s₂ s₃ : Set (Set Ω)} [MeasurableSpace Ω]
-    {μ : Measure Ω} (h_indep : IndepSetsCat s₁ s₂ μ) (h31 : s₃ ⊆ s₁) : IndepSetsCat s₃ s₂ μ :=
+#print ProbabilityTheory.indepSets_of_indepSets_of_le_left /-
+theorem indepSets_of_indepSets_of_le_left {s₁ s₂ s₃ : Set (Set Ω)} [MeasurableSpace Ω]
+    {μ : Measure Ω} (h_indep : IndepSets s₁ s₂ μ) (h31 : s₃ ⊆ s₁) : IndepSets s₃ s₂ μ :=
   fun t1 t2 ht1 ht2 => h_indep t1 t2 (Set.mem_of_subset_of_mem h31 ht1) ht2
-#align probability_theory.indep_sets_of_indep_sets_of_le_left ProbabilityTheory.indepSetsCat_of_indepSetsCat_of_le_left
+#align probability_theory.indep_sets_of_indep_sets_of_le_left ProbabilityTheory.indepSets_of_indepSets_of_le_left
+-/
 
-theorem indepSetsCat_of_indepSetsCat_of_le_right {s₁ s₂ s₃ : Set (Set Ω)} [MeasurableSpace Ω]
-    {μ : Measure Ω} (h_indep : IndepSetsCat s₁ s₂ μ) (h32 : s₃ ⊆ s₂) : IndepSetsCat s₁ s₃ μ :=
+#print ProbabilityTheory.indepSets_of_indepSets_of_le_right /-
+theorem indepSets_of_indepSets_of_le_right {s₁ s₂ s₃ : Set (Set Ω)} [MeasurableSpace Ω]
+    {μ : Measure Ω} (h_indep : IndepSets s₁ s₂ μ) (h32 : s₃ ⊆ s₂) : IndepSets s₁ s₃ μ :=
   fun t1 t2 ht1 ht2 => h_indep t1 t2 ht1 (Set.mem_of_subset_of_mem h32 ht2)
-#align probability_theory.indep_sets_of_indep_sets_of_le_right ProbabilityTheory.indepSetsCat_of_indepSetsCat_of_le_right
+#align probability_theory.indep_sets_of_indep_sets_of_le_right ProbabilityTheory.indepSets_of_indepSets_of_le_right
+-/
 
-theorem indepCat_of_indepCat_of_le_left {m₁ m₂ m₃ : MeasurableSpace Ω} [MeasurableSpace Ω]
-    {μ : Measure Ω} (h_indep : IndepCat m₁ m₂ μ) (h31 : m₃ ≤ m₁) : IndepCat m₃ m₂ μ :=
-  fun t1 t2 ht1 ht2 => h_indep t1 t2 (h31 _ ht1) ht2
-#align probability_theory.indep_of_indep_of_le_left ProbabilityTheory.indepCat_of_indepCat_of_le_left
+#print ProbabilityTheory.indep_of_indep_of_le_left /-
+theorem indep_of_indep_of_le_left {m₁ m₂ m₃ : MeasurableSpace Ω} [MeasurableSpace Ω] {μ : Measure Ω}
+    (h_indep : Indep m₁ m₂ μ) (h31 : m₃ ≤ m₁) : Indep m₃ m₂ μ := fun t1 t2 ht1 ht2 =>
+  h_indep t1 t2 (h31 _ ht1) ht2
+#align probability_theory.indep_of_indep_of_le_left ProbabilityTheory.indep_of_indep_of_le_left
+-/
 
-theorem indepCat_of_indepCat_of_le_right {m₁ m₂ m₃ : MeasurableSpace Ω} [MeasurableSpace Ω]
-    {μ : Measure Ω} (h_indep : IndepCat m₁ m₂ μ) (h32 : m₃ ≤ m₂) : IndepCat m₁ m₃ μ :=
+#print ProbabilityTheory.indep_of_indep_of_le_right /-
+theorem indep_of_indep_of_le_right {m₁ m₂ m₃ : MeasurableSpace Ω} [MeasurableSpace Ω]
+    {μ : Measure Ω} (h_indep : Indep m₁ m₂ μ) (h32 : m₃ ≤ m₂) : Indep m₁ m₃ μ :=
   fun t1 t2 ht1 ht2 => h_indep t1 t2 ht1 (h32 _ ht2)
-#align probability_theory.indep_of_indep_of_le_right ProbabilityTheory.indepCat_of_indepCat_of_le_right
+#align probability_theory.indep_of_indep_of_le_right ProbabilityTheory.indep_of_indep_of_le_right
+-/
 
-theorem IndepSetsCat.union [MeasurableSpace Ω] {s₁ s₂ s' : Set (Set Ω)} {μ : Measure Ω}
-    (h₁ : IndepSetsCat s₁ s' μ) (h₂ : IndepSetsCat s₂ s' μ) : IndepSetsCat (s₁ ∪ s₂) s' μ :=
+theorem IndepSets.union [MeasurableSpace Ω] {s₁ s₂ s' : Set (Set Ω)} {μ : Measure Ω}
+    (h₁ : IndepSets s₁ s' μ) (h₂ : IndepSets s₂ s' μ) : IndepSets (s₁ ∪ s₂) s' μ :=
   by
   intro t1 t2 ht1 ht2
   cases' (Set.mem_union _ _ _).mp ht1 with ht1₁ ht1₂
   · exact h₁ t1 t2 ht1₁ ht2
   · exact h₂ t1 t2 ht1₂ ht2
-#align probability_theory.indep_sets.union ProbabilityTheory.IndepSetsCat.union
+#align probability_theory.indep_sets.union ProbabilityTheory.IndepSets.union
 
 @[simp]
-theorem IndepSetsCat.union_iff [MeasurableSpace Ω] {s₁ s₂ s' : Set (Set Ω)} {μ : Measure Ω} :
-    IndepSetsCat (s₁ ∪ s₂) s' μ ↔ IndepSetsCat s₁ s' μ ∧ IndepSetsCat s₂ s' μ :=
+theorem IndepSets.union_iff [MeasurableSpace Ω] {s₁ s₂ s' : Set (Set Ω)} {μ : Measure Ω} :
+    IndepSets (s₁ ∪ s₂) s' μ ↔ IndepSets s₁ s' μ ∧ IndepSets s₂ s' μ :=
   ⟨fun h =>
-    ⟨indepSetsCat_of_indepSetsCat_of_le_left h (Set.subset_union_left s₁ s₂),
-      indepSetsCat_of_indepSetsCat_of_le_left h (Set.subset_union_right s₁ s₂)⟩,
-    fun h => IndepSetsCat.union h.left h.right⟩
-#align probability_theory.indep_sets.union_iff ProbabilityTheory.IndepSetsCat.union_iff
+    ⟨indepSets_of_indepSets_of_le_left h (Set.subset_union_left s₁ s₂),
+      indepSets_of_indepSets_of_le_left h (Set.subset_union_right s₁ s₂)⟩,
+    fun h => IndepSets.union h.left h.right⟩
+#align probability_theory.indep_sets.union_iff ProbabilityTheory.IndepSets.union_iff
 
-theorem IndepSetsCat.iUnion [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
-    {μ : Measure Ω} (hyp : ∀ n, IndepSetsCat (s n) s' μ) : IndepSetsCat (⋃ n, s n) s' μ :=
+theorem IndepSets.iUnion [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
+    {μ : Measure Ω} (hyp : ∀ n, IndepSets (s n) s' μ) : IndepSets (⋃ n, s n) s' μ :=
   by
   intro t1 t2 ht1 ht2
   rw [Set.mem_iUnion] at ht1 
   cases' ht1 with n ht1
   exact hyp n t1 t2 ht1 ht2
-#align probability_theory.indep_sets.Union ProbabilityTheory.IndepSetsCat.iUnion
+#align probability_theory.indep_sets.Union ProbabilityTheory.IndepSets.iUnion
 
-theorem IndepSetsCat.bUnion [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
-    {μ : Measure Ω} {u : Set ι} (hyp : ∀ n ∈ u, IndepSetsCat (s n) s' μ) :
-    IndepSetsCat (⋃ n ∈ u, s n) s' μ := by
+theorem IndepSets.bUnion [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
+    {μ : Measure Ω} {u : Set ι} (hyp : ∀ n ∈ u, IndepSets (s n) s' μ) :
+    IndepSets (⋃ n ∈ u, s n) s' μ := by
   intro t1 t2 ht1 ht2
   simp_rw [Set.mem_iUnion] at ht1 
   rcases ht1 with ⟨n, hpn, ht1⟩
   exact hyp n hpn t1 t2 ht1 ht2
-#align probability_theory.indep_sets.bUnion ProbabilityTheory.IndepSetsCat.bUnion
+#align probability_theory.indep_sets.bUnion ProbabilityTheory.IndepSets.bUnion
 
-theorem IndepSetsCat.inter [MeasurableSpace Ω] {s₁ s' : Set (Set Ω)} (s₂ : Set (Set Ω))
-    {μ : Measure Ω} (h₁ : IndepSetsCat s₁ s' μ) : IndepSetsCat (s₁ ∩ s₂) s' μ :=
-  fun t1 t2 ht1 ht2 => h₁ t1 t2 ((Set.mem_inter_iff _ _ _).mp ht1).left ht2
-#align probability_theory.indep_sets.inter ProbabilityTheory.IndepSetsCat.inter
+theorem IndepSets.inter [MeasurableSpace Ω] {s₁ s' : Set (Set Ω)} (s₂ : Set (Set Ω)) {μ : Measure Ω}
+    (h₁ : IndepSets s₁ s' μ) : IndepSets (s₁ ∩ s₂) s' μ := fun t1 t2 ht1 ht2 =>
+  h₁ t1 t2 ((Set.mem_inter_iff _ _ _).mp ht1).left ht2
+#align probability_theory.indep_sets.inter ProbabilityTheory.IndepSets.inter
 
-theorem IndepSetsCat.iInter [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
-    {μ : Measure Ω} (h : ∃ n, IndepSetsCat (s n) s' μ) : IndepSetsCat (⋂ n, s n) s' μ := by
+theorem IndepSets.iInter [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
+    {μ : Measure Ω} (h : ∃ n, IndepSets (s n) s' μ) : IndepSets (⋂ n, s n) s' μ := by
   intro t1 t2 ht1 ht2; cases' h with n h; exact h t1 t2 (set.mem_Inter.mp ht1 n) ht2
-#align probability_theory.indep_sets.Inter ProbabilityTheory.IndepSetsCat.iInter
+#align probability_theory.indep_sets.Inter ProbabilityTheory.IndepSets.iInter
 
-theorem IndepSetsCat.bInter [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
-    {μ : Measure Ω} {u : Set ι} (h : ∃ n ∈ u, IndepSetsCat (s n) s' μ) :
-    IndepSetsCat (⋂ n ∈ u, s n) s' μ := by
+theorem IndepSets.bInter [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
+    {μ : Measure Ω} {u : Set ι} (h : ∃ n ∈ u, IndepSets (s n) s' μ) :
+    IndepSets (⋂ n ∈ u, s n) s' μ := by
   intro t1 t2 ht1 ht2
   rcases h with ⟨n, hn, h⟩
   exact h t1 t2 (Set.biInter_subset_of_mem hn ht1) ht2
-#align probability_theory.indep_sets.bInter ProbabilityTheory.IndepSetsCat.bInter
+#align probability_theory.indep_sets.bInter ProbabilityTheory.IndepSets.bInter
 
-theorem indepSetsCat_singleton_iff [MeasurableSpace Ω] {s t : Set Ω} {μ : Measure Ω} :
-    IndepSetsCat {s} {t} μ ↔ μ (s ∩ t) = μ s * μ t :=
+theorem indepSets_singleton_iff [MeasurableSpace Ω] {s t : Set Ω} {μ : Measure Ω} :
+    IndepSets {s} {t} μ ↔ μ (s ∩ t) = μ s * μ t :=
   ⟨fun h => h s t rfl rfl, fun h s1 t1 hs1 ht1 => by
     rwa [set.mem_singleton_iff.mp hs1, set.mem_singleton_iff.mp ht1]⟩
-#align probability_theory.indep_sets_singleton_iff ProbabilityTheory.indepSetsCat_singleton_iff
+#align probability_theory.indep_sets_singleton_iff ProbabilityTheory.indepSets_singleton_iff
 
 end Indep
 
@@ -271,46 +306,45 @@ end Indep
 
 section FromIndepToIndep
 
-theorem IndepSets.indepSetsCat {s : ι → Set (Set Ω)} [MeasurableSpace Ω] {μ : Measure Ω}
-    (h_indep : IndepSets s μ) {i j : ι} (hij : i ≠ j) : IndepSetsCat (s i) (s j) μ := by
+theorem iIndepSets.indepSets {s : ι → Set (Set Ω)} [MeasurableSpace Ω] {μ : Measure Ω}
+    (h_indep : iIndepSets s μ) {i j : ι} (hij : i ≠ j) : IndepSets (s i) (s j) μ := by
   classical
-    intro t₁ t₂ ht₁ ht₂
-    have hf_m : ∀ x : ι, x ∈ {i, j} → ite (x = i) t₁ t₂ ∈ s x :=
-      by
-      intro x hx
-      cases' finset.mem_insert.mp hx with hx hx
-      · simp [hx, ht₁]
-      · simp [finset.mem_singleton.mp hx, hij.symm, ht₂]
-    have h1 : t₁ = ite (i = i) t₁ t₂ := by simp only [if_true, eq_self_iff_true]
-    have h2 : t₂ = ite (j = i) t₁ t₂ := by simp only [hij.symm, if_false]
-    have h_inter :
-      (⋂ (t : ι) (H : t ∈ ({i, j} : Finset ι)), ite (t = i) t₁ t₂) =
-        ite (i = i) t₁ t₂ ∩ ite (j = i) t₁ t₂ :=
-      by simp only [Finset.set_biInter_singleton, Finset.set_biInter_insert]
-    have h_prod :
-      (∏ t : ι in ({i, j} : Finset ι), μ (ite (t = i) t₁ t₂)) =
-        μ (ite (i = i) t₁ t₂) * μ (ite (j = i) t₁ t₂) :=
-      by
-      simp only [hij, Finset.prod_singleton, Finset.prod_insert, not_false_iff,
-        Finset.mem_singleton]
-    rw [h1]
-    nth_rw 2 [h2]
-    nth_rw 4 [h2]
-    rw [← h_inter, ← h_prod, h_indep {i, j} hf_m]
-#align probability_theory.Indep_sets.indep_sets ProbabilityTheory.IndepSets.indepSetsCat
-
-theorem Indep.indepCat {m : ι → MeasurableSpace Ω} [MeasurableSpace Ω] {μ : Measure Ω}
-    (h_indep : Indep m μ) {i j : ι} (hij : i ≠ j) : IndepCat (m i) (m j) μ :=
+  intro t₁ t₂ ht₁ ht₂
+  have hf_m : ∀ x : ι, x ∈ {i, j} → ite (x = i) t₁ t₂ ∈ s x :=
+    by
+    intro x hx
+    cases' finset.mem_insert.mp hx with hx hx
+    · simp [hx, ht₁]
+    · simp [finset.mem_singleton.mp hx, hij.symm, ht₂]
+  have h1 : t₁ = ite (i = i) t₁ t₂ := by simp only [if_true, eq_self_iff_true]
+  have h2 : t₂ = ite (j = i) t₁ t₂ := by simp only [hij.symm, if_false]
+  have h_inter :
+    (⋂ (t : ι) (H : t ∈ ({i, j} : Finset ι)), ite (t = i) t₁ t₂) =
+      ite (i = i) t₁ t₂ ∩ ite (j = i) t₁ t₂ :=
+    by simp only [Finset.set_biInter_singleton, Finset.set_biInter_insert]
+  have h_prod :
+    (∏ t : ι in ({i, j} : Finset ι), μ (ite (t = i) t₁ t₂)) =
+      μ (ite (i = i) t₁ t₂) * μ (ite (j = i) t₁ t₂) :=
+    by
+    simp only [hij, Finset.prod_singleton, Finset.prod_insert, not_false_iff, Finset.mem_singleton]
+  rw [h1]
+  nth_rw 2 [h2]
+  nth_rw 4 [h2]
+  rw [← h_inter, ← h_prod, h_indep {i, j} hf_m]
+#align probability_theory.Indep_sets.indep_sets ProbabilityTheory.iIndepSets.indepSets
+
+theorem iIndep.indep {m : ι → MeasurableSpace Ω} [MeasurableSpace Ω] {μ : Measure Ω}
+    (h_indep : iIndep m μ) {i j : ι} (hij : i ≠ j) : Indep (m i) (m j) μ :=
   by
   change indep_sets ((fun x => measurable_set[m x]) i) ((fun x => measurable_set[m x]) j) μ
   exact Indep_sets.indep_sets h_indep hij
-#align probability_theory.Indep.indep ProbabilityTheory.Indep.indepCat
+#align probability_theory.Indep.indep ProbabilityTheory.iIndep.indep
 
-theorem IndepFun.indepFunCat {m₀ : MeasurableSpace Ω} {μ : Measure Ω} {β : ι → Type _}
-    {m : ∀ x, MeasurableSpace (β x)} {f : ∀ i, Ω → β i} (hf_Indep : IndepFun m f μ) {i j : ι}
-    (hij : i ≠ j) : IndepFunCat (f i) (f j) μ :=
+theorem iIndepFun.indepFun {m₀ : MeasurableSpace Ω} {μ : Measure Ω} {β : ι → Type _}
+    {m : ∀ x, MeasurableSpace (β x)} {f : ∀ i, Ω → β i} (hf_Indep : iIndepFun m f μ) {i j : ι}
+    (hij : i ≠ j) : IndepFun (f i) (f j) μ :=
   hf_Indep.indep hij
-#align probability_theory.Indep_fun.indep_fun ProbabilityTheory.IndepFun.indepFunCat
+#align probability_theory.Indep_fun.indep_fun ProbabilityTheory.iIndepFun.indepFun
 
 end FromIndepToIndep
 
@@ -326,18 +360,20 @@ section FromMeasurableSpacesToSetsOfSets
 /-! ### Independence of measurable space structures implies independence of generating π-systems -/
 
 
-theorem Indep.indepSets [MeasurableSpace Ω] {μ : Measure Ω} {m : ι → MeasurableSpace Ω}
-    {s : ι → Set (Set Ω)} (hms : ∀ n, m n = generateFrom (s n)) (h_indep : Indep m μ) :
-    IndepSets s μ := fun S f hfs =>
+theorem iIndep.iIndepSets [MeasurableSpace Ω] {μ : Measure Ω} {m : ι → MeasurableSpace Ω}
+    {s : ι → Set (Set Ω)} (hms : ∀ n, m n = generateFrom (s n)) (h_indep : iIndep m μ) :
+    iIndepSets s μ := fun S f hfs =>
   h_indep S fun x hxS =>
     ((hms x).symm ▸ measurableSet_generateFrom (hfs x hxS) : measurable_set[m x] (f x))
-#align probability_theory.Indep.Indep_sets ProbabilityTheory.Indep.indepSets
+#align probability_theory.Indep.Indep_sets ProbabilityTheory.iIndep.iIndepSets
 
-theorem IndepCat.indepSetsCat [MeasurableSpace Ω] {μ : Measure Ω} {s1 s2 : Set (Set Ω)}
-    (h_indep : IndepCat (generateFrom s1) (generateFrom s2) μ) : IndepSetsCat s1 s2 μ :=
+#print ProbabilityTheory.Indep.indepSets /-
+theorem Indep.indepSets [MeasurableSpace Ω] {μ : Measure Ω} {s1 s2 : Set (Set Ω)}
+    (h_indep : Indep (generateFrom s1) (generateFrom s2) μ) : IndepSets s1 s2 μ :=
   fun t1 t2 ht1 ht2 =>
   h_indep t1 t2 (measurableSet_generateFrom ht1) (measurableSet_generateFrom ht2)
-#align probability_theory.indep.indep_sets ProbabilityTheory.IndepCat.indepSetsCat
+#align probability_theory.indep.indep_sets ProbabilityTheory.Indep.indepSets
+-/
 
 end FromMeasurableSpacesToSetsOfSets
 
@@ -347,9 +383,9 @@ section FromPiSystemsToMeasurableSpaces
 
 
 private theorem indep_sets.indep_aux {m2 : MeasurableSpace Ω} {m : MeasurableSpace Ω}
-    {μ : Measure Ω} [ProbabilityMeasure μ] {p1 p2 : Set (Set Ω)} (h2 : m2 ≤ m) (hp2 : IsPiSystem p2)
-    (hpm2 : m2 = generateFrom p2) (hyp : IndepSetsCat p1 p2 μ) {t1 t2 : Set Ω} (ht1 : t1 ∈ p1)
-    (ht2m : measurable_set[m2] t2) : μ (t1 ∩ t2) = μ t1 * μ t2 :=
+    {μ : Measure Ω} [IsProbabilityMeasure μ] {p1 p2 : Set (Set Ω)} (h2 : m2 ≤ m)
+    (hp2 : IsPiSystem p2) (hpm2 : m2 = generateFrom p2) (hyp : IndepSets p1 p2 μ) {t1 t2 : Set Ω}
+    (ht1 : t1 ∈ p1) (ht2m : measurable_set[m2] t2) : μ (t1 ∩ t2) = μ t1 * μ t2 :=
   by
   let μ_inter := μ.restrict t1
   let ν := μ t1 • μ
@@ -365,10 +401,11 @@ private theorem indep_sets.indep_aux {m2 : MeasurableSpace Ω} {m : MeasurableSp
   rw [measure.restrict_apply ht2, measure.smul_apply, Set.inter_comm]
   exact hyp t1 t ht1 ht
 
-theorem IndepSetsCat.indepCat {m1 m2 : MeasurableSpace Ω} {m : MeasurableSpace Ω} {μ : Measure Ω}
-    [ProbabilityMeasure μ] {p1 p2 : Set (Set Ω)} (h1 : m1 ≤ m) (h2 : m2 ≤ m) (hp1 : IsPiSystem p1)
+#print ProbabilityTheory.IndepSets.indep /-
+theorem IndepSets.indep {m1 m2 : MeasurableSpace Ω} {m : MeasurableSpace Ω} {μ : Measure Ω}
+    [IsProbabilityMeasure μ] {p1 p2 : Set (Set Ω)} (h1 : m1 ≤ m) (h2 : m2 ≤ m) (hp1 : IsPiSystem p1)
     (hp2 : IsPiSystem p2) (hpm1 : m1 = generateFrom p1) (hpm2 : m2 = generateFrom p2)
-    (hyp : IndepSetsCat p1 p2 μ) : IndepCat m1 m2 μ :=
+    (hyp : IndepSets p1 p2 μ) : Indep m1 m2 μ :=
   by
   intro t1 t2 ht1 ht2
   let μ_inter := μ.restrict t2
@@ -384,63 +421,66 @@ theorem IndepSetsCat.indepCat {m1 m2 : MeasurableSpace Ω} {m : MeasurableSpace
     exact measurable_set_generate_from ht
   rw [measure.restrict_apply ht1, measure.smul_apply, smul_eq_mul, mul_comm]
   exact indep_sets.indep_aux h2 hp2 hpm2 hyp ht ht2
-#align probability_theory.indep_sets.indep ProbabilityTheory.IndepSetsCat.indepCat
+#align probability_theory.indep_sets.indep ProbabilityTheory.IndepSets.indep
+-/
 
-theorem IndepSetsCat.indep' {m : MeasurableSpace Ω} {μ : Measure Ω} [ProbabilityMeasure μ]
+#print ProbabilityTheory.IndepSets.indep' /-
+theorem IndepSets.indep' {m : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
     {p1 p2 : Set (Set Ω)} (hp1m : ∀ s ∈ p1, MeasurableSet s) (hp2m : ∀ s ∈ p2, MeasurableSet s)
-    (hp1 : IsPiSystem p1) (hp2 : IsPiSystem p2) (hyp : IndepSetsCat p1 p2 μ) :
-    IndepCat (generateFrom p1) (generateFrom p2) μ :=
+    (hp1 : IsPiSystem p1) (hp2 : IsPiSystem p2) (hyp : IndepSets p1 p2 μ) :
+    Indep (generateFrom p1) (generateFrom p2) μ :=
   hyp.indep (generateFrom_le hp1m) (generateFrom_le hp2m) hp1 hp2 rfl rfl
-#align probability_theory.indep_sets.indep' ProbabilityTheory.IndepSetsCat.indep'
+#align probability_theory.indep_sets.indep' ProbabilityTheory.IndepSets.indep'
+-/
 
 variable {m0 : MeasurableSpace Ω} {μ : Measure Ω}
 
-theorem indepSetsCat_piiUnionInter_of_disjoint [ProbabilityMeasure μ] {s : ι → Set (Set Ω)}
-    {S T : Set ι} (h_indep : IndepSets s μ) (hST : Disjoint S T) :
-    IndepSetsCat (piiUnionInter s S) (piiUnionInter s T) μ :=
+theorem indepSets_piiUnionInter_of_disjoint [IsProbabilityMeasure μ] {s : ι → Set (Set Ω)}
+    {S T : Set ι} (h_indep : iIndepSets s μ) (hST : Disjoint S T) :
+    IndepSets (piiUnionInter s S) (piiUnionInter s T) μ :=
   by
   rintro t1 t2 ⟨p1, hp1, f1, ht1_m, ht1_eq⟩ ⟨p2, hp2, f2, ht2_m, ht2_eq⟩
   classical
-    let g i := ite (i ∈ p1) (f1 i) Set.univ ∩ ite (i ∈ p2) (f2 i) Set.univ
-    have h_P_inter : μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, μ (g n) :=
+  let g i := ite (i ∈ p1) (f1 i) Set.univ ∩ ite (i ∈ p2) (f2 i) Set.univ
+  have h_P_inter : μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, μ (g n) :=
+    by
+    have hgm : ∀ i ∈ p1 ∪ p2, g i ∈ s i :=
       by
-      have hgm : ∀ i ∈ p1 ∪ p2, g i ∈ s i :=
-        by
-        intro i hi_mem_union
-        rw [Finset.mem_union] at hi_mem_union 
-        cases' hi_mem_union with hi1 hi2
-        · have hi2 : i ∉ p2 := fun hip2 => set.disjoint_left.mp hST (hp1 hi1) (hp2 hip2)
-          simp_rw [g, if_pos hi1, if_neg hi2, Set.inter_univ]
-          exact ht1_m i hi1
-        · have hi1 : i ∉ p1 := fun hip1 => set.disjoint_right.mp hST (hp2 hi2) (hp1 hip1)
-          simp_rw [g, if_neg hi1, if_pos hi2, Set.univ_inter]
-          exact ht2_m i hi2
-      have h_p1_inter_p2 :
-        ((⋂ x ∈ p1, f1 x) ∩ ⋂ x ∈ p2, f2 x) =
-          ⋂ i ∈ p1 ∪ p2, ite (i ∈ p1) (f1 i) Set.univ ∩ ite (i ∈ p2) (f2 i) Set.univ :=
-        by
-        ext1 x
-        simp only [Set.mem_ite_univ_right, Set.mem_inter_iff, Set.mem_iInter, Finset.mem_union]
-        exact
-          ⟨fun h i _ => ⟨h.1 i, h.2 i⟩, fun h =>
-            ⟨fun i hi => (h i (Or.inl hi)).1 hi, fun i hi => (h i (Or.inr hi)).2 hi⟩⟩
-      rw [ht1_eq, ht2_eq, h_p1_inter_p2, ← h_indep _ hgm]
-    have h_μg : ∀ n, μ (g n) = ite (n ∈ p1) (μ (f1 n)) 1 * ite (n ∈ p2) (μ (f2 n)) 1 :=
+      intro i hi_mem_union
+      rw [Finset.mem_union] at hi_mem_union 
+      cases' hi_mem_union with hi1 hi2
+      · have hi2 : i ∉ p2 := fun hip2 => set.disjoint_left.mp hST (hp1 hi1) (hp2 hip2)
+        simp_rw [g, if_pos hi1, if_neg hi2, Set.inter_univ]
+        exact ht1_m i hi1
+      · have hi1 : i ∉ p1 := fun hip1 => set.disjoint_right.mp hST (hp2 hi2) (hp1 hip1)
+        simp_rw [g, if_neg hi1, if_pos hi2, Set.univ_inter]
+        exact ht2_m i hi2
+    have h_p1_inter_p2 :
+      ((⋂ x ∈ p1, f1 x) ∩ ⋂ x ∈ p2, f2 x) =
+        ⋂ i ∈ p1 ∪ p2, ite (i ∈ p1) (f1 i) Set.univ ∩ ite (i ∈ p2) (f2 i) Set.univ :=
       by
-      intro n
-      simp_rw [g]
-      split_ifs
-      · exact absurd rfl (set.disjoint_iff_forall_ne.mp hST _ (hp1 h) _ (hp2 h_1))
-      all_goals simp only [measure_univ, one_mul, mul_one, Set.inter_univ, Set.univ_inter]
-    simp_rw [h_P_inter, h_μg, Finset.prod_mul_distrib,
-      Finset.prod_ite_mem (p1 ∪ p2) p1 fun x => μ (f1 x), Finset.union_inter_cancel_left,
-      Finset.prod_ite_mem (p1 ∪ p2) p2 fun x => μ (f2 x), Finset.union_inter_cancel_right, ht1_eq, ←
-      h_indep p1 ht1_m, ht2_eq, ← h_indep p2 ht2_m]
-#align probability_theory.indep_sets_pi_Union_Inter_of_disjoint ProbabilityTheory.indepSetsCat_piiUnionInter_of_disjoint
-
-theorem IndepSet.indepCat_generateFrom_of_disjoint [ProbabilityMeasure μ] {s : ι → Set Ω}
-    (hsm : ∀ n, MeasurableSet (s n)) (hs : IndepSet s μ) (S T : Set ι) (hST : Disjoint S T) :
-    IndepCat (generateFrom { t | ∃ n ∈ S, s n = t }) (generateFrom { t | ∃ k ∈ T, s k = t }) μ :=
+      ext1 x
+      simp only [Set.mem_ite_univ_right, Set.mem_inter_iff, Set.mem_iInter, Finset.mem_union]
+      exact
+        ⟨fun h i _ => ⟨h.1 i, h.2 i⟩, fun h =>
+          ⟨fun i hi => (h i (Or.inl hi)).1 hi, fun i hi => (h i (Or.inr hi)).2 hi⟩⟩
+    rw [ht1_eq, ht2_eq, h_p1_inter_p2, ← h_indep _ hgm]
+  have h_μg : ∀ n, μ (g n) = ite (n ∈ p1) (μ (f1 n)) 1 * ite (n ∈ p2) (μ (f2 n)) 1 :=
+    by
+    intro n
+    simp_rw [g]
+    split_ifs
+    · exact absurd rfl (set.disjoint_iff_forall_ne.mp hST _ (hp1 h) _ (hp2 h_1))
+    all_goals simp only [measure_univ, one_mul, mul_one, Set.inter_univ, Set.univ_inter]
+  simp_rw [h_P_inter, h_μg, Finset.prod_mul_distrib,
+    Finset.prod_ite_mem (p1 ∪ p2) p1 fun x => μ (f1 x), Finset.union_inter_cancel_left,
+    Finset.prod_ite_mem (p1 ∪ p2) p2 fun x => μ (f2 x), Finset.union_inter_cancel_right, ht1_eq, ←
+    h_indep p1 ht1_m, ht2_eq, ← h_indep p2 ht2_m]
+#align probability_theory.indep_sets_pi_Union_Inter_of_disjoint ProbabilityTheory.indepSets_piiUnionInter_of_disjoint
+
+theorem iIndepSet.indep_generateFrom_of_disjoint [IsProbabilityMeasure μ] {s : ι → Set Ω}
+    (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (S T : Set ι) (hST : Disjoint S T) :
+    Indep (generateFrom {t | ∃ n ∈ S, s n = t}) (generateFrom {t | ∃ k ∈ T, s k = t}) μ :=
   by
   rw [← generateFrom_piiUnionInter_singleton_left, ← generateFrom_piiUnionInter_singleton_left]
   refine'
@@ -452,11 +492,11 @@ theorem IndepSet.indepCat_generateFrom_of_disjoint [ProbabilityMeasure μ] {s :
   · exact isPiSystem_piiUnionInter _ (fun k => IsPiSystem.singleton _) _
   · exact isPiSystem_piiUnionInter _ (fun k => IsPiSystem.singleton _) _
   · classical exact indep_sets_pi_Union_Inter_of_disjoint (Indep.Indep_sets (fun n => rfl) hs) hST
-#align probability_theory.Indep_set.indep_generate_from_of_disjoint ProbabilityTheory.IndepSet.indepCat_generateFrom_of_disjoint
+#align probability_theory.Indep_set.indep_generate_from_of_disjoint ProbabilityTheory.iIndepSet.indep_generateFrom_of_disjoint
 
-theorem indepCat_iSup_of_disjoint [ProbabilityMeasure μ] {m : ι → MeasurableSpace Ω}
-    (h_le : ∀ i, m i ≤ m0) (h_indep : Indep m μ) {S T : Set ι} (hST : Disjoint S T) :
-    IndepCat (⨆ i ∈ S, m i) (⨆ i ∈ T, m i) μ :=
+theorem indep_iSup_of_disjoint [IsProbabilityMeasure μ] {m : ι → MeasurableSpace Ω}
+    (h_le : ∀ i, m i ≤ m0) (h_indep : iIndep m μ) {S T : Set ι} (hST : Disjoint S T) :
+    Indep (⨆ i ∈ S, m i) (⨆ i ∈ T, m i) μ :=
   by
   refine'
     indep_sets.indep (iSup₂_le fun i _ => h_le i) (iSup₂_le fun i _ => h_le i) _ _
@@ -465,18 +505,18 @@ theorem indepCat_iSup_of_disjoint [ProbabilityMeasure μ] {m : ι → Measurable
   · exact isPiSystem_piiUnionInter _ (fun n => @is_pi_system_measurable_set Ω (m n)) _
   · exact isPiSystem_piiUnionInter _ (fun n => @is_pi_system_measurable_set Ω (m n)) _
   · classical exact indep_sets_pi_Union_Inter_of_disjoint h_indep hST
-#align probability_theory.indep_supr_of_disjoint ProbabilityTheory.indepCat_iSup_of_disjoint
+#align probability_theory.indep_supr_of_disjoint ProbabilityTheory.indep_iSup_of_disjoint
 
-theorem indepCat_iSup_of_directed_le {Ω} {m : ι → MeasurableSpace Ω} {m' m0 : MeasurableSpace Ω}
-    {μ : Measure Ω} [ProbabilityMeasure μ] (h_indep : ∀ i, IndepCat (m i) m' μ)
-    (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0) (hm : Directed (· ≤ ·) m) : IndepCat (⨆ i, m i) m' μ :=
+theorem indep_iSup_of_directed_le {Ω} {m : ι → MeasurableSpace Ω} {m' m0 : MeasurableSpace Ω}
+    {μ : Measure Ω} [IsProbabilityMeasure μ] (h_indep : ∀ i, Indep (m i) m' μ)
+    (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0) (hm : Directed (· ≤ ·) m) : Indep (⨆ i, m i) m' μ :=
   by
-  let p : ι → Set (Set Ω) := fun n => { t | measurable_set[m n] t }
+  let p : ι → Set (Set Ω) := fun n => {t | measurable_set[m n] t}
   have hp : ∀ n, IsPiSystem (p n) := fun n => @is_pi_system_measurable_set Ω (m n)
   have h_gen_n : ∀ n, m n = generate_from (p n) := fun n =>
     (@generate_from_measurable_set Ω (m n)).symm
   have hp_supr_pi : IsPiSystem (⋃ n, p n) := isPiSystem_iUnion_of_directed_le p hp hm
-  let p' := { t : Set Ω | measurable_set[m'] t }
+  let p' := {t : Set Ω | measurable_set[m'] t}
   have hp'_pi : IsPiSystem p' := @is_pi_system_measurable_set Ω m'
   have h_gen' : m' = generate_from p' := (@generate_from_measurable_set Ω m').symm
   -- the π-systems defined are independent
@@ -484,116 +524,120 @@ theorem indepCat_iSup_of_directed_le {Ω} {m : ι → MeasurableSpace Ω} {m' m0
     by
     refine' indep_sets.Union _
     simp_rw [h_gen_n, h_gen'] at h_indep 
-    exact fun n => (h_indep n).IndepSetsCat
+    exact fun n => (h_indep n).IndepSets
   -- now go from π-systems to σ-algebras
   refine' indep_sets.indep (iSup_le h_le) h_le' hp_supr_pi hp'_pi _ h_gen' h_pi_system_indep
   exact (generate_from_Union_measurable_set _).symm
-#align probability_theory.indep_supr_of_directed_le ProbabilityTheory.indepCat_iSup_of_directed_le
+#align probability_theory.indep_supr_of_directed_le ProbabilityTheory.indep_iSup_of_directed_le
 
-theorem IndepSet.indepCat_generateFrom_lt [Preorder ι] [ProbabilityMeasure μ] {s : ι → Set Ω}
-    (hsm : ∀ n, MeasurableSet (s n)) (hs : IndepSet s μ) (i : ι) :
-    IndepCat (generateFrom {s i}) (generateFrom { t | ∃ j < i, s j = t }) μ :=
+theorem iIndepSet.indep_generateFrom_lt [Preorder ι] [IsProbabilityMeasure μ] {s : ι → Set Ω}
+    (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (i : ι) :
+    Indep (generateFrom {s i}) (generateFrom {t | ∃ j < i, s j = t}) μ :=
   by
-  convert hs.indep_generate_from_of_disjoint hsm {i} { j | j < i }
+  convert
+    hs.indep_generate_from_of_disjoint hsm {i} {j | j < i}
       (set.disjoint_singleton_left.mpr (lt_irrefl _))
   simp only [Set.mem_singleton_iff, exists_prop, exists_eq_left, Set.setOf_eq_eq_singleton']
-#align probability_theory.Indep_set.indep_generate_from_lt ProbabilityTheory.IndepSet.indepCat_generateFrom_lt
+#align probability_theory.Indep_set.indep_generate_from_lt ProbabilityTheory.iIndepSet.indep_generateFrom_lt
 
-theorem IndepSet.indepCat_generateFrom_le [LinearOrder ι] [ProbabilityMeasure μ] {s : ι → Set Ω}
-    (hsm : ∀ n, MeasurableSet (s n)) (hs : IndepSet s μ) (i : ι) {k : ι} (hk : i < k) :
-    IndepCat (generateFrom {s k}) (generateFrom { t | ∃ j ≤ i, s j = t }) μ :=
+theorem iIndepSet.indep_generateFrom_le [LinearOrder ι] [IsProbabilityMeasure μ] {s : ι → Set Ω}
+    (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (i : ι) {k : ι} (hk : i < k) :
+    Indep (generateFrom {s k}) (generateFrom {t | ∃ j ≤ i, s j = t}) μ :=
   by
-  convert hs.indep_generate_from_of_disjoint hsm {k} { j | j ≤ i }
+  convert
+    hs.indep_generate_from_of_disjoint hsm {k} {j | j ≤ i}
       (set.disjoint_singleton_left.mpr hk.not_le)
   simp only [Set.mem_singleton_iff, exists_prop, exists_eq_left, Set.setOf_eq_eq_singleton']
-#align probability_theory.Indep_set.indep_generate_from_le ProbabilityTheory.IndepSet.indepCat_generateFrom_le
-
-theorem IndepSet.indepCat_generateFrom_le_nat [ProbabilityMeasure μ] {s : ℕ → Set Ω}
-    (hsm : ∀ n, MeasurableSet (s n)) (hs : IndepSet s μ) (n : ℕ) :
-    IndepCat (generateFrom {s (n + 1)}) (generateFrom { t | ∃ k ≤ n, s k = t }) μ :=
-  hs.indepCat_generateFrom_le hsm _ n.lt_succ_self
-#align probability_theory.Indep_set.indep_generate_from_le_nat ProbabilityTheory.IndepSet.indepCat_generateFrom_le_nat
-
-theorem indepCat_iSup_of_monotone [SemilatticeSup ι] {Ω} {m : ι → MeasurableSpace Ω}
-    {m' m0 : MeasurableSpace Ω} {μ : Measure Ω} [ProbabilityMeasure μ]
-    (h_indep : ∀ i, IndepCat (m i) m' μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0)
-    (hm : Monotone m) : IndepCat (⨆ i, m i) m' μ :=
-  indepCat_iSup_of_directed_le h_indep h_le h_le' (Monotone.directed_le hm)
-#align probability_theory.indep_supr_of_monotone ProbabilityTheory.indepCat_iSup_of_monotone
-
-theorem indepCat_iSup_of_antitone [SemilatticeInf ι] {Ω} {m : ι → MeasurableSpace Ω}
-    {m' m0 : MeasurableSpace Ω} {μ : Measure Ω} [ProbabilityMeasure μ]
-    (h_indep : ∀ i, IndepCat (m i) m' μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0)
-    (hm : Antitone m) : IndepCat (⨆ i, m i) m' μ :=
-  indepCat_iSup_of_directed_le h_indep h_le h_le' (directed_of_inf hm)
-#align probability_theory.indep_supr_of_antitone ProbabilityTheory.indepCat_iSup_of_antitone
-
-theorem IndepSets.piiUnionInter_of_not_mem {π : ι → Set (Set Ω)} {a : ι} {S : Finset ι}
-    (hp_ind : IndepSets π μ) (haS : a ∉ S) : IndepSetsCat (piiUnionInter π S) (π a) μ :=
+#align probability_theory.Indep_set.indep_generate_from_le ProbabilityTheory.iIndepSet.indep_generateFrom_le
+
+theorem iIndepSet.indep_generateFrom_le_nat [IsProbabilityMeasure μ] {s : ℕ → Set Ω}
+    (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (n : ℕ) :
+    Indep (generateFrom {s (n + 1)}) (generateFrom {t | ∃ k ≤ n, s k = t}) μ :=
+  hs.indep_generateFrom_le hsm _ n.lt_succ_self
+#align probability_theory.Indep_set.indep_generate_from_le_nat ProbabilityTheory.iIndepSet.indep_generateFrom_le_nat
+
+theorem indep_iSup_of_monotone [SemilatticeSup ι] {Ω} {m : ι → MeasurableSpace Ω}
+    {m' m0 : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
+    (h_indep : ∀ i, Indep (m i) m' μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0) (hm : Monotone m) :
+    Indep (⨆ i, m i) m' μ :=
+  indep_iSup_of_directed_le h_indep h_le h_le' (Monotone.directed_le hm)
+#align probability_theory.indep_supr_of_monotone ProbabilityTheory.indep_iSup_of_monotone
+
+theorem indep_iSup_of_antitone [SemilatticeInf ι] {Ω} {m : ι → MeasurableSpace Ω}
+    {m' m0 : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
+    (h_indep : ∀ i, Indep (m i) m' μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0) (hm : Antitone m) :
+    Indep (⨆ i, m i) m' μ :=
+  indep_iSup_of_directed_le h_indep h_le h_le' (directed_of_inf hm)
+#align probability_theory.indep_supr_of_antitone ProbabilityTheory.indep_iSup_of_antitone
+
+theorem iIndepSets.piiUnionInter_of_not_mem {π : ι → Set (Set Ω)} {a : ι} {S : Finset ι}
+    (hp_ind : iIndepSets π μ) (haS : a ∉ S) : IndepSets (piiUnionInter π S) (π a) μ :=
   by
   rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
   rw [Finset.coe_subset] at hs_mem 
   classical
-    let f n := ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
-    have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n :=
-      by
-      intro n hn_mem_insert
-      simp_rw [f]
-      cases' finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
-      · simp [hn_mem, ht2_mem_pia]
-      · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
-        simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
-    have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
-    have h_t1 : t1 = ⋂ n ∈ s, f n :=
-      by
-      suffices h_forall : ∀ n ∈ s, f n = ft1 n
-      · rw [ht1_eq]
-        congr with (n x)
-        congr with (hns y)
-        simp only [(h_forall n hns).symm]
-      intro n hnS
-      have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hnS)
-      simp_rw [f, if_pos hnS, if_neg hn_ne_a]
-    have h_μ_t1 : μ t1 = ∏ n in s, μ (f n) := by rw [h_t1, ← hp_ind s h_f_mem_pi]
-    have h_t2 : t2 = f a := by simp_rw [f]; simp
-    have h_μ_inter : μ (t1 ∩ t2) = ∏ n in insert a s, μ (f n) :=
-      by
-      have h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n := by
-        rw [h_t1, h_t2, Finset.set_biInter_insert, Set.inter_comm]
-      rw [h_t1_inter_t2, ← hp_ind (insert a s) h_f_mem]
-    have has : a ∉ s := fun has_mem => haS (hs_mem Membership)
-    rw [h_μ_inter, Finset.prod_insert has, h_t2, mul_comm, h_μ_t1]
-#align probability_theory.Indep_sets.pi_Union_Inter_of_not_mem ProbabilityTheory.IndepSets.piiUnionInter_of_not_mem
-
+  let f n := ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
+  have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n :=
+    by
+    intro n hn_mem_insert
+    simp_rw [f]
+    cases' finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
+    · simp [hn_mem, ht2_mem_pia]
+    · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
+      simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
+  have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
+  have h_t1 : t1 = ⋂ n ∈ s, f n :=
+    by
+    suffices h_forall : ∀ n ∈ s, f n = ft1 n
+    · rw [ht1_eq]
+      congr with (n x)
+      congr with (hns y)
+      simp only [(h_forall n hns).symm]
+    intro n hnS
+    have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hnS)
+    simp_rw [f, if_pos hnS, if_neg hn_ne_a]
+  have h_μ_t1 : μ t1 = ∏ n in s, μ (f n) := by rw [h_t1, ← hp_ind s h_f_mem_pi]
+  have h_t2 : t2 = f a := by simp_rw [f]; simp
+  have h_μ_inter : μ (t1 ∩ t2) = ∏ n in insert a s, μ (f n) :=
+    by
+    have h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n := by
+      rw [h_t1, h_t2, Finset.set_biInter_insert, Set.inter_comm]
+    rw [h_t1_inter_t2, ← hp_ind (insert a s) h_f_mem]
+  have has : a ∉ s := fun has_mem => haS (hs_mem Membership)
+  rw [h_μ_inter, Finset.prod_insert has, h_t2, mul_comm, h_μ_t1]
+#align probability_theory.Indep_sets.pi_Union_Inter_of_not_mem ProbabilityTheory.iIndepSets.piiUnionInter_of_not_mem
+
+/- warning: probability_theory.Indep_sets.Indep clashes with probability_theory.indep_sets.indep -> ProbabilityTheory.IndepSets.indep
+Case conversion may be inaccurate. Consider using '#align probability_theory.Indep_sets.Indep ProbabilityTheory.IndepSets.indepₓ'. -/
 /-- The measurable space structures generated by independent pi-systems are independent. -/
-theorem IndepSets.indep [ProbabilityMeasure μ] (m : ι → MeasurableSpace Ω) (h_le : ∀ i, m i ≤ m0)
+theorem IndepSets.indep [IsProbabilityMeasure μ] (m : ι → MeasurableSpace Ω) (h_le : ∀ i, m i ≤ m0)
     (π : ι → Set (Set Ω)) (h_pi : ∀ n, IsPiSystem (π n))
-    (h_generate : ∀ i, m i = generateFrom (π i)) (h_ind : IndepSets π μ) : Indep m μ := by
+    (h_generate : ∀ i, m i = generateFrom (π i)) (h_ind : iIndepSets π μ) : iIndep m μ := by
   classical
-    refine' Finset.induction _ _
-    ·
-      simp only [measure_univ, imp_true_iff, Set.iInter_false, Set.iInter_univ, Finset.prod_empty,
-        eq_self_iff_true]
-    intro a S ha_notin_S h_rec f hf_m
-    have hf_m_S : ∀ x ∈ S, measurable_set[m x] (f x) := fun x hx => hf_m x (by simp [hx])
-    rw [Finset.set_biInter_insert, Finset.prod_insert ha_notin_S, ← h_rec hf_m_S]
-    let p := piiUnionInter π S
-    set m_p := generate_from p with hS_eq_generate
-    have h_indep : indep m_p (m a) μ :=
-      by
-      have hp : IsPiSystem p := isPiSystem_piiUnionInter π h_pi S
-      have h_le' : ∀ i, generate_from (π i) ≤ m0 := fun i => (h_generate i).symm.trans_le (h_le i)
-      have hm_p : m_p ≤ m0 := generateFrom_piiUnionInter_le π h_le' S
-      exact
-        indep_sets.indep hm_p (h_le a) hp (h_pi a) hS_eq_generate (h_generate a)
-          (h_ind.pi_Union_Inter_of_not_mem ha_notin_S)
-    refine' h_indep.symm (f a) (⋂ n ∈ S, f n) (hf_m a (Finset.mem_insert_self a S)) _
-    have h_le_p : ∀ i ∈ S, m i ≤ m_p := by
-      intro n hn
-      rw [hS_eq_generate, h_generate n]
-      exact le_generateFrom_piiUnionInter S hn
-    have h_S_f : ∀ i ∈ S, measurable_set[m_p] (f i) := fun i hi => (h_le_p i hi) (f i) (hf_m_S i hi)
-    exact S.measurable_set_bInter h_S_f
+  refine' Finset.induction _ _
+  ·
+    simp only [measure_univ, imp_true_iff, Set.iInter_false, Set.iInter_univ, Finset.prod_empty,
+      eq_self_iff_true]
+  intro a S ha_notin_S h_rec f hf_m
+  have hf_m_S : ∀ x ∈ S, measurable_set[m x] (f x) := fun x hx => hf_m x (by simp [hx])
+  rw [Finset.set_biInter_insert, Finset.prod_insert ha_notin_S, ← h_rec hf_m_S]
+  let p := piiUnionInter π S
+  set m_p := generate_from p with hS_eq_generate
+  have h_indep : indep m_p (m a) μ :=
+    by
+    have hp : IsPiSystem p := isPiSystem_piiUnionInter π h_pi S
+    have h_le' : ∀ i, generate_from (π i) ≤ m0 := fun i => (h_generate i).symm.trans_le (h_le i)
+    have hm_p : m_p ≤ m0 := generateFrom_piiUnionInter_le π h_le' S
+    exact
+      indep_sets.indep hm_p (h_le a) hp (h_pi a) hS_eq_generate (h_generate a)
+        (h_ind.pi_Union_Inter_of_not_mem ha_notin_S)
+  refine' h_indep.symm (f a) (⋂ n ∈ S, f n) (hf_m a (Finset.mem_insert_self a S)) _
+  have h_le_p : ∀ i ∈ S, m i ≤ m_p := by
+    intro n hn
+    rw [hS_eq_generate, h_generate n]
+    exact le_generateFrom_piiUnionInter S hn
+  have h_S_f : ∀ i ∈ S, measurable_set[m_p] (f i) := fun i hi => (h_le_p i hi) (f i) (hf_m_S i hi)
+  exact S.measurable_set_bInter h_S_f
 #align probability_theory.Indep_sets.Indep ProbabilityTheory.IndepSets.indep
 
 end FromPiSystemsToMeasurableSpaces
@@ -610,31 +654,36 @@ We prove the following equivalences on `indep_set`, for measurable sets `s, t`.
 
 variable {s t : Set Ω} (S T : Set (Set Ω))
 
-theorem indepSetCat_iff_indepSetsCat_singleton {m0 : MeasurableSpace Ω} (hs_meas : MeasurableSet s)
+#print ProbabilityTheory.indepSet_iff_indepSets_singleton /-
+theorem indepSet_iff_indepSets_singleton {m0 : MeasurableSpace Ω} (hs_meas : MeasurableSet s)
     (ht_meas : MeasurableSet t) (μ : Measure Ω := by exact MeasureTheory.MeasureSpace.volume)
-    [ProbabilityMeasure μ] : IndepSetCat s t μ ↔ IndepSetsCat {s} {t} μ :=
-  ⟨IndepCat.indepSetsCat, fun h =>
-    IndepSetsCat.indepCat (generateFrom_le fun u hu => by rwa [set.mem_singleton_iff.mp hu])
+    [IsProbabilityMeasure μ] : IndepSet s t μ ↔ IndepSets {s} {t} μ :=
+  ⟨Indep.indepSets, fun h =>
+    IndepSets.indep (generateFrom_le fun u hu => by rwa [set.mem_singleton_iff.mp hu])
       (generateFrom_le fun u hu => by rwa [set.mem_singleton_iff.mp hu]) (IsPiSystem.singleton s)
       (IsPiSystem.singleton t) rfl rfl h⟩
-#align probability_theory.indep_set_iff_indep_sets_singleton ProbabilityTheory.indepSetCat_iff_indepSetsCat_singleton
+#align probability_theory.indep_set_iff_indep_sets_singleton ProbabilityTheory.indepSet_iff_indepSets_singleton
+-/
 
-theorem indepSetCat_iff_measure_inter_eq_mul {m0 : MeasurableSpace Ω} (hs_meas : MeasurableSet s)
+theorem indepSet_iff_measure_inter_eq_mul {m0 : MeasurableSpace Ω} (hs_meas : MeasurableSet s)
     (ht_meas : MeasurableSet t) (μ : Measure Ω := by exact MeasureTheory.MeasureSpace.volume)
-    [ProbabilityMeasure μ] : IndepSetCat s t μ ↔ μ (s ∩ t) = μ s * μ t :=
-  (indepSetCat_iff_indepSetsCat_singleton hs_meas ht_meas μ).trans indepSetsCat_singleton_iff
-#align probability_theory.indep_set_iff_measure_inter_eq_mul ProbabilityTheory.indepSetCat_iff_measure_inter_eq_mul
+    [IsProbabilityMeasure μ] : IndepSet s t μ ↔ μ (s ∩ t) = μ s * μ t :=
+  (indepSet_iff_indepSets_singleton hs_meas ht_meas μ).trans indepSets_singleton_iff
+#align probability_theory.indep_set_iff_measure_inter_eq_mul ProbabilityTheory.indepSet_iff_measure_inter_eq_mul
 
-theorem IndepSetsCat.indepSetCat_of_mem {m0 : MeasurableSpace Ω} (hs : s ∈ S) (ht : t ∈ T)
+#print ProbabilityTheory.IndepSets.indepSet_of_mem /-
+theorem IndepSets.indepSet_of_mem {m0 : MeasurableSpace Ω} (hs : s ∈ S) (ht : t ∈ T)
     (hs_meas : MeasurableSet s) (ht_meas : MeasurableSet t)
-    (μ : Measure Ω := by exact MeasureTheory.MeasureSpace.volume) [ProbabilityMeasure μ]
-    (h_indep : IndepSetsCat S T μ) : IndepSetCat s t μ :=
-  (indepSetCat_iff_measure_inter_eq_mul hs_meas ht_meas μ).mpr (h_indep s t hs ht)
-#align probability_theory.indep_sets.indep_set_of_mem ProbabilityTheory.IndepSetsCat.indepSetCat_of_mem
-
-theorem IndepCat.indepSetCat_of_measurableSet {m₁ m₂ m0 : MeasurableSpace Ω} {μ : Measure Ω}
-    (h_indep : IndepCat m₁ m₂ μ) {s t : Set Ω} (hs : measurable_set[m₁] s)
-    (ht : measurable_set[m₂] t) : IndepSetCat s t μ :=
+    (μ : Measure Ω := by exact MeasureTheory.MeasureSpace.volume) [IsProbabilityMeasure μ]
+    (h_indep : IndepSets S T μ) : IndepSet s t μ :=
+  (indepSet_iff_measure_inter_eq_mul hs_meas ht_meas μ).mpr (h_indep s t hs ht)
+#align probability_theory.indep_sets.indep_set_of_mem ProbabilityTheory.IndepSets.indepSet_of_mem
+-/
+
+#print ProbabilityTheory.Indep.indepSet_of_measurableSet /-
+theorem Indep.indepSet_of_measurableSet {m₁ m₂ m0 : MeasurableSpace Ω} {μ : Measure Ω}
+    (h_indep : Indep m₁ m₂ μ) {s t : Set Ω} (hs : measurable_set[m₁] s)
+    (ht : measurable_set[m₂] t) : IndepSet s t μ :=
   by
   refine' fun s' t' hs' ht' => h_indep s' t' _ _
   · refine' generate_from_induction (fun u => measurable_set[m₁] u) {s} _ _ _ _ hs'
@@ -647,15 +696,18 @@ theorem IndepCat.indepSetCat_of_measurableSet {m₁ m₂ m0 : MeasurableSpace Ω
     · exact @MeasurableSet.empty _ m₂
     · exact fun u hu => hu.compl
     · exact fun f hf => MeasurableSet.iUnion hf
-#align probability_theory.indep.indep_set_of_measurable_set ProbabilityTheory.IndepCat.indepSetCat_of_measurableSet
+#align probability_theory.indep.indep_set_of_measurable_set ProbabilityTheory.Indep.indepSet_of_measurableSet
+-/
 
-theorem indepCat_iff_forall_indepSetCat (m₁ m₂ : MeasurableSpace Ω) {m0 : MeasurableSpace Ω}
+#print ProbabilityTheory.indep_iff_forall_indepSet /-
+theorem indep_iff_forall_indepSet (m₁ m₂ : MeasurableSpace Ω) {m0 : MeasurableSpace Ω}
     (μ : Measure Ω) :
-    IndepCat m₁ m₂ μ ↔ ∀ s t, measurable_set[m₁] s → measurable_set[m₂] t → IndepSetCat s t μ :=
-  ⟨fun h => fun s t hs ht => h.indepSetCat_of_measurableSet hs ht, fun h s t hs ht =>
+    Indep m₁ m₂ μ ↔ ∀ s t, measurable_set[m₁] s → measurable_set[m₂] t → IndepSet s t μ :=
+  ⟨fun h => fun s t hs ht => h.indepSet_of_measurableSet hs ht, fun h s t hs ht =>
     h s t hs ht s t (measurableSet_generateFrom (Set.mem_singleton s))
       (measurableSet_generateFrom (Set.mem_singleton t))⟩
-#align probability_theory.indep_iff_forall_indep_set ProbabilityTheory.indepCat_iff_forall_indepSetCat
+#align probability_theory.indep_iff_forall_indep_set ProbabilityTheory.indep_iff_forall_indepSet
+-/
 
 end IndepSet
 
@@ -668,104 +720,106 @@ section IndepFun
 
 variable {β β' γ γ' : Type _} {mΩ : MeasurableSpace Ω} {μ : Measure Ω} {f : Ω → β} {g : Ω → β'}
 
-theorem indepFunCat_iff_measure_inter_preimage_eq_mul {mβ : MeasurableSpace β}
+theorem indepFun_iff_measure_inter_preimage_eq_mul {mβ : MeasurableSpace β}
     {mβ' : MeasurableSpace β'} :
-    IndepFunCat f g μ ↔
+    IndepFun f g μ ↔
       ∀ s t,
         MeasurableSet s → MeasurableSet t → μ (f ⁻¹' s ∩ g ⁻¹' t) = μ (f ⁻¹' s) * μ (g ⁻¹' t) :=
   by
   constructor <;> intro h
   · refine' fun s t hs ht => h (f ⁻¹' s) (g ⁻¹' t) ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩
   · rintro _ _ ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩; exact h s t hs ht
-#align probability_theory.indep_fun_iff_measure_inter_preimage_eq_mul ProbabilityTheory.indepFunCat_iff_measure_inter_preimage_eq_mul
+#align probability_theory.indep_fun_iff_measure_inter_preimage_eq_mul ProbabilityTheory.indepFun_iff_measure_inter_preimage_eq_mul
 
-theorem indepFun_iff_measure_inter_preimage_eq_mul {ι : Type _} {β : ι → Type _}
+theorem iIndepFun_iff_measure_inter_preimage_eq_mul {ι : Type _} {β : ι → Type _}
     (m : ∀ x, MeasurableSpace (β x)) (f : ∀ i, Ω → β i) :
-    IndepFun m f μ ↔
+    iIndepFun m f μ ↔
       ∀ (S : Finset ι) {sets : ∀ i : ι, Set (β i)} (H : ∀ i, i ∈ S → measurable_set[m i] (sets i)),
         μ (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, μ (f i ⁻¹' sets i) :=
   by
   refine' ⟨fun h S sets h_meas => h _ fun i hi_mem => ⟨sets i, h_meas i hi_mem, rfl⟩, _⟩
   intro h S setsΩ h_meas
   classical
-    let setsβ : ∀ i : ι, Set (β i) := fun i =>
-      dite (i ∈ S) (fun hi_mem => (h_meas i hi_mem).some) fun _ => Set.univ
-    have h_measβ : ∀ i ∈ S, measurable_set[m i] (setsβ i) :=
-      by
-      intro i hi_mem
-      simp_rw [setsβ, dif_pos hi_mem]
-      exact (h_meas i hi_mem).choose_spec.1
-    have h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i :=
-      by
-      intro i hi_mem
-      simp_rw [setsβ, dif_pos hi_mem]
-      exact (h_meas i hi_mem).choose_spec.2.symm
-    have h_left_eq : μ (⋂ i ∈ S, setsΩ i) = μ (⋂ i ∈ S, f i ⁻¹' setsβ i) :=
-      by
-      congr with (i x)
-      simp only [Set.mem_iInter]
-      constructor <;> intro h hi_mem <;> specialize h hi_mem
-      · rwa [h_preim i hi_mem] at h 
-      · rwa [h_preim i hi_mem]
-    have h_right_eq : (∏ i in S, μ (setsΩ i)) = ∏ i in S, μ (f i ⁻¹' setsβ i) :=
-      by
-      refine' Finset.prod_congr rfl fun i hi_mem => _
-      rw [h_preim i hi_mem]
-    rw [h_left_eq, h_right_eq]
-    exact h S h_measβ
-#align probability_theory.Indep_fun_iff_measure_inter_preimage_eq_mul ProbabilityTheory.indepFun_iff_measure_inter_preimage_eq_mul
-
-theorem indepFunCat_iff_indepSetCat_preimage {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
-    [ProbabilityMeasure μ] (hf : Measurable f) (hg : Measurable g) :
-    IndepFunCat f g μ ↔
-      ∀ s t, MeasurableSet s → MeasurableSet t → IndepSetCat (f ⁻¹' s) (g ⁻¹' t) μ :=
+  let setsβ : ∀ i : ι, Set (β i) := fun i =>
+    dite (i ∈ S) (fun hi_mem => (h_meas i hi_mem).some) fun _ => Set.univ
+  have h_measβ : ∀ i ∈ S, measurable_set[m i] (setsβ i) :=
+    by
+    intro i hi_mem
+    simp_rw [setsβ, dif_pos hi_mem]
+    exact (h_meas i hi_mem).choose_spec.1
+  have h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i :=
+    by
+    intro i hi_mem
+    simp_rw [setsβ, dif_pos hi_mem]
+    exact (h_meas i hi_mem).choose_spec.2.symm
+  have h_left_eq : μ (⋂ i ∈ S, setsΩ i) = μ (⋂ i ∈ S, f i ⁻¹' setsβ i) :=
+    by
+    congr with (i x)
+    simp only [Set.mem_iInter]
+    constructor <;> intro h hi_mem <;> specialize h hi_mem
+    · rwa [h_preim i hi_mem] at h 
+    · rwa [h_preim i hi_mem]
+  have h_right_eq : (∏ i in S, μ (setsΩ i)) = ∏ i in S, μ (f i ⁻¹' setsβ i) :=
+    by
+    refine' Finset.prod_congr rfl fun i hi_mem => _
+    rw [h_preim i hi_mem]
+  rw [h_left_eq, h_right_eq]
+  exact h S h_measβ
+#align probability_theory.Indep_fun_iff_measure_inter_preimage_eq_mul ProbabilityTheory.iIndepFun_iff_measure_inter_preimage_eq_mul
+
+theorem indepFun_iff_indepSet_preimage {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
+    [IsProbabilityMeasure μ] (hf : Measurable f) (hg : Measurable g) :
+    IndepFun f g μ ↔ ∀ s t, MeasurableSet s → MeasurableSet t → IndepSet (f ⁻¹' s) (g ⁻¹' t) μ :=
   by
   refine' indep_fun_iff_measure_inter_preimage_eq_mul.trans _
   constructor <;> intro h s t hs ht <;> specialize h s t hs ht
   · rwa [indep_set_iff_measure_inter_eq_mul (hf hs) (hg ht) μ]
   · rwa [← indep_set_iff_measure_inter_eq_mul (hf hs) (hg ht) μ]
-#align probability_theory.indep_fun_iff_indep_set_preimage ProbabilityTheory.indepFunCat_iff_indepSetCat_preimage
+#align probability_theory.indep_fun_iff_indep_set_preimage ProbabilityTheory.indepFun_iff_indepSet_preimage
 
+#print ProbabilityTheory.IndepFun.symm /-
 @[symm]
-theorem IndepFunCat.symm {mβ : MeasurableSpace β} {f g : Ω → β} (hfg : IndepFunCat f g μ) :
-    IndepFunCat g f μ :=
+theorem IndepFun.symm {mβ : MeasurableSpace β} {f g : Ω → β} (hfg : IndepFun f g μ) :
+    IndepFun g f μ :=
   hfg.symm
-#align probability_theory.indep_fun.symm ProbabilityTheory.IndepFunCat.symm
+#align probability_theory.indep_fun.symm ProbabilityTheory.IndepFun.symm
+-/
 
-theorem IndepFunCat.ae_eq {mβ : MeasurableSpace β} {f g f' g' : Ω → β} (hfg : IndepFunCat f g μ)
-    (hf : f =ᵐ[μ] f') (hg : g =ᵐ[μ] g') : IndepFunCat f' g' μ :=
+#print ProbabilityTheory.IndepFun.ae_eq /-
+theorem IndepFun.ae_eq {mβ : MeasurableSpace β} {f g f' g' : Ω → β} (hfg : IndepFun f g μ)
+    (hf : f =ᵐ[μ] f') (hg : g =ᵐ[μ] g') : IndepFun f' g' μ :=
   by
   rintro _ _ ⟨A, hA, rfl⟩ ⟨B, hB, rfl⟩
   have h1 : f ⁻¹' A =ᵐ[μ] f' ⁻¹' A := hf.fun_comp A
   have h2 : g ⁻¹' B =ᵐ[μ] g' ⁻¹' B := hg.fun_comp B
   rw [← measure_congr h1, ← measure_congr h2, ← measure_congr (h1.inter h2)]
   exact hfg _ _ ⟨_, hA, rfl⟩ ⟨_, hB, rfl⟩
-#align probability_theory.indep_fun.ae_eq ProbabilityTheory.IndepFunCat.ae_eq
+#align probability_theory.indep_fun.ae_eq ProbabilityTheory.IndepFun.ae_eq
+-/
 
-theorem IndepFunCat.comp {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
-    {mγ : MeasurableSpace γ} {mγ' : MeasurableSpace γ'} {φ : β → γ} {ψ : β' → γ'}
-    (hfg : IndepFunCat f g μ) (hφ : Measurable φ) (hψ : Measurable ψ) :
-    IndepFunCat (φ ∘ f) (ψ ∘ g) μ :=
+theorem IndepFun.comp {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'} {mγ : MeasurableSpace γ}
+    {mγ' : MeasurableSpace γ'} {φ : β → γ} {ψ : β' → γ'} (hfg : IndepFun f g μ) (hφ : Measurable φ)
+    (hψ : Measurable ψ) : IndepFun (φ ∘ f) (ψ ∘ g) μ :=
   by
   rintro _ _ ⟨A, hA, rfl⟩ ⟨B, hB, rfl⟩
   apply hfg
   · exact ⟨φ ⁻¹' A, hφ hA, set.preimage_comp.symm⟩
   · exact ⟨ψ ⁻¹' B, hψ hB, set.preimage_comp.symm⟩
-#align probability_theory.indep_fun.comp ProbabilityTheory.IndepFunCat.comp
+#align probability_theory.indep_fun.comp ProbabilityTheory.IndepFun.comp
 
 /-- If `f` is a family of mutually independent random variables (`Indep_fun m f μ`) and `S, T` are
 two disjoint finite index sets, then the tuple formed by `f i` for `i ∈ S` is independent of the
 tuple `(f i)_i` for `i ∈ T`. -/
-theorem IndepFun.indepFunCat_finset [ProbabilityMeasure μ] {ι : Type _} {β : ι → Type _}
+theorem iIndepFun.indepFun_finset [IsProbabilityMeasure μ] {ι : Type _} {β : ι → Type _}
     {m : ∀ i, MeasurableSpace (β i)} {f : ∀ i, Ω → β i} (S T : Finset ι) (hST : Disjoint S T)
-    (hf_Indep : IndepFun m f μ) (hf_meas : ∀ i, Measurable (f i)) :
-    IndepFunCat (fun a (i : S) => f i a) (fun a (i : T) => f i a) μ :=
+    (hf_Indep : iIndepFun m f μ) (hf_meas : ∀ i, Measurable (f i)) :
+    IndepFun (fun a (i : S) => f i a) (fun a (i : T) => f i a) μ :=
   by
   -- We introduce π-systems, build from the π-system of boxes which generates `measurable_space.pi`.
   let πSβ :=
     Set.pi (Set.univ : Set S) ''
-      Set.pi (Set.univ : Set S) fun i => { s : Set (β i) | measurable_set[m i] s }
-  let πS := { s : Set Ω | ∃ t ∈ πSβ, (fun a (i : S) => f i a) ⁻¹' t = s }
+      Set.pi (Set.univ : Set S) fun i => {s : Set (β i) | measurable_set[m i] s}
+  let πS := {s : Set Ω | ∃ t ∈ πSβ, (fun a (i : S) => f i a) ⁻¹' t = s}
   have hπS_pi : IsPiSystem πS := is_pi_system_pi.comap fun a i => f i a
   have hπS_gen : (measurable_space.pi.comap fun a (i : S) => f i a) = generate_from πS :=
     by
@@ -775,8 +829,8 @@ theorem IndepFun.indepFunCat_finset [ProbabilityMeasure μ] {ι : Type _} {β :
     · infer_instance
   let πTβ :=
     Set.pi (Set.univ : Set T) ''
-      Set.pi (Set.univ : Set T) fun i => { s : Set (β i) | measurable_set[m i] s }
-  let πT := { s : Set Ω | ∃ t ∈ πTβ, (fun a (i : T) => f i a) ⁻¹' t = s }
+      Set.pi (Set.univ : Set T) fun i => {s : Set (β i) | measurable_set[m i] s}
+  let πT := {s : Set Ω | ∃ t ∈ πTβ, (fun a (i : T) => f i a) ⁻¹' t = s}
   have hπT_pi : IsPiSystem πT := is_pi_system_pi.comap fun a i => f i a
   have hπT_gen : (measurable_space.pi.comap fun a (i : T) => f i a) = generate_from πT :=
     by
@@ -793,168 +847,168 @@ theorem IndepFun.indepFunCat_finset [ProbabilityMeasure μ] {ι : Type _} {β :
   simp only [Set.mem_univ_pi, Set.mem_setOf_eq] at hs1 ht1 
   rw [← hs2, ← ht2]
   classical
-    let sets_s' : ∀ i : ι, Set (β i) := fun i =>
-      dite (i ∈ S) (fun hi => sets_s ⟨i, hi⟩) fun _ => Set.univ
-    have h_sets_s'_eq : ∀ {i} (hi : i ∈ S), sets_s' i = sets_s ⟨i, hi⟩ := by intro i hi;
-      simp_rw [sets_s', dif_pos hi]
-    have h_sets_s'_univ : ∀ {i} (hi : i ∈ T), sets_s' i = Set.univ := by intro i hi;
-      simp_rw [sets_s', dif_neg (finset.disjoint_right.mp hST hi)]
-    let sets_t' : ∀ i : ι, Set (β i) := fun i =>
-      dite (i ∈ T) (fun hi => sets_t ⟨i, hi⟩) fun _ => Set.univ
-    have h_sets_t'_univ : ∀ {i} (hi : i ∈ S), sets_t' i = Set.univ := by intro i hi;
-      simp_rw [sets_t', dif_neg (finset.disjoint_left.mp hST hi)]
-    have h_meas_s' : ∀ i ∈ S, MeasurableSet (sets_s' i) := by intro i hi; rw [h_sets_s'_eq hi];
-      exact hs1 _
-    have h_meas_t' : ∀ i ∈ T, MeasurableSet (sets_t' i) := by intro i hi;
-      simp_rw [sets_t', dif_pos hi]; exact ht1 _
-    have h_eq_inter_S :
-      (fun (ω : Ω) (i : ↥S) => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i :=
-      by
-      ext1 x
-      simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
-      constructor <;> intro h
-      · intro i hi; rw [h_sets_s'_eq hi]; exact h ⟨i, hi⟩
-      · rintro ⟨i, hi⟩; specialize h i hi; rw [h_sets_s'_eq hi] at h ; exact h
-    have h_eq_inter_T :
-      (fun (ω : Ω) (i : ↥T) => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i :=
-      by
-      ext1 x
-      simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
-      constructor <;> intro h
-      · intro i hi; simp_rw [sets_t', dif_pos hi]; exact h ⟨i, hi⟩
-      · rintro ⟨i, hi⟩; specialize h i hi; simp_rw [sets_t', dif_pos hi] at h ; exact h
-    rw [Indep_fun_iff_measure_inter_preimage_eq_mul] at hf_Indep 
-    rw [h_eq_inter_S, h_eq_inter_T, hf_Indep S h_meas_s', hf_Indep T h_meas_t']
-    have h_Inter_inter :
-      ((⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i) =
-        ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) :=
-      by
-      ext1 x
-      simp only [Set.mem_inter_iff, Set.mem_iInter, Set.mem_preimage, Finset.mem_union]
-      constructor <;> intro h
-      · intro i hi
-        cases hi
-        · rw [h_sets_t'_univ hi]; exact ⟨h.1 i hi, Set.mem_univ _⟩
-        · rw [h_sets_s'_univ hi]; exact ⟨Set.mem_univ _, h.2 i hi⟩
-      · exact ⟨fun i hi => (h i (Or.inl hi)).1, fun i hi => (h i (Or.inr hi)).2⟩
-    rw [h_Inter_inter, hf_Indep (S ∪ T)]
-    swap
-    · intro i hi_mem
-      rw [Finset.mem_union] at hi_mem 
-      cases hi_mem
-      · rw [h_sets_t'_univ hi_mem, Set.inter_univ]; exact h_meas_s' i hi_mem
-      · rw [h_sets_s'_univ hi_mem, Set.univ_inter]; exact h_meas_t' i hi_mem
-    rw [Finset.prod_union hST]
-    congr 1
-    · refine' Finset.prod_congr rfl fun i hi => _
-      rw [h_sets_t'_univ hi, Set.inter_univ]
-    · refine' Finset.prod_congr rfl fun i hi => _
-      rw [h_sets_s'_univ hi, Set.univ_inter]
-#align probability_theory.Indep_fun.indep_fun_finset ProbabilityTheory.IndepFun.indepFunCat_finset
-
-theorem IndepFun.indepFunCat_prod [ProbabilityMeasure μ] {ι : Type _} {β : ι → Type _}
-    {m : ∀ i, MeasurableSpace (β i)} {f : ∀ i, Ω → β i} (hf_Indep : IndepFun m f μ)
+  let sets_s' : ∀ i : ι, Set (β i) := fun i =>
+    dite (i ∈ S) (fun hi => sets_s ⟨i, hi⟩) fun _ => Set.univ
+  have h_sets_s'_eq : ∀ {i} (hi : i ∈ S), sets_s' i = sets_s ⟨i, hi⟩ := by intro i hi;
+    simp_rw [sets_s', dif_pos hi]
+  have h_sets_s'_univ : ∀ {i} (hi : i ∈ T), sets_s' i = Set.univ := by intro i hi;
+    simp_rw [sets_s', dif_neg (finset.disjoint_right.mp hST hi)]
+  let sets_t' : ∀ i : ι, Set (β i) := fun i =>
+    dite (i ∈ T) (fun hi => sets_t ⟨i, hi⟩) fun _ => Set.univ
+  have h_sets_t'_univ : ∀ {i} (hi : i ∈ S), sets_t' i = Set.univ := by intro i hi;
+    simp_rw [sets_t', dif_neg (finset.disjoint_left.mp hST hi)]
+  have h_meas_s' : ∀ i ∈ S, MeasurableSet (sets_s' i) := by intro i hi; rw [h_sets_s'_eq hi];
+    exact hs1 _
+  have h_meas_t' : ∀ i ∈ T, MeasurableSet (sets_t' i) := by intro i hi;
+    simp_rw [sets_t', dif_pos hi]; exact ht1 _
+  have h_eq_inter_S :
+    (fun (ω : Ω) (i : ↥S) => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i :=
+    by
+    ext1 x
+    simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
+    constructor <;> intro h
+    · intro i hi; rw [h_sets_s'_eq hi]; exact h ⟨i, hi⟩
+    · rintro ⟨i, hi⟩; specialize h i hi; rw [h_sets_s'_eq hi] at h ; exact h
+  have h_eq_inter_T :
+    (fun (ω : Ω) (i : ↥T) => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i :=
+    by
+    ext1 x
+    simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
+    constructor <;> intro h
+    · intro i hi; simp_rw [sets_t', dif_pos hi]; exact h ⟨i, hi⟩
+    · rintro ⟨i, hi⟩; specialize h i hi; simp_rw [sets_t', dif_pos hi] at h ; exact h
+  rw [Indep_fun_iff_measure_inter_preimage_eq_mul] at hf_Indep 
+  rw [h_eq_inter_S, h_eq_inter_T, hf_Indep S h_meas_s', hf_Indep T h_meas_t']
+  have h_Inter_inter :
+    ((⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i) =
+      ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) :=
+    by
+    ext1 x
+    simp only [Set.mem_inter_iff, Set.mem_iInter, Set.mem_preimage, Finset.mem_union]
+    constructor <;> intro h
+    · intro i hi
+      cases hi
+      · rw [h_sets_t'_univ hi]; exact ⟨h.1 i hi, Set.mem_univ _⟩
+      · rw [h_sets_s'_univ hi]; exact ⟨Set.mem_univ _, h.2 i hi⟩
+    · exact ⟨fun i hi => (h i (Or.inl hi)).1, fun i hi => (h i (Or.inr hi)).2⟩
+  rw [h_Inter_inter, hf_Indep (S ∪ T)]
+  swap
+  · intro i hi_mem
+    rw [Finset.mem_union] at hi_mem 
+    cases hi_mem
+    · rw [h_sets_t'_univ hi_mem, Set.inter_univ]; exact h_meas_s' i hi_mem
+    · rw [h_sets_s'_univ hi_mem, Set.univ_inter]; exact h_meas_t' i hi_mem
+  rw [Finset.prod_union hST]
+  congr 1
+  · refine' Finset.prod_congr rfl fun i hi => _
+    rw [h_sets_t'_univ hi, Set.inter_univ]
+  · refine' Finset.prod_congr rfl fun i hi => _
+    rw [h_sets_s'_univ hi, Set.univ_inter]
+#align probability_theory.Indep_fun.indep_fun_finset ProbabilityTheory.iIndepFun.indepFun_finset
+
+theorem iIndepFun.indepFun_prod [IsProbabilityMeasure μ] {ι : Type _} {β : ι → Type _}
+    {m : ∀ i, MeasurableSpace (β i)} {f : ∀ i, Ω → β i} (hf_Indep : iIndepFun m f μ)
     (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
-    IndepFunCat (fun a => (f i a, f j a)) (f k) μ := by
+    IndepFun (fun a => (f i a, f j a)) (f k) μ := by
   classical
-    have h_right :
-      f k =
-        (fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩) ∘
-          fun a (j : ({k} : Finset ι)) => f j a :=
-      rfl
-    have h_meas_right :
-      Measurable fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩ :=
-      measurable_pi_apply ⟨k, Finset.mem_singleton_self k⟩
-    let s : Finset ι := {i, j}
-    have h_left :
-      (fun ω => (f i ω, f j ω)) =
-        (fun p : ∀ l : s, β l =>
-            (p ⟨i, Finset.mem_insert_self i _⟩,
-              p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩)) ∘
-          fun a (j : s) => f j a :=
-      by
-      ext1 a
-      simp only [Prod.mk.inj_iff]
-      constructor <;> rfl
-    have h_meas_left :
-      Measurable fun p : ∀ l : s, β l =>
-        (p ⟨i, Finset.mem_insert_self i _⟩,
-          p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩) :=
-      Measurable.prod (measurable_pi_apply ⟨i, Finset.mem_insert_self i {j}⟩)
-        (measurable_pi_apply ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self j)⟩)
-    rw [h_left, h_right]
-    refine' (hf_Indep.indep_fun_finset s {k} _ hf_meas).comp h_meas_left h_meas_right
-    rw [Finset.disjoint_singleton_right]
-    simp only [Finset.mem_insert, Finset.mem_singleton, not_or]
-    exact ⟨hik.symm, hjk.symm⟩
-#align probability_theory.Indep_fun.indep_fun_prod ProbabilityTheory.IndepFun.indepFunCat_prod
+  have h_right :
+    f k =
+      (fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩) ∘
+        fun a (j : ({k} : Finset ι)) => f j a :=
+    rfl
+  have h_meas_right :
+    Measurable fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩ :=
+    measurable_pi_apply ⟨k, Finset.mem_singleton_self k⟩
+  let s : Finset ι := {i, j}
+  have h_left :
+    (fun ω => (f i ω, f j ω)) =
+      (fun p : ∀ l : s, β l =>
+          (p ⟨i, Finset.mem_insert_self i _⟩,
+            p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩)) ∘
+        fun a (j : s) => f j a :=
+    by
+    ext1 a
+    simp only [Prod.mk.inj_iff]
+    constructor <;> rfl
+  have h_meas_left :
+    Measurable fun p : ∀ l : s, β l =>
+      (p ⟨i, Finset.mem_insert_self i _⟩,
+        p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩) :=
+    Measurable.prod (measurable_pi_apply ⟨i, Finset.mem_insert_self i {j}⟩)
+      (measurable_pi_apply ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self j)⟩)
+  rw [h_left, h_right]
+  refine' (hf_Indep.indep_fun_finset s {k} _ hf_meas).comp h_meas_left h_meas_right
+  rw [Finset.disjoint_singleton_right]
+  simp only [Finset.mem_insert, Finset.mem_singleton, not_or]
+  exact ⟨hik.symm, hjk.symm⟩
+#align probability_theory.Indep_fun.indep_fun_prod ProbabilityTheory.iIndepFun.indepFun_prod
 
 @[to_additive]
-theorem IndepFun.mul [ProbabilityMeasure μ] {ι : Type _} {β : Type _} {m : MeasurableSpace β}
-    [Mul β] [MeasurableMul₂ β] {f : ι → Ω → β} (hf_Indep : IndepFun (fun _ => m) f μ)
+theorem iIndepFun.mul [IsProbabilityMeasure μ] {ι : Type _} {β : Type _} {m : MeasurableSpace β}
+    [Mul β] [MeasurableMul₂ β] {f : ι → Ω → β} (hf_Indep : iIndepFun (fun _ => m) f μ)
     (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
-    IndepFunCat (f i * f j) (f k) μ :=
+    IndepFun (f i * f j) (f k) μ :=
   by
   have : indep_fun (fun ω => (f i ω, f j ω)) (f k) μ :=
     hf_Indep.indep_fun_prod hf_meas i j k hik hjk
   change indep_fun ((fun p : β × β => p.fst * p.snd) ∘ fun ω => (f i ω, f j ω)) (id ∘ f k) μ
   exact indep_fun.comp this (measurable_fst.mul measurable_snd) measurable_id
-#align probability_theory.Indep_fun.mul ProbabilityTheory.IndepFun.mul
-#align probability_theory.Indep_fun.add ProbabilityTheory.IndepFun.add
+#align probability_theory.Indep_fun.mul ProbabilityTheory.iIndepFun.mul
+#align probability_theory.Indep_fun.add ProbabilityTheory.iIndepFun.add
 
 @[to_additive]
-theorem IndepFun.indepFunCat_finset_prod_of_not_mem [ProbabilityMeasure μ] {ι : Type _} {β : Type _}
+theorem iIndepFun.indepFun_finset_prod_of_not_mem [IsProbabilityMeasure μ] {ι : Type _} {β : Type _}
     {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂ β] {f : ι → Ω → β}
-    (hf_Indep : IndepFun (fun _ => m) f μ) (hf_meas : ∀ i, Measurable (f i)) {s : Finset ι} {i : ι}
-    (hi : i ∉ s) : IndepFunCat (∏ j in s, f j) (f i) μ := by
+    (hf_Indep : iIndepFun (fun _ => m) f μ) (hf_meas : ∀ i, Measurable (f i)) {s : Finset ι} {i : ι}
+    (hi : i ∉ s) : IndepFun (∏ j in s, f j) (f i) μ := by
   classical
-    have h_right :
-      f i =
-        (fun p : ∀ j : ({i} : Finset ι), β => p ⟨i, Finset.mem_singleton_self i⟩) ∘
-          fun a (j : ({i} : Finset ι)) => f j a :=
-      rfl
-    have h_meas_right :
-      Measurable fun p : ∀ j : ({i} : Finset ι), β => p ⟨i, Finset.mem_singleton_self i⟩ :=
-      measurable_pi_apply ⟨i, Finset.mem_singleton_self i⟩
-    have h_left : (∏ j in s, f j) = (fun p : ∀ j : s, β => ∏ j, p j) ∘ fun a (j : s) => f j a :=
-      by
-      ext1 a
-      simp only [Function.comp_apply]
-      have : (∏ j : ↥s, f (↑j) a) = (∏ j : ↥s, f ↑j) a := by rw [Finset.prod_apply]
-      rw [this, Finset.prod_coe_sort]
-    have h_meas_left : Measurable fun p : ∀ j : s, β => ∏ j, p j :=
-      finset.univ.measurable_prod fun (j : ↥s) (H : j ∈ Finset.univ) => measurable_pi_apply j
-    rw [h_left, h_right]
-    exact
-      (hf_Indep.indep_fun_finset s {i} (finset.disjoint_singleton_left.mpr hi).symm hf_meas).comp
-        h_meas_left h_meas_right
-#align probability_theory.Indep_fun.indep_fun_finset_prod_of_not_mem ProbabilityTheory.IndepFun.indepFunCat_finset_prod_of_not_mem
-#align probability_theory.Indep_fun.indep_fun_finset_sum_of_not_mem ProbabilityTheory.IndepFun.indepFunCat_finset_sum_of_not_mem
+  have h_right :
+    f i =
+      (fun p : ∀ j : ({i} : Finset ι), β => p ⟨i, Finset.mem_singleton_self i⟩) ∘
+        fun a (j : ({i} : Finset ι)) => f j a :=
+    rfl
+  have h_meas_right :
+    Measurable fun p : ∀ j : ({i} : Finset ι), β => p ⟨i, Finset.mem_singleton_self i⟩ :=
+    measurable_pi_apply ⟨i, Finset.mem_singleton_self i⟩
+  have h_left : (∏ j in s, f j) = (fun p : ∀ j : s, β => ∏ j, p j) ∘ fun a (j : s) => f j a :=
+    by
+    ext1 a
+    simp only [Function.comp_apply]
+    have : (∏ j : ↥s, f (↑j) a) = (∏ j : ↥s, f ↑j) a := by rw [Finset.prod_apply]
+    rw [this, Finset.prod_coe_sort]
+  have h_meas_left : Measurable fun p : ∀ j : s, β => ∏ j, p j :=
+    finset.univ.measurable_prod fun (j : ↥s) (H : j ∈ Finset.univ) => measurable_pi_apply j
+  rw [h_left, h_right]
+  exact
+    (hf_Indep.indep_fun_finset s {i} (finset.disjoint_singleton_left.mpr hi).symm hf_meas).comp
+      h_meas_left h_meas_right
+#align probability_theory.Indep_fun.indep_fun_finset_prod_of_not_mem ProbabilityTheory.iIndepFun.indepFun_finset_prod_of_not_mem
+#align probability_theory.Indep_fun.indep_fun_finset_sum_of_not_mem ProbabilityTheory.iIndepFun.indepFun_finset_sum_of_not_mem
 
 @[to_additive]
-theorem IndepFun.indepFunCat_prod_range_succ [ProbabilityMeasure μ] {β : Type _}
+theorem iIndepFun.indepFun_prod_range_succ [IsProbabilityMeasure μ] {β : Type _}
     {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂ β] {f : ℕ → Ω → β}
-    (hf_Indep : IndepFun (fun _ => m) f μ) (hf_meas : ∀ i, Measurable (f i)) (n : ℕ) :
-    IndepFunCat (∏ j in Finset.range n, f j) (f n) μ :=
-  hf_Indep.indepFunCat_finset_prod_of_not_mem hf_meas Finset.not_mem_range_self
-#align probability_theory.Indep_fun.indep_fun_prod_range_succ ProbabilityTheory.IndepFun.indepFunCat_prod_range_succ
-#align probability_theory.Indep_fun.indep_fun_sum_range_succ ProbabilityTheory.IndepFun.indepFunCat_sum_range_succ
-
-theorem IndepSet.indepFun_indicator [Zero β] [One β] {m : MeasurableSpace β} {s : ι → Set Ω}
-    (hs : IndepSet s μ) : IndepFun (fun n => m) (fun n => (s n).indicator fun ω => 1) μ := by
+    (hf_Indep : iIndepFun (fun _ => m) f μ) (hf_meas : ∀ i, Measurable (f i)) (n : ℕ) :
+    IndepFun (∏ j in Finset.range n, f j) (f n) μ :=
+  hf_Indep.indepFun_finset_prod_of_not_mem hf_meas Finset.not_mem_range_self
+#align probability_theory.Indep_fun.indep_fun_prod_range_succ ProbabilityTheory.iIndepFun.indepFun_prod_range_succ
+#align probability_theory.Indep_fun.indep_fun_sum_range_succ ProbabilityTheory.iIndepFun.indepFun_sum_range_succ
+
+theorem iIndepSet.iIndepFun_indicator [Zero β] [One β] {m : MeasurableSpace β} {s : ι → Set Ω}
+    (hs : iIndepSet s μ) : iIndepFun (fun n => m) (fun n => (s n).indicator fun ω => 1) μ := by
   classical
-    rw [Indep_fun_iff_measure_inter_preimage_eq_mul]
-    rintro S π hπ
-    simp_rw [Set.indicator_const_preimage_eq_union]
-    refine' @hs S (fun i => ite (1 ∈ π i) (s i) ∅ ∪ ite ((0 : β) ∈ π i) (s iᶜ) ∅) fun i hi => _
-    have hsi : measurable_set[generate_from {s i}] (s i) :=
-      measurable_set_generate_from (Set.mem_singleton _)
-    refine'
-      MeasurableSet.union (MeasurableSet.ite' (fun _ => hsi) fun _ => _)
-        (MeasurableSet.ite' (fun _ => hsi.compl) fun _ => _)
-    · exact @MeasurableSet.empty _ (generate_from {s i})
-    · exact @MeasurableSet.empty _ (generate_from {s i})
-#align probability_theory.Indep_set.Indep_fun_indicator ProbabilityTheory.IndepSet.indepFun_indicator
+  rw [Indep_fun_iff_measure_inter_preimage_eq_mul]
+  rintro S π hπ
+  simp_rw [Set.indicator_const_preimage_eq_union]
+  refine' @hs S (fun i => ite (1 ∈ π i) (s i) ∅ ∪ ite ((0 : β) ∈ π i) (s iᶜ) ∅) fun i hi => _
+  have hsi : measurable_set[generate_from {s i}] (s i) :=
+    measurable_set_generate_from (Set.mem_singleton _)
+  refine'
+    MeasurableSet.union (MeasurableSet.ite' (fun _ => hsi) fun _ => _)
+      (MeasurableSet.ite' (fun _ => hsi.compl) fun _ => _)
+  · exact @MeasurableSet.empty _ (generate_from {s i})
+  · exact @MeasurableSet.empty _ (generate_from {s i})
+#align probability_theory.Indep_set.Indep_fun_indicator ProbabilityTheory.iIndepSet.iIndepFun_indicator
 
 end IndepFun
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -163,7 +163,7 @@ theorem indepCat_bot_right (m' : MeasurableSpace Ω) {m : MeasurableSpace Ω} {
     [ProbabilityMeasure μ] : IndepCat m' ⊥ μ :=
   by
   intro s t hs ht
-  rw [Set.mem_setOf_eq, MeasurableSpace.measurableSet_bot_iff] at ht
+  rw [Set.mem_setOf_eq, MeasurableSpace.measurableSet_bot_iff] at ht 
   cases ht
   · rw [ht, Set.inter_empty, measure_empty, MulZeroClass.mul_zero]
   · rw [ht, Set.inter_univ, measure_univ, mul_one]
@@ -226,7 +226,7 @@ theorem IndepSetsCat.iUnion [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' :
     {μ : Measure Ω} (hyp : ∀ n, IndepSetsCat (s n) s' μ) : IndepSetsCat (⋃ n, s n) s' μ :=
   by
   intro t1 t2 ht1 ht2
-  rw [Set.mem_iUnion] at ht1
+  rw [Set.mem_iUnion] at ht1 
   cases' ht1 with n ht1
   exact hyp n t1 t2 ht1 ht2
 #align probability_theory.indep_sets.Union ProbabilityTheory.IndepSetsCat.iUnion
@@ -235,7 +235,7 @@ theorem IndepSetsCat.bUnion [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' :
     {μ : Measure Ω} {u : Set ι} (hyp : ∀ n ∈ u, IndepSetsCat (s n) s' μ) :
     IndepSetsCat (⋃ n ∈ u, s n) s' μ := by
   intro t1 t2 ht1 ht2
-  simp_rw [Set.mem_iUnion] at ht1
+  simp_rw [Set.mem_iUnion] at ht1 
   rcases ht1 with ⟨n, hpn, ht1⟩
   exact hyp n hpn t1 t2 ht1 ht2
 #align probability_theory.indep_sets.bUnion ProbabilityTheory.IndepSetsCat.bUnion
@@ -407,7 +407,7 @@ theorem indepSetsCat_piiUnionInter_of_disjoint [ProbabilityMeasure μ] {s : ι 
       have hgm : ∀ i ∈ p1 ∪ p2, g i ∈ s i :=
         by
         intro i hi_mem_union
-        rw [Finset.mem_union] at hi_mem_union
+        rw [Finset.mem_union] at hi_mem_union 
         cases' hi_mem_union with hi1 hi2
         · have hi2 : i ∉ p2 := fun hip2 => set.disjoint_left.mp hST (hp1 hi1) (hp2 hip2)
           simp_rw [g, if_pos hi1, if_neg hi2, Set.inter_univ]
@@ -483,7 +483,7 @@ theorem indepCat_iSup_of_directed_le {Ω} {m : ι → MeasurableSpace Ω} {m' m0
   have h_pi_system_indep : indep_sets (⋃ n, p n) p' μ :=
     by
     refine' indep_sets.Union _
-    simp_rw [h_gen_n, h_gen'] at h_indep
+    simp_rw [h_gen_n, h_gen'] at h_indep 
     exact fun n => (h_indep n).IndepSetsCat
   -- now go from π-systems to σ-algebras
   refine' indep_sets.indep (iSup_le h_le) h_le' hp_supr_pi hp'_pi _ h_gen' h_pi_system_indep
@@ -532,7 +532,7 @@ theorem IndepSets.piiUnionInter_of_not_mem {π : ι → Set (Set Ω)} {a : ι} {
     (hp_ind : IndepSets π μ) (haS : a ∉ S) : IndepSetsCat (piiUnionInter π S) (π a) μ :=
   by
   rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
-  rw [Finset.coe_subset] at hs_mem
+  rw [Finset.coe_subset] at hs_mem 
   classical
     let f n := ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
     have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n :=
@@ -705,7 +705,7 @@ theorem indepFun_iff_measure_inter_preimage_eq_mul {ι : Type _} {β : ι → Ty
       congr with (i x)
       simp only [Set.mem_iInter]
       constructor <;> intro h hi_mem <;> specialize h hi_mem
-      · rwa [h_preim i hi_mem] at h
+      · rwa [h_preim i hi_mem] at h 
       · rwa [h_preim i hi_mem]
     have h_right_eq : (∏ i in S, μ (setsΩ i)) = ∏ i in S, μ (f i ⁻¹' setsβ i) :=
       by
@@ -790,7 +790,7 @@ theorem IndepFun.indepFunCat_finset [ProbabilityMeasure μ] {ι : Type _} {β :
       (Measurable.comap_le (measurable_pi_iff.mpr fun i => hf_meas i)) hπS_pi hπT_pi hπS_gen hπT_gen
       _
   rintro _ _ ⟨s, ⟨sets_s, hs1, hs2⟩, rfl⟩ ⟨t, ⟨sets_t, ht1, ht2⟩, rfl⟩
-  simp only [Set.mem_univ_pi, Set.mem_setOf_eq] at hs1 ht1
+  simp only [Set.mem_univ_pi, Set.mem_setOf_eq] at hs1 ht1 
   rw [← hs2, ← ht2]
   classical
     let sets_s' : ∀ i : ι, Set (β i) := fun i =>
@@ -814,7 +814,7 @@ theorem IndepFun.indepFunCat_finset [ProbabilityMeasure μ] {ι : Type _} {β :
       simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
       constructor <;> intro h
       · intro i hi; rw [h_sets_s'_eq hi]; exact h ⟨i, hi⟩
-      · rintro ⟨i, hi⟩; specialize h i hi; rw [h_sets_s'_eq hi] at h; exact h
+      · rintro ⟨i, hi⟩; specialize h i hi; rw [h_sets_s'_eq hi] at h ; exact h
     have h_eq_inter_T :
       (fun (ω : Ω) (i : ↥T) => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i :=
       by
@@ -822,8 +822,8 @@ theorem IndepFun.indepFunCat_finset [ProbabilityMeasure μ] {ι : Type _} {β :
       simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
       constructor <;> intro h
       · intro i hi; simp_rw [sets_t', dif_pos hi]; exact h ⟨i, hi⟩
-      · rintro ⟨i, hi⟩; specialize h i hi; simp_rw [sets_t', dif_pos hi] at h; exact h
-    rw [Indep_fun_iff_measure_inter_preimage_eq_mul] at hf_Indep
+      · rintro ⟨i, hi⟩; specialize h i hi; simp_rw [sets_t', dif_pos hi] at h ; exact h
+    rw [Indep_fun_iff_measure_inter_preimage_eq_mul] at hf_Indep 
     rw [h_eq_inter_S, h_eq_inter_T, hf_Indep S h_meas_s', hf_Indep T h_meas_t']
     have h_Inter_inter :
       ((⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i) =
@@ -840,7 +840,7 @@ theorem IndepFun.indepFunCat_finset [ProbabilityMeasure μ] {ι : Type _} {β :
     rw [h_Inter_inter, hf_Indep (S ∪ T)]
     swap
     · intro i hi_mem
-      rw [Finset.mem_union] at hi_mem
+      rw [Finset.mem_union] at hi_mem 
       cases hi_mem
       · rw [h_sets_t'_univ hi_mem, Set.inter_univ]; exact h_meas_s' i hi_mem
       · rw [h_sets_s'_univ hi_mem, Set.univ_inter]; exact h_meas_t' i hi_mem

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -69,7 +69,7 @@ Part A, Chapter 4.
 
 open MeasureTheory MeasurableSpace
 
-open BigOperators MeasureTheory ENNReal
+open scoped BigOperators MeasureTheory ENNReal
 
 namespace ProbabilityTheory
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -149,11 +149,8 @@ section Indep
 
 @[symm]
 theorem IndepSetsCat.symm {s₁ s₂ : Set (Set Ω)} [MeasurableSpace Ω] {μ : Measure Ω}
-    (h : IndepSetsCat s₁ s₂ μ) : IndepSetsCat s₂ s₁ μ :=
-  by
-  intro t1 t2 ht1 ht2
-  rw [Set.inter_comm, mul_comm]
-  exact h t2 t1 ht2 ht1
+    (h : IndepSetsCat s₁ s₂ μ) : IndepSetsCat s₂ s₁ μ := by intro t1 t2 ht1 ht2;
+  rw [Set.inter_comm, mul_comm]; exact h t2 t1 ht2 ht1
 #align probability_theory.indep_sets.symm ProbabilityTheory.IndepSetsCat.symm
 
 @[symm]
@@ -178,9 +175,7 @@ theorem indepCat_bot_left (m' : MeasurableSpace Ω) {m : MeasurableSpace Ω} {μ
 #align probability_theory.indep_bot_left ProbabilityTheory.indepCat_bot_left
 
 theorem indepSetCat_empty_right {m : MeasurableSpace Ω} {μ : Measure Ω} [ProbabilityMeasure μ]
-    (s : Set Ω) : IndepSetCat s ∅ μ :=
-  by
-  simp only [indep_set, generate_from_singleton_empty]
+    (s : Set Ω) : IndepSetCat s ∅ μ := by simp only [indep_set, generate_from_singleton_empty];
   exact indep_bot_right _
 #align probability_theory.indep_set_empty_right ProbabilityTheory.indepSetCat_empty_right
 
@@ -251,11 +246,8 @@ theorem IndepSetsCat.inter [MeasurableSpace Ω] {s₁ s' : Set (Set Ω)} (s₂ :
 #align probability_theory.indep_sets.inter ProbabilityTheory.IndepSetsCat.inter
 
 theorem IndepSetsCat.iInter [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
-    {μ : Measure Ω} (h : ∃ n, IndepSetsCat (s n) s' μ) : IndepSetsCat (⋂ n, s n) s' μ :=
-  by
-  intro t1 t2 ht1 ht2
-  cases' h with n h
-  exact h t1 t2 (set.mem_Inter.mp ht1 n) ht2
+    {μ : Measure Ω} (h : ∃ n, IndepSetsCat (s n) s' μ) : IndepSetsCat (⋂ n, s n) s' μ := by
+  intro t1 t2 ht1 ht2; cases' h with n h; exact h t1 t2 (set.mem_Inter.mp ht1 n) ht2
 #align probability_theory.indep_sets.Inter ProbabilityTheory.IndepSetsCat.iInter
 
 theorem IndepSetsCat.bInter [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
@@ -549,9 +541,7 @@ theorem IndepSets.piiUnionInter_of_not_mem {π : ι → Set (Set Ω)} {a : ι} {
       simp_rw [f]
       cases' finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
       · simp [hn_mem, ht2_mem_pia]
-      · have hn_ne_a : n ≠ a := by
-          rintro rfl
-          exact haS (hs_mem hn_mem)
+      · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
         simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
     have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
     have h_t1 : t1 = ⋂ n ∈ s, f n :=
@@ -562,14 +552,10 @@ theorem IndepSets.piiUnionInter_of_not_mem {π : ι → Set (Set Ω)} {a : ι} {
         congr with (hns y)
         simp only [(h_forall n hns).symm]
       intro n hnS
-      have hn_ne_a : n ≠ a := by
-        rintro rfl
-        exact haS (hs_mem hnS)
+      have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hnS)
       simp_rw [f, if_pos hnS, if_neg hn_ne_a]
     have h_μ_t1 : μ t1 = ∏ n in s, μ (f n) := by rw [h_t1, ← hp_ind s h_f_mem_pi]
-    have h_t2 : t2 = f a := by
-      simp_rw [f]
-      simp
+    have h_t2 : t2 = f a := by simp_rw [f]; simp
     have h_μ_inter : μ (t1 ∩ t2) = ∏ n in insert a s, μ (f n) :=
       by
       have h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n := by
@@ -690,8 +676,7 @@ theorem indepFunCat_iff_measure_inter_preimage_eq_mul {mβ : MeasurableSpace β}
   by
   constructor <;> intro h
   · refine' fun s t hs ht => h (f ⁻¹' s) (g ⁻¹' t) ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩
-  · rintro _ _ ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩
-    exact h s t hs ht
+  · rintro _ _ ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩; exact h s t hs ht
 #align probability_theory.indep_fun_iff_measure_inter_preimage_eq_mul ProbabilityTheory.indepFunCat_iff_measure_inter_preimage_eq_mul
 
 theorem indepFun_iff_measure_inter_preimage_eq_mul {ι : Type _} {β : ι → Type _}
@@ -810,56 +795,34 @@ theorem IndepFun.indepFunCat_finset [ProbabilityMeasure μ] {ι : Type _} {β :
   classical
     let sets_s' : ∀ i : ι, Set (β i) := fun i =>
       dite (i ∈ S) (fun hi => sets_s ⟨i, hi⟩) fun _ => Set.univ
-    have h_sets_s'_eq : ∀ {i} (hi : i ∈ S), sets_s' i = sets_s ⟨i, hi⟩ :=
-      by
-      intro i hi
+    have h_sets_s'_eq : ∀ {i} (hi : i ∈ S), sets_s' i = sets_s ⟨i, hi⟩ := by intro i hi;
       simp_rw [sets_s', dif_pos hi]
-    have h_sets_s'_univ : ∀ {i} (hi : i ∈ T), sets_s' i = Set.univ :=
-      by
-      intro i hi
+    have h_sets_s'_univ : ∀ {i} (hi : i ∈ T), sets_s' i = Set.univ := by intro i hi;
       simp_rw [sets_s', dif_neg (finset.disjoint_right.mp hST hi)]
     let sets_t' : ∀ i : ι, Set (β i) := fun i =>
       dite (i ∈ T) (fun hi => sets_t ⟨i, hi⟩) fun _ => Set.univ
-    have h_sets_t'_univ : ∀ {i} (hi : i ∈ S), sets_t' i = Set.univ :=
-      by
-      intro i hi
+    have h_sets_t'_univ : ∀ {i} (hi : i ∈ S), sets_t' i = Set.univ := by intro i hi;
       simp_rw [sets_t', dif_neg (finset.disjoint_left.mp hST hi)]
-    have h_meas_s' : ∀ i ∈ S, MeasurableSet (sets_s' i) :=
-      by
-      intro i hi
-      rw [h_sets_s'_eq hi]
+    have h_meas_s' : ∀ i ∈ S, MeasurableSet (sets_s' i) := by intro i hi; rw [h_sets_s'_eq hi];
       exact hs1 _
-    have h_meas_t' : ∀ i ∈ T, MeasurableSet (sets_t' i) :=
-      by
-      intro i hi
-      simp_rw [sets_t', dif_pos hi]
-      exact ht1 _
+    have h_meas_t' : ∀ i ∈ T, MeasurableSet (sets_t' i) := by intro i hi;
+      simp_rw [sets_t', dif_pos hi]; exact ht1 _
     have h_eq_inter_S :
       (fun (ω : Ω) (i : ↥S) => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i :=
       by
       ext1 x
       simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
       constructor <;> intro h
-      · intro i hi
-        rw [h_sets_s'_eq hi]
-        exact h ⟨i, hi⟩
-      · rintro ⟨i, hi⟩
-        specialize h i hi
-        rw [h_sets_s'_eq hi] at h
-        exact h
+      · intro i hi; rw [h_sets_s'_eq hi]; exact h ⟨i, hi⟩
+      · rintro ⟨i, hi⟩; specialize h i hi; rw [h_sets_s'_eq hi] at h; exact h
     have h_eq_inter_T :
       (fun (ω : Ω) (i : ↥T) => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i :=
       by
       ext1 x
       simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
       constructor <;> intro h
-      · intro i hi
-        simp_rw [sets_t', dif_pos hi]
-        exact h ⟨i, hi⟩
-      · rintro ⟨i, hi⟩
-        specialize h i hi
-        simp_rw [sets_t', dif_pos hi] at h
-        exact h
+      · intro i hi; simp_rw [sets_t', dif_pos hi]; exact h ⟨i, hi⟩
+      · rintro ⟨i, hi⟩; specialize h i hi; simp_rw [sets_t', dif_pos hi] at h; exact h
     rw [Indep_fun_iff_measure_inter_preimage_eq_mul] at hf_Indep
     rw [h_eq_inter_S, h_eq_inter_T, hf_Indep S h_meas_s', hf_Indep T h_meas_t']
     have h_Inter_inter :
@@ -871,20 +834,16 @@ theorem IndepFun.indepFunCat_finset [ProbabilityMeasure μ] {ι : Type _} {β :
       constructor <;> intro h
       · intro i hi
         cases hi
-        · rw [h_sets_t'_univ hi]
-          exact ⟨h.1 i hi, Set.mem_univ _⟩
-        · rw [h_sets_s'_univ hi]
-          exact ⟨Set.mem_univ _, h.2 i hi⟩
+        · rw [h_sets_t'_univ hi]; exact ⟨h.1 i hi, Set.mem_univ _⟩
+        · rw [h_sets_s'_univ hi]; exact ⟨Set.mem_univ _, h.2 i hi⟩
       · exact ⟨fun i hi => (h i (Or.inl hi)).1, fun i hi => (h i (Or.inr hi)).2⟩
     rw [h_Inter_inter, hf_Indep (S ∪ T)]
     swap
     · intro i hi_mem
       rw [Finset.mem_union] at hi_mem
       cases hi_mem
-      · rw [h_sets_t'_univ hi_mem, Set.inter_univ]
-        exact h_meas_s' i hi_mem
-      · rw [h_sets_s'_univ hi_mem, Set.univ_inter]
-        exact h_meas_t' i hi_mem
+      · rw [h_sets_t'_univ hi_mem, Set.inter_univ]; exact h_meas_s' i hi_mem
+      · rw [h_sets_s'_univ hi_mem, Set.univ_inter]; exact h_meas_t' i hi_mem
     rw [Finset.prod_union hST]
     congr 1
     · refine' Finset.prod_congr rfl fun i hi => _

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -372,7 +372,6 @@ private theorem indep_sets.indep_aux {m2 : MeasurableSpace Ω} {m : MeasurableSp
     exact measurable_set_generate_from ht
   rw [measure.restrict_apply ht2, measure.smul_apply, Set.inter_comm]
   exact hyp t1 t ht1 ht
-#align probability_theory.indep_sets.indep_aux probability_theory.indep_sets.indep_aux
 
 theorem IndepSetsCat.indepCat {m1 m2 : MeasurableSpace Ω} {m : MeasurableSpace Ω} {μ : Measure Ω}
     [ProbabilityMeasure μ] {p1 p2 : Set (Set Ω)} (h1 : m1 ≤ m) (h2 : m2 ≤ m) (hp1 : IsPiSystem p1)

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

@@ -227,20 +227,20 @@ theorem IndepSetsCat.union_iff [MeasurableSpace Ω] {s₁ s₂ s' : Set (Set Ω)
     fun h => IndepSetsCat.union h.left h.right⟩
 #align probability_theory.indep_sets.union_iff ProbabilityTheory.IndepSetsCat.union_iff
 
-theorem IndepSetsCat.unionᵢ [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
+theorem IndepSetsCat.iUnion [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
     {μ : Measure Ω} (hyp : ∀ n, IndepSetsCat (s n) s' μ) : IndepSetsCat (⋃ n, s n) s' μ :=
   by
   intro t1 t2 ht1 ht2
-  rw [Set.mem_unionᵢ] at ht1
+  rw [Set.mem_iUnion] at ht1
   cases' ht1 with n ht1
   exact hyp n t1 t2 ht1 ht2
-#align probability_theory.indep_sets.Union ProbabilityTheory.IndepSetsCat.unionᵢ
+#align probability_theory.indep_sets.Union ProbabilityTheory.IndepSetsCat.iUnion
 
 theorem IndepSetsCat.bUnion [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
     {μ : Measure Ω} {u : Set ι} (hyp : ∀ n ∈ u, IndepSetsCat (s n) s' μ) :
     IndepSetsCat (⋃ n ∈ u, s n) s' μ := by
   intro t1 t2 ht1 ht2
-  simp_rw [Set.mem_unionᵢ] at ht1
+  simp_rw [Set.mem_iUnion] at ht1
   rcases ht1 with ⟨n, hpn, ht1⟩
   exact hyp n hpn t1 t2 ht1 ht2
 #align probability_theory.indep_sets.bUnion ProbabilityTheory.IndepSetsCat.bUnion
@@ -250,20 +250,20 @@ theorem IndepSetsCat.inter [MeasurableSpace Ω] {s₁ s' : Set (Set Ω)} (s₂ :
   fun t1 t2 ht1 ht2 => h₁ t1 t2 ((Set.mem_inter_iff _ _ _).mp ht1).left ht2
 #align probability_theory.indep_sets.inter ProbabilityTheory.IndepSetsCat.inter
 
-theorem IndepSetsCat.interᵢ [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
+theorem IndepSetsCat.iInter [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
     {μ : Measure Ω} (h : ∃ n, IndepSetsCat (s n) s' μ) : IndepSetsCat (⋂ n, s n) s' μ :=
   by
   intro t1 t2 ht1 ht2
   cases' h with n h
   exact h t1 t2 (set.mem_Inter.mp ht1 n) ht2
-#align probability_theory.indep_sets.Inter ProbabilityTheory.IndepSetsCat.interᵢ
+#align probability_theory.indep_sets.Inter ProbabilityTheory.IndepSetsCat.iInter
 
 theorem IndepSetsCat.bInter [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
     {μ : Measure Ω} {u : Set ι} (h : ∃ n ∈ u, IndepSetsCat (s n) s' μ) :
     IndepSetsCat (⋂ n ∈ u, s n) s' μ := by
   intro t1 t2 ht1 ht2
   rcases h with ⟨n, hn, h⟩
-  exact h t1 t2 (Set.binterᵢ_subset_of_mem hn ht1) ht2
+  exact h t1 t2 (Set.biInter_subset_of_mem hn ht1) ht2
 #align probability_theory.indep_sets.bInter ProbabilityTheory.IndepSetsCat.bInter
 
 theorem indepSetsCat_singleton_iff [MeasurableSpace Ω] {s t : Set Ω} {μ : Measure Ω} :
@@ -294,7 +294,7 @@ theorem IndepSets.indepSetsCat {s : ι → Set (Set Ω)} [MeasurableSpace Ω] {
     have h_inter :
       (⋂ (t : ι) (H : t ∈ ({i, j} : Finset ι)), ite (t = i) t₁ t₂) =
         ite (i = i) t₁ t₂ ∩ ite (j = i) t₁ t₂ :=
-      by simp only [Finset.set_binterᵢ_singleton, Finset.set_binterᵢ_insert]
+      by simp only [Finset.set_biInter_singleton, Finset.set_biInter_insert]
     have h_prod :
       (∏ t : ι in ({i, j} : Finset ι), μ (ite (t = i) t₁ t₂)) =
         μ (ite (i = i) t₁ t₂) * μ (ite (j = i) t₁ t₂) :=
@@ -404,9 +404,9 @@ theorem IndepSetsCat.indep' {m : MeasurableSpace Ω} {μ : Measure Ω} [Probabil
 
 variable {m0 : MeasurableSpace Ω} {μ : Measure Ω}
 
-theorem indepSetsCat_piUnionᵢInter_of_disjoint [ProbabilityMeasure μ] {s : ι → Set (Set Ω)}
+theorem indepSetsCat_piiUnionInter_of_disjoint [ProbabilityMeasure μ] {s : ι → Set (Set Ω)}
     {S T : Set ι} (h_indep : IndepSets s μ) (hST : Disjoint S T) :
-    IndepSetsCat (piUnionᵢInter s S) (piUnionᵢInter s T) μ :=
+    IndepSetsCat (piiUnionInter s S) (piiUnionInter s T) μ :=
   by
   rintro t1 t2 ⟨p1, hp1, f1, ht1_m, ht1_eq⟩ ⟨p2, hp2, f2, ht2_m, ht2_eq⟩
   classical
@@ -429,7 +429,7 @@ theorem indepSetsCat_piUnionᵢInter_of_disjoint [ProbabilityMeasure μ] {s : ι
           ⋂ i ∈ p1 ∪ p2, ite (i ∈ p1) (f1 i) Set.univ ∩ ite (i ∈ p2) (f2 i) Set.univ :=
         by
         ext1 x
-        simp only [Set.mem_ite_univ_right, Set.mem_inter_iff, Set.mem_interᵢ, Finset.mem_union]
+        simp only [Set.mem_ite_univ_right, Set.mem_inter_iff, Set.mem_iInter, Finset.mem_union]
         exact
           ⟨fun h i _ => ⟨h.1 i, h.2 i⟩, fun h =>
             ⟨fun i hi => (h i (Or.inl hi)).1 hi, fun i hi => (h i (Or.inr hi)).2 hi⟩⟩
@@ -445,38 +445,38 @@ theorem indepSetsCat_piUnionᵢInter_of_disjoint [ProbabilityMeasure μ] {s : ι
       Finset.prod_ite_mem (p1 ∪ p2) p1 fun x => μ (f1 x), Finset.union_inter_cancel_left,
       Finset.prod_ite_mem (p1 ∪ p2) p2 fun x => μ (f2 x), Finset.union_inter_cancel_right, ht1_eq, ←
       h_indep p1 ht1_m, ht2_eq, ← h_indep p2 ht2_m]
-#align probability_theory.indep_sets_pi_Union_Inter_of_disjoint ProbabilityTheory.indepSetsCat_piUnionᵢInter_of_disjoint
+#align probability_theory.indep_sets_pi_Union_Inter_of_disjoint ProbabilityTheory.indepSetsCat_piiUnionInter_of_disjoint
 
 theorem IndepSet.indepCat_generateFrom_of_disjoint [ProbabilityMeasure μ] {s : ι → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : IndepSet s μ) (S T : Set ι) (hST : Disjoint S T) :
     IndepCat (generateFrom { t | ∃ n ∈ S, s n = t }) (generateFrom { t | ∃ k ∈ T, s k = t }) μ :=
   by
-  rw [← generateFrom_piUnionᵢInter_singleton_left, ← generateFrom_piUnionᵢInter_singleton_left]
+  rw [← generateFrom_piiUnionInter_singleton_left, ← generateFrom_piiUnionInter_singleton_left]
   refine'
     indep_sets.indep'
-      (fun t ht => generateFrom_piUnionᵢInter_le _ _ _ _ (measurable_set_generate_from ht))
-      (fun t ht => generateFrom_piUnionᵢInter_le _ _ _ _ (measurable_set_generate_from ht)) _ _ _
+      (fun t ht => generateFrom_piiUnionInter_le _ _ _ _ (measurable_set_generate_from ht))
+      (fun t ht => generateFrom_piiUnionInter_le _ _ _ _ (measurable_set_generate_from ht)) _ _ _
   · exact fun k => generate_from_le fun t ht => (Set.mem_singleton_iff.1 ht).symm ▸ hsm k
   · exact fun k => generate_from_le fun t ht => (Set.mem_singleton_iff.1 ht).symm ▸ hsm k
-  · exact isPiSystem_piUnionᵢInter _ (fun k => IsPiSystem.singleton _) _
-  · exact isPiSystem_piUnionᵢInter _ (fun k => IsPiSystem.singleton _) _
+  · exact isPiSystem_piiUnionInter _ (fun k => IsPiSystem.singleton _) _
+  · exact isPiSystem_piiUnionInter _ (fun k => IsPiSystem.singleton _) _
   · classical exact indep_sets_pi_Union_Inter_of_disjoint (Indep.Indep_sets (fun n => rfl) hs) hST
 #align probability_theory.Indep_set.indep_generate_from_of_disjoint ProbabilityTheory.IndepSet.indepCat_generateFrom_of_disjoint
 
-theorem indepCat_supᵢ_of_disjoint [ProbabilityMeasure μ] {m : ι → MeasurableSpace Ω}
+theorem indepCat_iSup_of_disjoint [ProbabilityMeasure μ] {m : ι → MeasurableSpace Ω}
     (h_le : ∀ i, m i ≤ m0) (h_indep : Indep m μ) {S T : Set ι} (hST : Disjoint S T) :
     IndepCat (⨆ i ∈ S, m i) (⨆ i ∈ T, m i) μ :=
   by
   refine'
-    indep_sets.indep (supᵢ₂_le fun i _ => h_le i) (supᵢ₂_le fun i _ => h_le i) _ _
-      (generateFrom_piUnionᵢInter_measurableSet m S).symm
-      (generateFrom_piUnionᵢInter_measurableSet m T).symm _
-  · exact isPiSystem_piUnionᵢInter _ (fun n => @is_pi_system_measurable_set Ω (m n)) _
-  · exact isPiSystem_piUnionᵢInter _ (fun n => @is_pi_system_measurable_set Ω (m n)) _
+    indep_sets.indep (iSup₂_le fun i _ => h_le i) (iSup₂_le fun i _ => h_le i) _ _
+      (generateFrom_piiUnionInter_measurableSet m S).symm
+      (generateFrom_piiUnionInter_measurableSet m T).symm _
+  · exact isPiSystem_piiUnionInter _ (fun n => @is_pi_system_measurable_set Ω (m n)) _
+  · exact isPiSystem_piiUnionInter _ (fun n => @is_pi_system_measurable_set Ω (m n)) _
   · classical exact indep_sets_pi_Union_Inter_of_disjoint h_indep hST
-#align probability_theory.indep_supr_of_disjoint ProbabilityTheory.indepCat_supᵢ_of_disjoint
+#align probability_theory.indep_supr_of_disjoint ProbabilityTheory.indepCat_iSup_of_disjoint
 
-theorem indepCat_supᵢ_of_directed_le {Ω} {m : ι → MeasurableSpace Ω} {m' m0 : MeasurableSpace Ω}
+theorem indepCat_iSup_of_directed_le {Ω} {m : ι → MeasurableSpace Ω} {m' m0 : MeasurableSpace Ω}
     {μ : Measure Ω} [ProbabilityMeasure μ] (h_indep : ∀ i, IndepCat (m i) m' μ)
     (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0) (hm : Directed (· ≤ ·) m) : IndepCat (⨆ i, m i) m' μ :=
   by
@@ -484,7 +484,7 @@ theorem indepCat_supᵢ_of_directed_le {Ω} {m : ι → MeasurableSpace Ω} {m'
   have hp : ∀ n, IsPiSystem (p n) := fun n => @is_pi_system_measurable_set Ω (m n)
   have h_gen_n : ∀ n, m n = generate_from (p n) := fun n =>
     (@generate_from_measurable_set Ω (m n)).symm
-  have hp_supr_pi : IsPiSystem (⋃ n, p n) := isPiSystem_unionᵢ_of_directed_le p hp hm
+  have hp_supr_pi : IsPiSystem (⋃ n, p n) := isPiSystem_iUnion_of_directed_le p hp hm
   let p' := { t : Set Ω | measurable_set[m'] t }
   have hp'_pi : IsPiSystem p' := @is_pi_system_measurable_set Ω m'
   have h_gen' : m' = generate_from p' := (@generate_from_measurable_set Ω m').symm
@@ -495,9 +495,9 @@ theorem indepCat_supᵢ_of_directed_le {Ω} {m : ι → MeasurableSpace Ω} {m'
     simp_rw [h_gen_n, h_gen'] at h_indep
     exact fun n => (h_indep n).IndepSetsCat
   -- now go from π-systems to σ-algebras
-  refine' indep_sets.indep (supᵢ_le h_le) h_le' hp_supr_pi hp'_pi _ h_gen' h_pi_system_indep
+  refine' indep_sets.indep (iSup_le h_le) h_le' hp_supr_pi hp'_pi _ h_gen' h_pi_system_indep
   exact (generate_from_Union_measurable_set _).symm
-#align probability_theory.indep_supr_of_directed_le ProbabilityTheory.indepCat_supᵢ_of_directed_le
+#align probability_theory.indep_supr_of_directed_le ProbabilityTheory.indepCat_iSup_of_directed_le
 
 theorem IndepSet.indepCat_generateFrom_lt [Preorder ι] [ProbabilityMeasure μ] {s : ι → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : IndepSet s μ) (i : ι) :
@@ -523,22 +523,22 @@ theorem IndepSet.indepCat_generateFrom_le_nat [ProbabilityMeasure μ] {s : ℕ 
   hs.indepCat_generateFrom_le hsm _ n.lt_succ_self
 #align probability_theory.Indep_set.indep_generate_from_le_nat ProbabilityTheory.IndepSet.indepCat_generateFrom_le_nat
 
-theorem indepCat_supᵢ_of_monotone [SemilatticeSup ι] {Ω} {m : ι → MeasurableSpace Ω}
+theorem indepCat_iSup_of_monotone [SemilatticeSup ι] {Ω} {m : ι → MeasurableSpace Ω}
     {m' m0 : MeasurableSpace Ω} {μ : Measure Ω} [ProbabilityMeasure μ]
     (h_indep : ∀ i, IndepCat (m i) m' μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0)
     (hm : Monotone m) : IndepCat (⨆ i, m i) m' μ :=
-  indepCat_supᵢ_of_directed_le h_indep h_le h_le' (Monotone.directed_le hm)
-#align probability_theory.indep_supr_of_monotone ProbabilityTheory.indepCat_supᵢ_of_monotone
+  indepCat_iSup_of_directed_le h_indep h_le h_le' (Monotone.directed_le hm)
+#align probability_theory.indep_supr_of_monotone ProbabilityTheory.indepCat_iSup_of_monotone
 
-theorem indepCat_supᵢ_of_antitone [SemilatticeInf ι] {Ω} {m : ι → MeasurableSpace Ω}
+theorem indepCat_iSup_of_antitone [SemilatticeInf ι] {Ω} {m : ι → MeasurableSpace Ω}
     {m' m0 : MeasurableSpace Ω} {μ : Measure Ω} [ProbabilityMeasure μ]
     (h_indep : ∀ i, IndepCat (m i) m' μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0)
     (hm : Antitone m) : IndepCat (⨆ i, m i) m' μ :=
-  indepCat_supᵢ_of_directed_le h_indep h_le h_le' (directed_of_inf hm)
-#align probability_theory.indep_supr_of_antitone ProbabilityTheory.indepCat_supᵢ_of_antitone
+  indepCat_iSup_of_directed_le h_indep h_le h_le' (directed_of_inf hm)
+#align probability_theory.indep_supr_of_antitone ProbabilityTheory.indepCat_iSup_of_antitone
 
-theorem IndepSets.piUnionᵢInter_of_not_mem {π : ι → Set (Set Ω)} {a : ι} {S : Finset ι}
-    (hp_ind : IndepSets π μ) (haS : a ∉ S) : IndepSetsCat (piUnionᵢInter π S) (π a) μ :=
+theorem IndepSets.piiUnionInter_of_not_mem {π : ι → Set (Set Ω)} {a : ι} {S : Finset ι}
+    (hp_ind : IndepSets π μ) (haS : a ∉ S) : IndepSetsCat (piiUnionInter π S) (π a) μ :=
   by
   rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
   rw [Finset.coe_subset] at hs_mem
@@ -574,11 +574,11 @@ theorem IndepSets.piUnionᵢInter_of_not_mem {π : ι → Set (Set Ω)} {a : ι}
     have h_μ_inter : μ (t1 ∩ t2) = ∏ n in insert a s, μ (f n) :=
       by
       have h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n := by
-        rw [h_t1, h_t2, Finset.set_binterᵢ_insert, Set.inter_comm]
+        rw [h_t1, h_t2, Finset.set_biInter_insert, Set.inter_comm]
       rw [h_t1_inter_t2, ← hp_ind (insert a s) h_f_mem]
     have has : a ∉ s := fun has_mem => haS (hs_mem Membership)
     rw [h_μ_inter, Finset.prod_insert has, h_t2, mul_comm, h_μ_t1]
-#align probability_theory.Indep_sets.pi_Union_Inter_of_not_mem ProbabilityTheory.IndepSets.piUnionᵢInter_of_not_mem
+#align probability_theory.Indep_sets.pi_Union_Inter_of_not_mem ProbabilityTheory.IndepSets.piiUnionInter_of_not_mem
 
 /-- The measurable space structures generated by independent pi-systems are independent. -/
 theorem IndepSets.indep [ProbabilityMeasure μ] (m : ι → MeasurableSpace Ω) (h_le : ∀ i, m i ≤ m0)
@@ -587,18 +587,18 @@ theorem IndepSets.indep [ProbabilityMeasure μ] (m : ι → MeasurableSpace Ω)
   classical
     refine' Finset.induction _ _
     ·
-      simp only [measure_univ, imp_true_iff, Set.interᵢ_false, Set.interᵢ_univ, Finset.prod_empty,
+      simp only [measure_univ, imp_true_iff, Set.iInter_false, Set.iInter_univ, Finset.prod_empty,
         eq_self_iff_true]
     intro a S ha_notin_S h_rec f hf_m
     have hf_m_S : ∀ x ∈ S, measurable_set[m x] (f x) := fun x hx => hf_m x (by simp [hx])
-    rw [Finset.set_binterᵢ_insert, Finset.prod_insert ha_notin_S, ← h_rec hf_m_S]
-    let p := piUnionᵢInter π S
+    rw [Finset.set_biInter_insert, Finset.prod_insert ha_notin_S, ← h_rec hf_m_S]
+    let p := piiUnionInter π S
     set m_p := generate_from p with hS_eq_generate
     have h_indep : indep m_p (m a) μ :=
       by
-      have hp : IsPiSystem p := isPiSystem_piUnionᵢInter π h_pi S
+      have hp : IsPiSystem p := isPiSystem_piiUnionInter π h_pi S
       have h_le' : ∀ i, generate_from (π i) ≤ m0 := fun i => (h_generate i).symm.trans_le (h_le i)
-      have hm_p : m_p ≤ m0 := generateFrom_piUnionᵢInter_le π h_le' S
+      have hm_p : m_p ≤ m0 := generateFrom_piiUnionInter_le π h_le' S
       exact
         indep_sets.indep hm_p (h_le a) hp (h_pi a) hS_eq_generate (h_generate a)
           (h_ind.pi_Union_Inter_of_not_mem ha_notin_S)
@@ -606,7 +606,7 @@ theorem IndepSets.indep [ProbabilityMeasure μ] (m : ι → MeasurableSpace Ω)
     have h_le_p : ∀ i ∈ S, m i ≤ m_p := by
       intro n hn
       rw [hS_eq_generate, h_generate n]
-      exact le_generateFrom_piUnionᵢInter S hn
+      exact le_generateFrom_piiUnionInter S hn
     have h_S_f : ∀ i ∈ S, measurable_set[m_p] (f i) := fun i hi => (h_le_p i hi) (f i) (hf_m_S i hi)
     exact S.measurable_set_bInter h_S_f
 #align probability_theory.Indep_sets.Indep ProbabilityTheory.IndepSets.indep
@@ -656,12 +656,12 @@ theorem IndepCat.indepSetCat_of_measurableSet {m₁ m₂ m0 : MeasurableSpace Ω
     · simp only [hs, Set.mem_singleton_iff, Set.mem_setOf_eq, forall_eq]
     · exact @MeasurableSet.empty _ m₁
     · exact fun u hu => hu.compl
-    · exact fun f hf => MeasurableSet.unionᵢ hf
+    · exact fun f hf => MeasurableSet.iUnion hf
   · refine' generate_from_induction (fun u => measurable_set[m₂] u) {t} _ _ _ _ ht'
     · simp only [ht, Set.mem_singleton_iff, Set.mem_setOf_eq, forall_eq]
     · exact @MeasurableSet.empty _ m₂
     · exact fun u hu => hu.compl
-    · exact fun f hf => MeasurableSet.unionᵢ hf
+    · exact fun f hf => MeasurableSet.iUnion hf
 #align probability_theory.indep.indep_set_of_measurable_set ProbabilityTheory.IndepCat.indepSetCat_of_measurableSet
 
 theorem indepCat_iff_forall_indepSetCat (m₁ m₂ : MeasurableSpace Ω) {m0 : MeasurableSpace Ω}
@@ -719,7 +719,7 @@ theorem indepFun_iff_measure_inter_preimage_eq_mul {ι : Type _} {β : ι → Ty
     have h_left_eq : μ (⋂ i ∈ S, setsΩ i) = μ (⋂ i ∈ S, f i ⁻¹' setsβ i) :=
       by
       congr with (i x)
-      simp only [Set.mem_interᵢ]
+      simp only [Set.mem_iInter]
       constructor <;> intro h hi_mem <;> specialize h hi_mem
       · rwa [h_preim i hi_mem] at h
       · rwa [h_preim i hi_mem]
@@ -839,7 +839,7 @@ theorem IndepFun.indepFunCat_finset [ProbabilityMeasure μ] {ι : Type _} {β :
       (fun (ω : Ω) (i : ↥S) => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i :=
       by
       ext1 x
-      simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_interᵢ]
+      simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
       constructor <;> intro h
       · intro i hi
         rw [h_sets_s'_eq hi]
@@ -852,7 +852,7 @@ theorem IndepFun.indepFunCat_finset [ProbabilityMeasure μ] {ι : Type _} {β :
       (fun (ω : Ω) (i : ↥T) => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i :=
       by
       ext1 x
-      simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_interᵢ]
+      simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
       constructor <;> intro h
       · intro i hi
         simp_rw [sets_t', dif_pos hi]
@@ -868,7 +868,7 @@ theorem IndepFun.indepFunCat_finset [ProbabilityMeasure μ] {ι : Type _} {β :
         ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) :=
       by
       ext1 x
-      simp only [Set.mem_inter_iff, Set.mem_interᵢ, Set.mem_preimage, Finset.mem_union]
+      simp only [Set.mem_inter_iff, Set.mem_iInter, Set.mem_preimage, Finset.mem_union]
       constructor <;> intro h
       · intro i hi
         cases hi

mathlib commit https://github.com/leanprover-community/mathlib/commit/2f8347015b12b0864dfaf366ec4909eb70c78740

@@ -3,8 +3,8 @@ Copyright (c) 2021 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 
-! This file was ported from Lean 3 source module probability.independence
-! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a
+! This file was ported from Lean 3 source module probability.independence.basic
+! leanprover-community/mathlib commit 2f8347015b12b0864dfaf366ec4909eb70c78740
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -33,9 +33,6 @@ import Mathbin.MeasureTheory.Constructions.Pi
 * `Indep_sets.Indep`: if π-systems are independent as sets of sets, then the
   measurable space structures they generate are independent.
 * `indep_sets.indep`: variant with two π-systems.
-* `measure_zero_or_one_of_measurable_set_limsup_at_top`: Kolmogorov's 0-1 law. Any set which is
-  measurable with respect to the tail σ-algebra `limsup s at_top` of an independent sequence of
-  σ-algebras `s` has probability 0 or 1.
 
 ## Implementation notes
 
@@ -1003,189 +1000,5 @@ theorem IndepSet.indepFun_indicator [Zero β] [One β] {m : MeasurableSpace β}
 
 end IndepFun
 
-/-! ### Kolmogorov's 0-1 law
-
-Let `s : ι → measurable_space Ω` be an independent sequence of sub-σ-algebras. Then any set which
-is measurable with respect to the tail σ-algebra `limsup s at_top` has probability 0 or 1.
--/
-
-
-section ZeroOneLaw
-
-variable {m m0 : MeasurableSpace Ω} {μ : Measure Ω}
-
-theorem measure_eq_zero_or_one_or_top_of_indepSetCat_self {t : Set Ω}
-    (h_indep : IndepSetCat t t μ) : μ t = 0 ∨ μ t = 1 ∨ μ t = ∞ :=
-  by
-  specialize
-    h_indep t t (measurable_set_generate_from (Set.mem_singleton t))
-      (measurable_set_generate_from (Set.mem_singleton t))
-  by_cases h0 : μ t = 0
-  · exact Or.inl h0
-  by_cases h_top : μ t = ∞
-  · exact Or.inr (Or.inr h_top)
-  rw [← one_mul (μ (t ∩ t)), Set.inter_self, ENNReal.mul_eq_mul_right h0 h_top] at h_indep
-  exact Or.inr (Or.inl h_indep.symm)
-#align probability_theory.measure_eq_zero_or_one_or_top_of_indep_set_self ProbabilityTheory.measure_eq_zero_or_one_or_top_of_indepSetCat_self
-
-theorem measure_eq_zero_or_one_of_indepSetCat_self [FiniteMeasure μ] {t : Set Ω}
-    (h_indep : IndepSetCat t t μ) : μ t = 0 ∨ μ t = 1 :=
-  by
-  have h_0_1_top := measure_eq_zero_or_one_or_top_of_indep_set_self h_indep
-  simpa [measure_ne_top μ] using h_0_1_top
-#align probability_theory.measure_eq_zero_or_one_of_indep_set_self ProbabilityTheory.measure_eq_zero_or_one_of_indepSetCat_self
-
-variable [ProbabilityMeasure μ] {s : ι → MeasurableSpace Ω}
-
-open Filter
-
-theorem indepCat_bsupr_compl (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) (t : Set ι) :
-    IndepCat (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) μ :=
-  indepCat_supᵢ_of_disjoint h_le h_indep disjoint_compl_right
-#align probability_theory.indep_bsupr_compl ProbabilityTheory.indepCat_bsupr_compl
-
-section Abstract
-
-variable {α : Type _} {p : Set ι → Prop} {f : Filter ι} {ns : α → Set ι}
-
-/-! We prove a version of Kolmogorov's 0-1 law for the σ-algebra `limsup s f` where `f` is a filter
-for which we can define the following two functions:
-* `p : set ι → Prop` such that for a set `t`, `p t → tᶜ ∈ f`,
-* `ns : α → set ι` a directed sequence of sets which all verify `p` and such that
-  `⋃ a, ns a = set.univ`.
-
-For the example of `f = at_top`, we can take `p = bdd_above` and `ns : ι → set ι := λ i, set.Iic i`.
--/
-
-
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic filter.is_bounded_default -/
-theorem indepCat_bsupr_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
-    {t : Set ι} (ht : p t) : IndepCat (⨆ n ∈ t, s n) (limsup s f) μ :=
-  by
-  refine' indep_of_indep_of_le_right (indep_bsupr_compl h_le h_indep t) _
-  refine'
-    Limsup_le_of_le
-      (by
-        run_tac
-          is_bounded_default)
-      _
-  simp only [Set.mem_compl_iff, eventually_map]
-  exact eventually_of_mem (hf t ht) le_supᵢ₂
-#align probability_theory.indep_bsupr_limsup ProbabilityTheory.indepCat_bsupr_limsup
-
-theorem indepCat_supᵢ_directed_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ)
-    (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) :
-    IndepCat (⨆ a, ⨆ n ∈ ns a, s n) (limsup s f) μ :=
-  by
-  refine' indep_supr_of_directed_le _ _ _ _
-  · exact fun a => indep_bsupr_limsup h_le h_indep hf (hnsp a)
-  · exact fun a => supᵢ₂_le fun n hn => h_le n
-  · exact limsup_le_supr.trans (supᵢ_le h_le)
-  · intro a b
-    obtain ⟨c, hc⟩ := hns a b
-    refine' ⟨c, _, _⟩ <;> refine' supᵢ_mono fun n => supᵢ_mono' fun hn => ⟨_, le_rfl⟩
-    · exact hc.1 hn
-    · exact hc.2 hn
-#align probability_theory.indep_supr_directed_limsup ProbabilityTheory.indepCat_supᵢ_directed_limsup
-
-theorem indepCat_supᵢ_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
-    (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
-    IndepCat (⨆ n, s n) (limsup s f) μ :=
-  by
-  suffices (⨆ a, ⨆ n ∈ ns a, s n) = ⨆ n, s n by
-    rw [← this]
-    exact indep_supr_directed_limsup h_le h_indep hf hns hnsp
-  rw [supᵢ_comm]
-  refine' supᵢ_congr fun n => _
-  have : (⨆ (i : α) (H : n ∈ ns i), s n) = ⨆ h : ∃ i, n ∈ ns i, s n := by rw [supᵢ_exists]
-  haveI : Nonempty (∃ i : α, n ∈ ns i) := ⟨hns_univ n⟩
-  rw [this, supᵢ_const]
-#align probability_theory.indep_supr_limsup ProbabilityTheory.indepCat_supᵢ_limsup
-
-theorem indepCat_limsup_self (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
-    (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
-    IndepCat (limsup s f) (limsup s f) μ :=
-  indepCat_of_indepCat_of_le_left (indepCat_supᵢ_limsup h_le h_indep hf hns hnsp hns_univ)
-    limsup_le_supᵢ
-#align probability_theory.indep_limsup_self ProbabilityTheory.indepCat_limsup_self
-
-theorem measure_zero_or_one_of_measurableSet_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ)
-    (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a))
-    (hns_univ : ∀ n, ∃ a, n ∈ ns a) {t : Set Ω} (ht_tail : measurable_set[limsup s f] t) :
-    μ t = 0 ∨ μ t = 1 :=
-  measure_eq_zero_or_one_of_indepSetCat_self
-    ((indepCat_limsup_self h_le h_indep hf hns hnsp hns_univ).indepSetCat_of_measurableSet ht_tail
-      ht_tail)
-#align probability_theory.measure_zero_or_one_of_measurable_set_limsup ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup
-
-end Abstract
-
-section AtTop
-
-variable [SemilatticeSup ι] [NoMaxOrder ι] [Nonempty ι]
-
-theorem indepCat_limsup_atTop_self (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) :
-    IndepCat (limsup s atTop) (limsup s atTop) μ :=
-  by
-  let ns : ι → Set ι := Set.Iic
-  have hnsp : ∀ i, BddAbove (ns i) := fun i => bddAbove_Iic
-  refine' indep_limsup_self h_le h_indep _ _ hnsp _
-  · simp only [mem_at_top_sets, ge_iff_le, Set.mem_compl_iff, BddAbove, upperBounds, Set.Nonempty]
-    rintro t ⟨a, ha⟩
-    obtain ⟨b, hb⟩ : ∃ b, a < b := exists_gt a
-    refine' ⟨b, fun c hc hct => _⟩
-    suffices : ∀ i ∈ t, i < c
-    exact lt_irrefl c (this c hct)
-    exact fun i hi => (ha hi).trans_lt (hb.trans_le hc)
-  · exact Monotone.directed_le fun i j hij k hki => le_trans hki hij
-  · exact fun n => ⟨n, le_rfl⟩
-#align probability_theory.indep_limsup_at_top_self ProbabilityTheory.indepCat_limsup_atTop_self
-
-/-- **Kolmogorov's 0-1 law** : any event in the tail σ-algebra of an independent sequence of
-sub-σ-algebras has probability 0 or 1.
-The tail σ-algebra `limsup s at_top` is the same as `⋂ n, ⋃ i ≥ n, s i`. -/
-theorem measure_zero_or_one_of_measurableSet_limsup_atTop (h_le : ∀ n, s n ≤ m0)
-    (h_indep : Indep s μ) {t : Set Ω} (ht_tail : measurable_set[limsup s atTop] t) :
-    μ t = 0 ∨ μ t = 1 :=
-  measure_eq_zero_or_one_of_indepSetCat_self
-    ((indepCat_limsup_atTop_self h_le h_indep).indepSetCat_of_measurableSet ht_tail ht_tail)
-#align probability_theory.measure_zero_or_one_of_measurable_set_limsup_at_top ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup_atTop
-
-end AtTop
-
-section AtBot
-
-variable [SemilatticeInf ι] [NoMinOrder ι] [Nonempty ι]
-
-theorem indepCat_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) :
-    IndepCat (limsup s atBot) (limsup s atBot) μ :=
-  by
-  let ns : ι → Set ι := Set.Ici
-  have hnsp : ∀ i, BddBelow (ns i) := fun i => bddBelow_Ici
-  refine' indep_limsup_self h_le h_indep _ _ hnsp _
-  · simp only [mem_at_bot_sets, ge_iff_le, Set.mem_compl_iff, BddBelow, lowerBounds, Set.Nonempty]
-    rintro t ⟨a, ha⟩
-    obtain ⟨b, hb⟩ : ∃ b, b < a := exists_lt a
-    refine' ⟨b, fun c hc hct => _⟩
-    suffices : ∀ i ∈ t, c < i
-    exact lt_irrefl c (this c hct)
-    exact fun i hi => hc.trans_lt (hb.trans_le (ha hi))
-  · exact directed_of_inf fun i j hij k hki => hij.trans hki
-  · exact fun n => ⟨n, le_rfl⟩
-#align probability_theory.indep_limsup_at_bot_self ProbabilityTheory.indepCat_limsup_atBot_self
-
-/-- **Kolmogorov's 0-1 law** : any event in the tail σ-algebra of an independent sequence of
-sub-σ-algebras has probability 0 or 1. -/
-theorem measure_zero_or_one_of_measurableSet_limsup_atBot (h_le : ∀ n, s n ≤ m0)
-    (h_indep : Indep s μ) {t : Set Ω} (ht_tail : measurable_set[limsup s atBot] t) :
-    μ t = 0 ∨ μ t = 1 :=
-  measure_eq_zero_or_one_of_indepSetCat_self
-    ((indepCat_limsup_atBot_self h_le h_indep).indepSetCat_of_measurableSet ht_tail ht_tail)
-#align probability_theory.measure_zero_or_one_of_measurable_set_limsup_at_bot ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup_atBot
-
-end AtBot
-
-end ZeroOneLaw
-
 end ProbabilityTheory
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/2f8347015b12b0864dfaf366ec4909eb70c78740

@@ -3,8 +3,8 @@ Copyright (c) 2021 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 
-! This file was ported from Lean 3 source module probability.independence
-! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a
+! This file was ported from Lean 3 source module probability.independence.basic
+! leanprover-community/mathlib commit 2f8347015b12b0864dfaf366ec4909eb70c78740
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -33,9 +33,6 @@ import Mathbin.MeasureTheory.Constructions.Pi
 * `Indep_sets.Indep`: if π-systems are independent as sets of sets, then the
   measurable space structures they generate are independent.
 * `indep_sets.indep`: variant with two π-systems.
-* `measure_zero_or_one_of_measurable_set_limsup_at_top`: Kolmogorov's 0-1 law. Any set which is
-  measurable with respect to the tail σ-algebra `limsup s at_top` of an independent sequence of
-  σ-algebras `s` has probability 0 or 1.
 
 ## Implementation notes
 
@@ -1003,189 +1000,5 @@ theorem IndepSet.indepFun_indicator [Zero β] [One β] {m : MeasurableSpace β}
 
 end IndepFun
 
-/-! ### Kolmogorov's 0-1 law
-
-Let `s : ι → measurable_space Ω` be an independent sequence of sub-σ-algebras. Then any set which
-is measurable with respect to the tail σ-algebra `limsup s at_top` has probability 0 or 1.
--/
-
-
-section ZeroOneLaw
-
-variable {m m0 : MeasurableSpace Ω} {μ : Measure Ω}
-
-theorem measure_eq_zero_or_one_or_top_of_indepSetCat_self {t : Set Ω}
-    (h_indep : IndepSetCat t t μ) : μ t = 0 ∨ μ t = 1 ∨ μ t = ∞ :=
-  by
-  specialize
-    h_indep t t (measurable_set_generate_from (Set.mem_singleton t))
-      (measurable_set_generate_from (Set.mem_singleton t))
-  by_cases h0 : μ t = 0
-  · exact Or.inl h0
-  by_cases h_top : μ t = ∞
-  · exact Or.inr (Or.inr h_top)
-  rw [← one_mul (μ (t ∩ t)), Set.inter_self, ENNReal.mul_eq_mul_right h0 h_top] at h_indep
-  exact Or.inr (Or.inl h_indep.symm)
-#align probability_theory.measure_eq_zero_or_one_or_top_of_indep_set_self ProbabilityTheory.measure_eq_zero_or_one_or_top_of_indepSetCat_self
-
-theorem measure_eq_zero_or_one_of_indepSetCat_self [FiniteMeasure μ] {t : Set Ω}
-    (h_indep : IndepSetCat t t μ) : μ t = 0 ∨ μ t = 1 :=
-  by
-  have h_0_1_top := measure_eq_zero_or_one_or_top_of_indep_set_self h_indep
-  simpa [measure_ne_top μ] using h_0_1_top
-#align probability_theory.measure_eq_zero_or_one_of_indep_set_self ProbabilityTheory.measure_eq_zero_or_one_of_indepSetCat_self
-
-variable [ProbabilityMeasure μ] {s : ι → MeasurableSpace Ω}
-
-open Filter
-
-theorem indepCat_bsupr_compl (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) (t : Set ι) :
-    IndepCat (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) μ :=
-  indepCat_supᵢ_of_disjoint h_le h_indep disjoint_compl_right
-#align probability_theory.indep_bsupr_compl ProbabilityTheory.indepCat_bsupr_compl
-
-section Abstract
-
-variable {α : Type _} {p : Set ι → Prop} {f : Filter ι} {ns : α → Set ι}
-
-/-! We prove a version of Kolmogorov's 0-1 law for the σ-algebra `limsup s f` where `f` is a filter
-for which we can define the following two functions:
-* `p : set ι → Prop` such that for a set `t`, `p t → tᶜ ∈ f`,
-* `ns : α → set ι` a directed sequence of sets which all verify `p` and such that
-  `⋃ a, ns a = set.univ`.
-
-For the example of `f = at_top`, we can take `p = bdd_above` and `ns : ι → set ι := λ i, set.Iic i`.
--/
-
-
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic filter.is_bounded_default -/
-theorem indepCat_bsupr_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
-    {t : Set ι} (ht : p t) : IndepCat (⨆ n ∈ t, s n) (limsup s f) μ :=
-  by
-  refine' indep_of_indep_of_le_right (indep_bsupr_compl h_le h_indep t) _
-  refine'
-    Limsup_le_of_le
-      (by
-        run_tac
-          is_bounded_default)
-      _
-  simp only [Set.mem_compl_iff, eventually_map]
-  exact eventually_of_mem (hf t ht) le_supᵢ₂
-#align probability_theory.indep_bsupr_limsup ProbabilityTheory.indepCat_bsupr_limsup
-
-theorem indepCat_supᵢ_directed_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ)
-    (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) :
-    IndepCat (⨆ a, ⨆ n ∈ ns a, s n) (limsup s f) μ :=
-  by
-  refine' indep_supr_of_directed_le _ _ _ _
-  · exact fun a => indep_bsupr_limsup h_le h_indep hf (hnsp a)
-  · exact fun a => supᵢ₂_le fun n hn => h_le n
-  · exact limsup_le_supr.trans (supᵢ_le h_le)
-  · intro a b
-    obtain ⟨c, hc⟩ := hns a b
-    refine' ⟨c, _, _⟩ <;> refine' supᵢ_mono fun n => supᵢ_mono' fun hn => ⟨_, le_rfl⟩
-    · exact hc.1 hn
-    · exact hc.2 hn
-#align probability_theory.indep_supr_directed_limsup ProbabilityTheory.indepCat_supᵢ_directed_limsup
-
-theorem indepCat_supᵢ_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
-    (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
-    IndepCat (⨆ n, s n) (limsup s f) μ :=
-  by
-  suffices (⨆ a, ⨆ n ∈ ns a, s n) = ⨆ n, s n by
-    rw [← this]
-    exact indep_supr_directed_limsup h_le h_indep hf hns hnsp
-  rw [supᵢ_comm]
-  refine' supᵢ_congr fun n => _
-  have : (⨆ (i : α) (H : n ∈ ns i), s n) = ⨆ h : ∃ i, n ∈ ns i, s n := by rw [supᵢ_exists]
-  haveI : Nonempty (∃ i : α, n ∈ ns i) := ⟨hns_univ n⟩
-  rw [this, supᵢ_const]
-#align probability_theory.indep_supr_limsup ProbabilityTheory.indepCat_supᵢ_limsup
-
-theorem indepCat_limsup_self (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
-    (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
-    IndepCat (limsup s f) (limsup s f) μ :=
-  indepCat_of_indepCat_of_le_left (indepCat_supᵢ_limsup h_le h_indep hf hns hnsp hns_univ)
-    limsup_le_supᵢ
-#align probability_theory.indep_limsup_self ProbabilityTheory.indepCat_limsup_self
-
-theorem measure_zero_or_one_of_measurableSet_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ)
-    (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a))
-    (hns_univ : ∀ n, ∃ a, n ∈ ns a) {t : Set Ω} (ht_tail : measurable_set[limsup s f] t) :
-    μ t = 0 ∨ μ t = 1 :=
-  measure_eq_zero_or_one_of_indepSetCat_self
-    ((indepCat_limsup_self h_le h_indep hf hns hnsp hns_univ).indepSetCat_of_measurableSet ht_tail
-      ht_tail)
-#align probability_theory.measure_zero_or_one_of_measurable_set_limsup ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup
-
-end Abstract
-
-section AtTop
-
-variable [SemilatticeSup ι] [NoMaxOrder ι] [Nonempty ι]
-
-theorem indepCat_limsup_atTop_self (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) :
-    IndepCat (limsup s atTop) (limsup s atTop) μ :=
-  by
-  let ns : ι → Set ι := Set.Iic
-  have hnsp : ∀ i, BddAbove (ns i) := fun i => bddAbove_Iic
-  refine' indep_limsup_self h_le h_indep _ _ hnsp _
-  · simp only [mem_at_top_sets, ge_iff_le, Set.mem_compl_iff, BddAbove, upperBounds, Set.Nonempty]
-    rintro t ⟨a, ha⟩
-    obtain ⟨b, hb⟩ : ∃ b, a < b := exists_gt a
-    refine' ⟨b, fun c hc hct => _⟩
-    suffices : ∀ i ∈ t, i < c
-    exact lt_irrefl c (this c hct)
-    exact fun i hi => (ha hi).trans_lt (hb.trans_le hc)
-  · exact Monotone.directed_le fun i j hij k hki => le_trans hki hij
-  · exact fun n => ⟨n, le_rfl⟩
-#align probability_theory.indep_limsup_at_top_self ProbabilityTheory.indepCat_limsup_atTop_self
-
-/-- **Kolmogorov's 0-1 law** : any event in the tail σ-algebra of an independent sequence of
-sub-σ-algebras has probability 0 or 1.
-The tail σ-algebra `limsup s at_top` is the same as `⋂ n, ⋃ i ≥ n, s i`. -/
-theorem measure_zero_or_one_of_measurableSet_limsup_atTop (h_le : ∀ n, s n ≤ m0)
-    (h_indep : Indep s μ) {t : Set Ω} (ht_tail : measurable_set[limsup s atTop] t) :
-    μ t = 0 ∨ μ t = 1 :=
-  measure_eq_zero_or_one_of_indepSetCat_self
-    ((indepCat_limsup_atTop_self h_le h_indep).indepSetCat_of_measurableSet ht_tail ht_tail)
-#align probability_theory.measure_zero_or_one_of_measurable_set_limsup_at_top ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup_atTop
-
-end AtTop
-
-section AtBot
-
-variable [SemilatticeInf ι] [NoMinOrder ι] [Nonempty ι]
-
-theorem indepCat_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) :
-    IndepCat (limsup s atBot) (limsup s atBot) μ :=
-  by
-  let ns : ι → Set ι := Set.Ici
-  have hnsp : ∀ i, BddBelow (ns i) := fun i => bddBelow_Ici
-  refine' indep_limsup_self h_le h_indep _ _ hnsp _
-  · simp only [mem_at_bot_sets, ge_iff_le, Set.mem_compl_iff, BddBelow, lowerBounds, Set.Nonempty]
-    rintro t ⟨a, ha⟩
-    obtain ⟨b, hb⟩ : ∃ b, b < a := exists_lt a
-    refine' ⟨b, fun c hc hct => _⟩
-    suffices : ∀ i ∈ t, c < i
-    exact lt_irrefl c (this c hct)
-    exact fun i hi => hc.trans_lt (hb.trans_le (ha hi))
-  · exact directed_of_inf fun i j hij k hki => hij.trans hki
-  · exact fun n => ⟨n, le_rfl⟩
-#align probability_theory.indep_limsup_at_bot_self ProbabilityTheory.indepCat_limsup_atBot_self
-
-/-- **Kolmogorov's 0-1 law** : any event in the tail σ-algebra of an independent sequence of
-sub-σ-algebras has probability 0 or 1. -/
-theorem measure_zero_or_one_of_measurableSet_limsup_atBot (h_le : ∀ n, s n ≤ m0)
-    (h_indep : Indep s μ) {t : Set Ω} (ht_tail : measurable_set[limsup s atBot] t) :
-    μ t = 0 ∨ μ t = 1 :=
-  measure_eq_zero_or_one_of_indepSetCat_self
-    ((indepCat_limsup_atBot_self h_le h_indep).indepSetCat_of_measurableSet ht_tail ht_tail)
-#align probability_theory.measure_zero_or_one_of_measurable_set_limsup_at_bot ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup_atBot
-
-end AtBot
-
-end ZeroOneLaw
-
 end ProbabilityTheory
 

See Zulip thread for the discussion.

@@ -704,7 +704,7 @@ lemma iIndepFun.indepFun_prod_mk_prod_mk (h_indep : iIndepFun m f μ) (hf : ∀
     IndepFun (fun a ↦ (f i a, f j a)) (fun a ↦ (f k a, f l a)) μ := by
   classical
   let g (i j : ι) (v : Π x : ({i, j} : Finset ι), β x) : β i × β j :=
-    ⟨v ⟨i, mem_insert_self _ _⟩, v ⟨j, mem_insert_of_mem $ mem_singleton_self _⟩⟩
+    ⟨v ⟨i, mem_insert_self _ _⟩, v ⟨j, mem_insert_of_mem <| mem_singleton_self _⟩⟩
   have hg (i j : ι) : Measurable (g i j) := by measurability
   exact (h_indep.indepFun_finset {i, j} {k, l} (by aesop) hf).comp (hg i j) (hg k l)
 

Co-authored-by: L Lllvvuu <git@llllvvuu.dev>

@@ -650,11 +650,12 @@ theorem indepFun_iff_map_prod_eq_prod_map_map {mβ : MeasurableSpace β} {mβ' :
     rw [(h₀ hs ht).1, (h₀ hs ht).2, h, Measure.prod_prod]
 
 @[symm]
-nonrec theorem IndepFun.symm {_ : MeasurableSpace β} {f g : Ω → β} (hfg : IndepFun f g μ) :
-    IndepFun g f μ := hfg.symm
+nonrec theorem IndepFun.symm {_ : MeasurableSpace β} {_ : MeasurableSpace β'}
+    (hfg : IndepFun f g μ) : IndepFun g f μ := hfg.symm
 #align probability_theory.indep_fun.symm ProbabilityTheory.IndepFun.symm
 
-theorem IndepFun.ae_eq {mβ : MeasurableSpace β} {f g f' g' : Ω → β} (hfg : IndepFun f g μ)
+theorem IndepFun.ae_eq {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
+    {f' : Ω → β} {g' : Ω → β'} (hfg : IndepFun f g μ)
     (hf : f =ᵐ[μ] f') (hg : g =ᵐ[μ] g') : IndepFun f' g' μ := by
   refine kernel.IndepFun.ae_eq hfg ?_ ?_ <;>
     simp only [ae_dirac_eq, Filter.eventually_pure, kernel.const_apply]

Also prove that a subsingleton family is always independent and that an independent family implies the measure is a probability measure.

This latter result means we can drop IsProbabilityMeasure μ assumptions from many theorems.

From PFR

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>

@@ -668,42 +668,113 @@ theorem IndepFun.comp {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'}
   kernel.IndepFun.comp hfg hφ hψ
 #align probability_theory.indep_fun.comp ProbabilityTheory.IndepFun.comp
 
+section iIndepFun
+variable {β : ι → Type*} {m : ∀ i, MeasurableSpace (β i)} {f : ∀ i, Ω → β i}
+
+@[nontriviality]
+lemma iIndepFun.of_subsingleton [IsProbabilityMeasure μ] [Subsingleton ι] : iIndepFun m f μ :=
+  kernel.iIndepFun.of_subsingleton
+
+lemma iIndepFun.isProbabilityMeasure (h : iIndepFun m f μ) : IsProbabilityMeasure μ :=
+  ⟨by simpa using h.meas_biInter (S := ∅) (s := fun _ ↦ univ)⟩
+
 /-- If `f` is a family of mutually independent random variables (`iIndepFun m f μ`) and `S, T` are
 two disjoint finite index sets, then the tuple formed by `f i` for `i ∈ S` is independent of the
 tuple `(f i)_i` for `i ∈ T`. -/
-theorem iIndepFun.indepFun_finset [IsProbabilityMeasure μ] {ι : Type*} {β : ι → Type*}
-    {m : ∀ i, MeasurableSpace (β i)} {f : ∀ i, Ω → β i} (S T : Finset ι) (hST : Disjoint S T)
-    (hf_Indep : iIndepFun m f μ) (hf_meas : ∀ i, Measurable (f i)) :
-    IndepFun (fun a (i : S) => f i a) (fun a (i : T) => f i a) μ :=
+lemma iIndepFun.indepFun_finset (S T : Finset ι) (hST : Disjoint S T) (hf_Indep : iIndepFun m f μ)
+    (hf_meas : ∀ i, Measurable (f i)) :
+    IndepFun (fun a (i : S) ↦ f i a) (fun a (i : T) ↦ f i a) μ :=
+  have := hf_Indep.isProbabilityMeasure
   kernel.iIndepFun.indepFun_finset S T hST hf_Indep hf_meas
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun.indep_fun_finset ProbabilityTheory.iIndepFun.indepFun_finset
 
-theorem iIndepFun.indepFun_prod [IsProbabilityMeasure μ] {ι : Type*} {β : ι → Type*}
-    {m : ∀ i, MeasurableSpace (β i)} {f : ∀ i, Ω → β i} (hf_Indep : iIndepFun m f μ)
-    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
+lemma iIndepFun.indepFun_prod_mk (hf_Indep : iIndepFun m f μ) (hf_meas : ∀ i, Measurable (f i))
+    (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
     IndepFun (fun a => (f i a, f j a)) (f k) μ :=
-  kernel.iIndepFun.indepFun_prod hf_Indep hf_meas i j k hik hjk
+  have := hf_Indep.isProbabilityMeasure
+  kernel.iIndepFun.indepFun_prod_mk hf_Indep hf_meas i j k hik hjk
 set_option linter.uppercaseLean3 false in
-#align probability_theory.Indep_fun.indep_fun_prod ProbabilityTheory.iIndepFun.indepFun_prod
+#align probability_theory.Indep_fun.indep_fun_prod ProbabilityTheory.iIndepFun.indepFun_prod_mk
+
+open Finset in
+lemma iIndepFun.indepFun_prod_mk_prod_mk (h_indep : iIndepFun m f μ) (hf : ∀ i, Measurable (f i))
+    (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l) :
+    IndepFun (fun a ↦ (f i a, f j a)) (fun a ↦ (f k a, f l a)) μ := by
+  classical
+  let g (i j : ι) (v : Π x : ({i, j} : Finset ι), β x) : β i × β j :=
+    ⟨v ⟨i, mem_insert_self _ _⟩, v ⟨j, mem_insert_of_mem $ mem_singleton_self _⟩⟩
+  have hg (i j : ι) : Measurable (g i j) := by measurability
+  exact (h_indep.indepFun_finset {i, j} {k, l} (by aesop) hf).comp (hg i j) (hg k l)
+
+end iIndepFun
+
+section Mul
+variable {β : Type*} {m : MeasurableSpace β} [Mul β] [MeasurableMul₂ β] {f : ι → Ω → β}
 
 @[to_additive]
-theorem iIndepFun.mul [IsProbabilityMeasure μ] {ι : Type*} {β : Type*} {m : MeasurableSpace β}
-    [Mul β] [MeasurableMul₂ β] {f : ι → Ω → β} (hf_Indep : iIndepFun (fun _ => m) f μ)
+lemma iIndepFun.indepFun_mul_left (hf_indep : iIndepFun (fun _ ↦ m) f μ)
     (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
     IndepFun (f i * f j) (f k) μ :=
-  kernel.iIndepFun.mul hf_Indep hf_meas i j k hik hjk
+  have := hf_indep.isProbabilityMeasure
+  kernel.iIndepFun.indepFun_mul_left hf_indep hf_meas i j k hik hjk
 set_option linter.uppercaseLean3 false in
-#align probability_theory.Indep_fun.mul ProbabilityTheory.iIndepFun.mul
+#align probability_theory.Indep_fun.mul ProbabilityTheory.iIndepFun.indepFun_mul_left
 set_option linter.uppercaseLean3 false in
-#align probability_theory.Indep_fun.add ProbabilityTheory.iIndepFun.add
+#align probability_theory.Indep_fun.add ProbabilityTheory.iIndepFun.indepFun_add_left
+
+@[to_additive]
+lemma iIndepFun.indepFun_mul_right (hf_indep : iIndepFun (fun _ ↦ m) f μ)
+    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hij : i ≠ j) (hik : i ≠ k) :
+    IndepFun (f i) (f j * f k) μ :=
+  have := hf_indep.isProbabilityMeasure
+  kernel.iIndepFun.indepFun_mul_right hf_indep hf_meas i j k hij hik
 
 @[to_additive]
-theorem iIndepFun.indepFun_finset_prod_of_not_mem [IsProbabilityMeasure μ] {ι : Type*} {β : Type*}
-    {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂ β] {f : ι → Ω → β}
-    (hf_Indep : iIndepFun (fun _ => m) f μ) (hf_meas : ∀ i, Measurable (f i)) {s : Finset ι} {i : ι}
-    (hi : i ∉ s) :
+lemma iIndepFun.indepFun_mul_mul (hf_indep : iIndepFun (fun _ ↦ m) f μ)
+    (hf_meas : ∀ i, Measurable (f i))
+    (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l) :
+    IndepFun (f i * f j) (f k * f l) μ :=
+  have := hf_indep.isProbabilityMeasure
+  kernel.iIndepFun.indepFun_mul_mul hf_indep hf_meas i j k l hik hil hjk hjl
+
+end Mul
+
+section Div
+variable {β : Type*} {m : MeasurableSpace β} [Div β] [MeasurableDiv₂ β] {f : ι → Ω → β}
+
+@[to_additive]
+lemma iIndepFun.indepFun_div_left (hf_indep : iIndepFun (fun _ ↦ m) f μ)
+    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
+    IndepFun (f i / f j) (f k) μ :=
+  have := hf_indep.isProbabilityMeasure
+  kernel.iIndepFun.indepFun_div_left hf_indep hf_meas i j k hik hjk
+
+@[to_additive]
+lemma iIndepFun.indepFun_div_right (hf_indep : iIndepFun (fun _ ↦ m) f μ)
+    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hij : i ≠ j) (hik : i ≠ k) :
+    IndepFun (f i) (f j / f k) μ :=
+  have := hf_indep.isProbabilityMeasure
+  kernel.iIndepFun.indepFun_div_right hf_indep hf_meas i j k hij hik
+
+@[to_additive]
+lemma iIndepFun.indepFun_div_div (hf_indep : iIndepFun (fun _ ↦ m) f μ)
+    (hf_meas : ∀ i, Measurable (f i))
+    (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l) :
+    IndepFun (f i / f j) (f k / f l) μ :=
+  have := hf_indep.isProbabilityMeasure
+  kernel.iIndepFun.indepFun_div_div hf_indep hf_meas i j k l hik hil hjk hjl
+
+end Div
+
+section CommMonoid
+variable {β : Type*} {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂ β] {f : ι → Ω → β}
+
+@[to_additive]
+lemma iIndepFun.indepFun_finset_prod_of_not_mem (hf_Indep : iIndepFun (fun _ ↦ m) f μ)
+    (hf_meas : ∀ i, Measurable (f i)) {s : Finset ι} {i : ι} (hi : i ∉ s) :
     IndepFun (∏ j in s, f j) (f i) μ :=
+  have := hf_Indep.isProbabilityMeasure
   kernel.iIndepFun.indepFun_finset_prod_of_not_mem hf_Indep hf_meas hi
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun.indep_fun_finset_prod_of_not_mem ProbabilityTheory.iIndepFun.indepFun_finset_prod_of_not_mem
@@ -711,16 +782,17 @@ set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun.indep_fun_finset_sum_of_not_mem ProbabilityTheory.iIndepFun.indepFun_finset_sum_of_not_mem
 
 @[to_additive]
-theorem iIndepFun.indepFun_prod_range_succ [IsProbabilityMeasure μ] {β : Type*}
-    {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂ β] {f : ℕ → Ω → β}
-    (hf_Indep : iIndepFun (fun _ => m) f μ) (hf_meas : ∀ i, Measurable (f i)) (n : ℕ) :
-    IndepFun (∏ j in Finset.range n, f j) (f n) μ :=
+lemma iIndepFun.indepFun_prod_range_succ {f : ℕ → Ω → β} (hf_Indep : iIndepFun (fun _ ↦ m) f μ)
+    (hf_meas : ∀ i, Measurable (f i)) (n : ℕ) : IndepFun (∏ j in Finset.range n, f j) (f n) μ :=
+  have := hf_Indep.isProbabilityMeasure
   kernel.iIndepFun.indepFun_prod_range_succ hf_Indep hf_meas n
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun.indep_fun_prod_range_succ ProbabilityTheory.iIndepFun.indepFun_prod_range_succ
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun.indep_fun_sum_range_succ ProbabilityTheory.iIndepFun.indepFun_sum_range_succ
 
+end CommMonoid
+
 theorem iIndepSet.iIndepFun_indicator [Zero β] [One β] {m : MeasurableSpace β} {s : ι → Set Ω}
     (hs : iIndepSet s μ) :
     iIndepFun (fun _n => m) (fun n => (s n).indicator fun _ω => 1) μ :=

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -156,7 +156,7 @@ lemma iIndepSets.meas_biInter (h : iIndepSets π μ) (s : Finset ι) {f : ι →
   (iIndepSets_iff _ _).1 h s hf
 
 lemma iIndepSets.meas_iInter [Fintype ι] (h : iIndepSets π μ) (hs : ∀ i, s i ∈ π i) :
-    μ (⋂ i, s i) = ∏ i, μ (s i) := by simp [←h.meas_biInter _ fun _i _ ↦ hs _]
+    μ (⋂ i, s i) = ∏ i, μ (s i) := by simp [← h.meas_biInter _ fun _i _ ↦ hs _]
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_sets.meas_Inter ProbabilityTheory.iIndepSets.meas_iInter
 
@@ -180,7 +180,7 @@ lemma iIndep.meas_biInter (hμ : iIndep m μ) (hs : ∀ i, i ∈ S → Measurabl
     μ (⋂ i ∈ S, s i) = ∏ i in S, μ (s i) := (iIndep_iff _ _).1 hμ _ hs
 
 lemma iIndep.meas_iInter [Fintype ι] (hμ : iIndep m μ) (hs : ∀ i, MeasurableSet[m i] (s i)) :
-    μ (⋂ i, s i) = ∏ i, μ (s i) := by simp [←hμ.meas_biInter fun _ _ ↦ hs _]
+    μ (⋂ i, s i) = ∏ i, μ (s i) := by simp [← hμ.meas_biInter fun _ _ ↦ hs _]
 
 lemma Indep_iff_IndepSets (m₁ m₂ : MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Measure Ω) :
     Indep m₁ m₂ μ ↔ IndepSets {s | MeasurableSet[m₁] s} {s | MeasurableSet[m₂] s} μ := by

Defines pdf in terms of rnDeriv.

Main definition change:

/-- A random variable `X : Ω → E` is said to `HasPDF` with respect to the measure `ℙ` on `Ω` and
`μ` on `E` if the push-forward measure of `ℙ` along `X` is absolutely continuous with respect to
`μ` and they `HaveLebesgueDecomposition`. -/
class HasPDF {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
    (μ : Measure E := by volume_tac) : Prop where
  pdf' : Measurable X ∧ HaveLebesgueDecomposition (map X ℙ) μ ∧ map X ℙ ≪ μ

/-- If `X` is a random variable that `HasPDF X ℙ μ`, then `pdf X` is the Radon–Nikodym
derivative of the push-forward measure of `ℙ` along `X` with respect to `μ`. -/
def pdf {_ : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω) (μ : Measure E := by volume_tac) :
    E → ℝ≥0∞ :=
  if HasPDF X ℙ μ then (map X ℙ).rnDeriv μ else 0

The law of the unconscious statistician is first generalized to rnDeriv on a general Banach space (∫ x, (μ.rnDeriv ν x).toReal • f x ∂ν = ∫ x, f x ∂μ), and then proven for PDFs.

Zulip thread

@@ -635,14 +635,14 @@ theorem indepFun_iff_indepSet_preimage {mβ : MeasurableSpace β} {mβ' : Measur
 #align probability_theory.indep_fun_iff_indep_set_preimage ProbabilityTheory.indepFun_iff_indepSet_preimage
 
 theorem indepFun_iff_map_prod_eq_prod_map_map {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
-    [IsFiniteMeasure μ] (hf : Measurable f) (hg : Measurable g) :
+    [IsFiniteMeasure μ] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
     IndepFun f g μ ↔ μ.map (fun ω ↦ (f ω, g ω)) = (μ.map f).prod (μ.map g) := by
   rw [indepFun_iff_measure_inter_preimage_eq_mul]
   have h₀ {s : Set β} {t : Set β'} (hs : MeasurableSet s) (ht : MeasurableSet t) :
       μ (f ⁻¹' s) * μ (g ⁻¹' t) = μ.map f s * μ.map g t ∧
       μ (f ⁻¹' s ∩ g ⁻¹' t) = μ.map (fun ω ↦ (f ω, g ω)) (s ×ˢ t) :=
-    ⟨by rw [Measure.map_apply hf hs, Measure.map_apply hg ht],
-      (Measure.map_apply (hf.prod_mk hg) (hs.prod ht)).symm⟩
+    ⟨by rw [Measure.map_apply_of_aemeasurable hf hs, Measure.map_apply_of_aemeasurable hg ht],
+      (Measure.map_apply_of_aemeasurable (hf.prod_mk hg) (hs.prod ht)).symm⟩
   constructor
   · refine fun h ↦ (Measure.prod_eq fun s t hs ht ↦ ?_).symm
     rw [← (h₀ hs ht).1, ← (h₀ hs ht).2, h s t hs ht]

This PR proves that two random variables are independent, iff their joint distribution is the product measure of marginal distributions, iff their joint density is a product of their marginal densities up to AE equality. It also uses lemmas stating that μ.withDensity is injective up to AE equality.

@@ -634,6 +634,21 @@ theorem indepFun_iff_indepSet_preimage {mβ : MeasurableSpace β} {mβ' : Measur
     Filter.eventually_pure, kernel.const_apply]
 #align probability_theory.indep_fun_iff_indep_set_preimage ProbabilityTheory.indepFun_iff_indepSet_preimage
 
+theorem indepFun_iff_map_prod_eq_prod_map_map {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
+    [IsFiniteMeasure μ] (hf : Measurable f) (hg : Measurable g) :
+    IndepFun f g μ ↔ μ.map (fun ω ↦ (f ω, g ω)) = (μ.map f).prod (μ.map g) := by
+  rw [indepFun_iff_measure_inter_preimage_eq_mul]
+  have h₀ {s : Set β} {t : Set β'} (hs : MeasurableSet s) (ht : MeasurableSet t) :
+      μ (f ⁻¹' s) * μ (g ⁻¹' t) = μ.map f s * μ.map g t ∧
+      μ (f ⁻¹' s ∩ g ⁻¹' t) = μ.map (fun ω ↦ (f ω, g ω)) (s ×ˢ t) :=
+    ⟨by rw [Measure.map_apply hf hs, Measure.map_apply hg ht],
+      (Measure.map_apply (hf.prod_mk hg) (hs.prod ht)).symm⟩
+  constructor
+  · refine fun h ↦ (Measure.prod_eq fun s t hs ht ↦ ?_).symm
+    rw [← (h₀ hs ht).1, ← (h₀ hs ht).2, h s t hs ht]
+  · intro h s t hs ht
+    rw [(h₀ hs ht).1, (h₀ hs ht).2, h, Measure.prod_prod]
+
 @[symm]
 nonrec theorem IndepFun.symm {_ : MeasurableSpace β} {f g : Ω → β} (hfg : IndepFun f g μ) :
     IndepFun g f μ := hfg.symm

Port a bit of https://github.com/leanprover-community/mathlib/pull/18506, but it's mostly handmade.

Co-authored-by: RemyDegenne <Remydegenne@gmail.com>

@@ -5,7 +5,7 @@ Authors: Rémy Degenne
 -/
 import Mathlib.Probability.Independence.Kernel
 
-#align_import probability.independence.basic from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
+#align_import probability.independence.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
 
 /-!
 # Independence of sets of sets and measure spaces (σ-algebras)
@@ -66,15 +66,13 @@ when defining `μ` in the example above, the measurable space used is the last o
 Part A, Chapter 4.
 -/
 
-set_option autoImplicit true
-
-open MeasureTheory MeasurableSpace
+open MeasureTheory MeasurableSpace Set
 
 open scoped BigOperators MeasureTheory ENNReal
 
 namespace ProbabilityTheory
 
-variable {Ω ι : Type*}
+variable {Ω ι β γ : Type*} {κ : ι → Type*}
 
 section Definitions
 
@@ -145,12 +143,23 @@ def IndepFun {β γ} [MeasurableSpace Ω] [MeasurableSpace β] [MeasurableSpace
 end Definitions
 
 section Definition_lemmas
+variable {π : ι → Set (Set Ω)} {m : ι → MeasurableSpace Ω} {_ : MeasurableSpace Ω} {μ : Measure Ω}
+  {S : Finset ι} {s : ι → Set Ω}
 
 lemma iIndepSets_iff [MeasurableSpace Ω] (π : ι → Set (Set Ω)) (μ : Measure Ω) :
     iIndepSets π μ ↔ ∀ (s : Finset ι) {f : ι → Set Ω} (_H : ∀ i, i ∈ s → f i ∈ π i),
       μ (⋂ i ∈ s, f i) = ∏ i in s, μ (f i) := by
   simp only [iIndepSets, kernel.iIndepSets, ae_dirac_eq, Filter.eventually_pure, kernel.const_apply]
 
+lemma iIndepSets.meas_biInter (h : iIndepSets π μ) (s : Finset ι) {f : ι → Set Ω}
+    (hf : ∀ i, i ∈ s → f i ∈ π i) : μ (⋂ i ∈ s, f i) = ∏ i in s, μ (f i) :=
+  (iIndepSets_iff _ _).1 h s hf
+
+lemma iIndepSets.meas_iInter [Fintype ι] (h : iIndepSets π μ) (hs : ∀ i, s i ∈ π i) :
+    μ (⋂ i, s i) = ∏ i, μ (s i) := by simp [←h.meas_biInter _ fun _i _ ↦ hs _]
+set_option linter.uppercaseLean3 false in
+#align probability_theory.Indep_sets.meas_Inter ProbabilityTheory.iIndepSets.meas_iInter
+
 lemma IndepSets_iff [MeasurableSpace Ω] (s1 s2 : Set (Set Ω)) (μ : Measure Ω) :
     IndepSets s1 s2 μ ↔ ∀ t1 t2 : Set Ω, t1 ∈ s1 → t2 ∈ s2 → (μ (t1 ∩ t2) = μ t1 * μ t2) := by
   simp only [IndepSets, kernel.IndepSets, ae_dirac_eq, Filter.eventually_pure, kernel.const_apply]
@@ -159,11 +168,20 @@ lemma iIndep_iff_iIndepSets (m : ι → MeasurableSpace Ω) [MeasurableSpace Ω]
     iIndep m μ ↔ iIndepSets (fun x ↦ {s | MeasurableSet[m x] s}) μ := by
   simp only [iIndep, iIndepSets, kernel.iIndep]
 
+lemma iIndep.iIndepSets' {m : ι → MeasurableSpace Ω} (hμ : iIndep m μ) :
+    iIndepSets (fun x ↦ {s | MeasurableSet[m x] s}) μ := (iIndep_iff_iIndepSets _ _).1 hμ
+
 lemma iIndep_iff (m : ι → MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Measure Ω) :
     iIndep m μ ↔ ∀ (s : Finset ι) {f : ι → Set Ω} (_H : ∀ i, i ∈ s → MeasurableSet[m i] (f i)),
       μ (⋂ i ∈ s, f i) = ∏ i in s, μ (f i) := by
   simp only [iIndep_iff_iIndepSets, iIndepSets_iff]; rfl
 
+lemma iIndep.meas_biInter (hμ : iIndep m μ) (hs : ∀ i, i ∈ S → MeasurableSet[m i] (s i)) :
+    μ (⋂ i ∈ S, s i) = ∏ i in S, μ (s i) := (iIndep_iff _ _).1 hμ _ hs
+
+lemma iIndep.meas_iInter [Fintype ι] (hμ : iIndep m μ) (hs : ∀ i, MeasurableSet[m i] (s i)) :
+    μ (⋂ i, s i) = ∏ i, μ (s i) := by simp [←hμ.meas_biInter fun _ _ ↦ hs _]
+
 lemma Indep_iff_IndepSets (m₁ m₂ : MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Measure Ω) :
     Indep m₁ m₂ μ ↔ IndepSets {s | MeasurableSet[m₁] s} {s | MeasurableSet[m₂] s} μ := by
   simp only [Indep, IndepSets, kernel.Indep]
@@ -197,6 +215,10 @@ lemma iIndepFun_iff_iIndep [MeasurableSpace Ω] {β : ι → Type*}
     iIndepFun m f μ ↔ iIndep (fun x ↦ (m x).comap (f x)) μ := by
   simp only [iIndepFun, iIndep, kernel.iIndepFun]
 
+protected lemma iIndepFun.iIndep {m : ∀ i, MeasurableSpace (κ i)} {f : ∀ x : ι, Ω → κ x}
+    (hf : iIndepFun m f μ) :
+    iIndep (fun x ↦ (m x).comap (f x)) μ := hf
+
 lemma iIndepFun_iff [MeasurableSpace Ω] {β : ι → Type*}
     (m : ∀ x : ι, MeasurableSpace (β x)) (f : ∀ x : ι, Ω → β x) (μ : Measure Ω) :
     iIndepFun m f μ ↔ ∀ (s : Finset ι) {f' : ι → Set Ω}
@@ -204,6 +226,14 @@ lemma iIndepFun_iff [MeasurableSpace Ω] {β : ι → Type*}
       μ (⋂ i ∈ s, f' i) = ∏ i in s, μ (f' i) := by
   simp only [iIndepFun_iff_iIndep, iIndep_iff]
 
+lemma iIndepFun.meas_biInter {m : ∀ i, MeasurableSpace (κ i)} {f : ∀ x : ι, Ω → κ x}
+    (hf : iIndepFun m f μ) (hs : ∀ i, i ∈ S → MeasurableSet[(m i).comap (f i)] (s i)) :
+    μ (⋂ i ∈ S, s i) = ∏ i in S, μ (s i) := hf.iIndep.meas_biInter hs
+
+lemma iIndepFun.meas_iInter [Fintype ι] {m : ∀ i, MeasurableSpace (κ i)} {f : ∀ x : ι, Ω → κ x}
+    (hf : iIndepFun m f μ) (hs : ∀ i, MeasurableSet[(m i).comap (f i)] (s i)) :
+    μ (⋂ i, s i) = ∏ i, μ (s i) := hf.iIndep.meas_iInter hs
+
 lemma IndepFun_iff_Indep [MeasurableSpace Ω] [mβ : MeasurableSpace β]
     [mγ : MeasurableSpace γ] (f : Ω → β) (g : Ω → γ) (μ : Measure Ω) :
     IndepFun f g μ ↔ Indep (MeasurableSpace.comap f mβ) (MeasurableSpace.comap g mγ) μ := by
@@ -215,6 +245,12 @@ lemma IndepFun_iff {β γ} [MeasurableSpace Ω] [mβ : MeasurableSpace β] [mγ
       → MeasurableSet[MeasurableSpace.comap g mγ] t2 → μ (t1 ∩ t2) = μ t1 * μ t2 := by
   rw [IndepFun_iff_Indep, Indep_iff]
 
+lemma IndepFun.meas_inter [mβ : MeasurableSpace β] [mγ : MeasurableSpace γ] {f : Ω → β} {g : Ω → γ}
+    (hfg : IndepFun f g μ) {s t : Set Ω} (hs : MeasurableSet[mβ.comap f] s)
+    (ht : MeasurableSet[mγ.comap g] t) :
+    μ (s ∩ t) = μ s * μ t :=
+  (IndepFun_iff _ _ _).1 hfg _ _ hs ht
+
 end Definition_lemmas
 
 section Indep
@@ -472,7 +508,7 @@ theorem iIndepSets.iIndep [IsProbabilityMeasure μ] (m : ι → MeasurableSpace
     iIndep m μ :=
   kernel.iIndepSets.iIndep m h_le π h_pi h_generate h_ind
 set_option linter.uppercaseLean3 false in
-#align probability_theory.Indep_sets.Indep ProbabilityTheory.IndepSets.indep
+#align probability_theory.Indep_sets.Indep ProbabilityTheory.iIndepSets.iIndep
 
 end FromPiSystemsToMeasurableSpaces
 
@@ -486,7 +522,7 @@ We prove the following equivalences on `IndepSet`, for measurable sets `s, t`.
 -/
 
 
-variable {s t : Set Ω} (S T : Set (Set Ω))
+variable {_ : MeasurableSpace Ω} {μ : Measure Ω} {s t : Set Ω} (S T : Set (Set Ω))
 
 theorem indepSet_iff_indepSets_singleton {_m0 : MeasurableSpace Ω} (hs_meas : MeasurableSet s)
     (ht_meas : MeasurableSet t) (μ : Measure Ω := by volume_tac)
@@ -521,6 +557,45 @@ theorem indep_iff_forall_indepSet (m₁ m₂ : MeasurableSpace Ω) {_m0 : Measur
   kernel.indep_iff_forall_indepSet m₁ m₂ _ _
 #align probability_theory.indep_iff_forall_indep_set ProbabilityTheory.indep_iff_forall_indepSet
 
+theorem iIndep_comap_mem_iff {f : ι → Set Ω} :
+    iIndep (fun i => MeasurableSpace.comap (· ∈ f i) ⊤) μ ↔ iIndepSet f μ :=
+  kernel.iIndep_comap_mem_iff
+set_option linter.uppercaseLean3 false in
+#align probability_theory.Indep_comap_mem_iff ProbabilityTheory.iIndep_comap_mem_iff
+
+alias ⟨_, iIndepSet.iIndep_comap_mem⟩ := iIndep_comap_mem_iff
+set_option linter.uppercaseLean3 false in
+#align probability_theory.Indep_set.Indep_comap_mem ProbabilityTheory.iIndepSet.iIndep_comap_mem
+
+theorem iIndepSets_singleton_iff {s : ι → Set Ω} :
+    iIndepSets (fun i ↦ {s i}) μ ↔ ∀ t, μ (⋂ i ∈ t, s i) = ∏ i in t, μ (s i) := by
+  simp_rw [iIndepSets, kernel.iIndepSets_singleton_iff, ae_dirac_eq, Filter.eventually_pure,
+    kernel.const_apply]
+set_option linter.uppercaseLean3 false in
+#align probability_theory.Indep_sets_singleton_iff ProbabilityTheory.iIndepSets_singleton_iff
+
+variable [IsProbabilityMeasure μ]
+
+theorem iIndepSet_iff_iIndepSets_singleton {f : ι → Set Ω} (hf : ∀ i, MeasurableSet (f i)) :
+    iIndepSet f μ ↔ iIndepSets (fun i ↦ {f i}) μ :=
+  kernel.iIndepSet_iff_iIndepSets_singleton hf
+set_option linter.uppercaseLean3 false in
+#align probability_theory.Indep_set_iff_Indep_sets_singleton ProbabilityTheory.iIndepSet_iff_iIndepSets_singleton
+
+theorem iIndepSet_iff_meas_biInter {f : ι → Set Ω} (hf : ∀ i, MeasurableSet (f i)) :
+    iIndepSet f μ ↔ ∀ s, μ (⋂ i ∈ s, f i) = ∏ i in s, μ (f i) := by
+  simp_rw [iIndepSet, kernel.iIndepSet_iff_meas_biInter hf, ae_dirac_eq, Filter.eventually_pure,
+    kernel.const_apply]
+set_option linter.uppercaseLean3 false in
+#align probability_theory.Indep_set_iff_measure_Inter_eq_prod ProbabilityTheory.iIndepSet_iff_meas_biInter
+
+theorem iIndepSets.iIndepSet_of_mem {π : ι → Set (Set Ω)} {f : ι → Set Ω}
+    (hfπ : ∀ i, f i ∈ π i) (hf : ∀ i, MeasurableSet (f i))
+    (hπ : iIndepSets π μ) : iIndepSet f μ :=
+  kernel.iIndepSets.iIndepSet_of_mem hfπ hf hπ
+set_option linter.uppercaseLean3 false in
+#align probability_theory.Indep_sets.Indep_set_of_mem ProbabilityTheory.iIndepSets.iIndepSet_of_mem
+
 end IndepSet
 
 section IndepFun

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -164,11 +164,11 @@ lemma iIndep_iff (m : ι → MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Meas
       μ (⋂ i ∈ s, f i) = ∏ i in s, μ (f i) := by
   simp only [iIndep_iff_iIndepSets, iIndepSets_iff]; rfl
 
-lemma Indep_iff_IndepSets (m₁ m₂ : MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Measure Ω ) :
+lemma Indep_iff_IndepSets (m₁ m₂ : MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Measure Ω) :
     Indep m₁ m₂ μ ↔ IndepSets {s | MeasurableSet[m₁] s} {s | MeasurableSet[m₂] s} μ := by
   simp only [Indep, IndepSets, kernel.Indep]
 
-lemma Indep_iff (m₁ m₂ : MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Measure Ω ) :
+lemma Indep_iff (m₁ m₂ : MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Measure Ω) :
     Indep m₁ m₂ μ
       ↔ ∀ t1 t2, MeasurableSet[m₁] t1 → MeasurableSet[m₂] t2 → μ (t1 ∩ t2) = μ t1 * μ t2 := by
   rw [Indep_iff_IndepSets, IndepSets_iff]; rfl

This will improve spaces in the mathlib4 docs.

@@ -12,7 +12,7 @@ import Mathlib.Probability.Independence.Kernel
 
 * A family of sets of sets `π : ι → Set (Set Ω)` is independent with respect to a measure `μ` if for
   any finite set of indices `s = {i_1, ..., i_n}`, for any sets `f i_1 ∈ π i_1, ..., f i_n ∈ π i_n`,
-  `μ (⋂ i in s, f i) = ∏ i in s, μ (f i) `. It will be used for families of π-systems.
+  `μ (⋂ i in s, f i) = ∏ i in s, μ (f i)`. It will be used for families of π-systems.
 * A family of measurable space structures (i.e. of σ-algebras) is independent with respect to a
   measure `μ` (typically defined on a finer σ-algebra) if the family of sets of measurable sets they
   define is independent. I.e., `m : ι → MeasurableSpace Ω` is independent with respect to a
@@ -97,7 +97,7 @@ def IndepSets [MeasurableSpace Ω] (s1 s2 : Set (Set Ω)) (μ : Measure Ω := by
 measure `μ` (typically defined on a finer σ-algebra) if the family of sets of measurable sets they
 define is independent. `m : ι → MeasurableSpace Ω` is independent with respect to measure `μ` if
 for any finite set of indices `s = {i_1, ..., i_n}`, for any sets
-`f i_1 ∈ m i_1, ..., f i_n ∈ m i_n`, then `μ (⋂ i in s, f i) = ∏ i in s, μ (f i) `. -/
+`f i_1 ∈ m i_1, ..., f i_n ∈ m i_n`, then `μ (⋂ i in s, f i) = ∏ i in s, μ (f i)`. -/
 def iIndep (m : ι → MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Measure Ω := by volume_tac) :
     Prop :=
   kernel.iIndep m (kernel.const Unit μ) (Measure.dirac () : Measure Unit)

Autoimplicits are highly controversial and also defeat the performance-improving work in #6474.

The intent of this PR is to make autoImplicit opt-in on a per-file basis, by disabling it in the lakefile and enabling it again with set_option autoImplicit true in the few files that rely on it.

That also keeps this PR small, as opposed to attempting to "fix" files to not need it any more.

I claim that many of the uses of autoImplicit in these files are accidental; situations such as:

• Assuming variables are in scope, but pasting the lemma in the wrong section
• Pasting in a lemma from a scratch file without checking to see if the variable names are consistent with the rest of the file
• Making a copy-paste error between lemmas and forgetting to add an explicit arguments.

Having set_option autoImplicit false as the default prevents these types of mistake being made in the 90% of files where autoImplicits are not used at all, and causes them to be caught by CI during review.

I think there were various points during the port where we encouraged porters to delete the universes u v lines; I think having autoparams for universe variables only would cover a lot of the cases we actually use them, while avoiding any real shortcomings.

A Zulip poll (after combining overlapping votes accordingly) was in favor of this change with 5:5:18 as the no:dontcare:yes vote ratio.

While this PR was being reviewed, a handful of files gained some more likely-accidental autoImplicits. In these places, set_option autoImplicit true has been placed locally within a section, rather than at the top of the file.

@@ -66,6 +66,8 @@ when defining `μ` in the example above, the measurable space used is the last o
 Part A, Chapter 4.
 -/
 
+set_option autoImplicit true
+
 open MeasureTheory MeasurableSpace
 
 open scoped BigOperators MeasureTheory ENNReal

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -72,7 +72,7 @@ open scoped BigOperators MeasureTheory ENNReal
 
 namespace ProbabilityTheory
 
-variable {Ω ι : Type _}
+variable {Ω ι : Type*}
 
 section Definitions
 
@@ -126,7 +126,7 @@ def IndepSet [MeasurableSpace Ω] (s t : Set Ω) (μ : Measure Ω := by volume_t
 spaces, each with a measurable space structure, is independent if the family of measurable space
 structures they generate on `Ω` is independent. For a function `g` with codomain having measurable
 space structure `m`, the generated measurable space structure is `MeasurableSpace.comap g m`. -/
-def iIndepFun [MeasurableSpace Ω] {β : ι → Type _} (m : ∀ x : ι, MeasurableSpace (β x))
+def iIndepFun [MeasurableSpace Ω] {β : ι → Type*} (m : ∀ x : ι, MeasurableSpace (β x))
     (f : ∀ x : ι, Ω → β x) (μ : Measure Ω := by volume_tac) : Prop :=
   kernel.iIndepFun m f (kernel.const Unit μ) (Measure.dirac () : Measure Unit)
 set_option linter.uppercaseLean3 false in
@@ -190,12 +190,12 @@ lemma IndepSet_iff [MeasurableSpace Ω] (s t : Set Ω) (μ : Measure Ω) :
       → MeasurableSet[generateFrom {t}] t2 → μ (t1 ∩ t2) = μ t1 * μ t2 := by
   simp only [IndepSet_iff_Indep, Indep_iff]
 
-lemma iIndepFun_iff_iIndep [MeasurableSpace Ω] {β : ι → Type _}
+lemma iIndepFun_iff_iIndep [MeasurableSpace Ω] {β : ι → Type*}
     (m : ∀ x : ι, MeasurableSpace (β x)) (f : ∀ x : ι, Ω → β x) (μ : Measure Ω) :
     iIndepFun m f μ ↔ iIndep (fun x ↦ (m x).comap (f x)) μ := by
   simp only [iIndepFun, iIndep, kernel.iIndepFun]
 
-lemma iIndepFun_iff [MeasurableSpace Ω] {β : ι → Type _}
+lemma iIndepFun_iff [MeasurableSpace Ω] {β : ι → Type*}
     (m : ∀ x : ι, MeasurableSpace (β x)) (f : ∀ x : ι, Ω → β x) (μ : Measure Ω) :
     iIndepFun m f μ ↔ ∀ (s : Finset ι) {f' : ι → Set Ω}
       (_H : ∀ i, i ∈ s → MeasurableSet[(m i).comap (f i)] (f' i)),
@@ -337,7 +337,7 @@ theorem iIndep.indep {m : ι → MeasurableSpace Ω} [MeasurableSpace Ω] {μ :
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep.indep ProbabilityTheory.iIndep.indep
 
-theorem iIndepFun.indepFun {_m₀ : MeasurableSpace Ω} {μ : Measure Ω} {β : ι → Type _}
+theorem iIndepFun.indepFun {_m₀ : MeasurableSpace Ω} {μ : Measure Ω} {β : ι → Type*}
     {m : ∀ x, MeasurableSpace (β x)} {f : ∀ i, Ω → β i} (hf_Indep : iIndepFun m f μ) {i j : ι}
     (hij : i ≠ j) :
     IndepFun (f i) (f j) μ :=
@@ -528,7 +528,7 @@ section IndepFun
 -/
 
 
-variable {β β' γ γ' : Type _} {mΩ : MeasurableSpace Ω} {μ : Measure Ω} {f : Ω → β} {g : Ω → β'}
+variable {β β' γ γ' : Type*} {mΩ : MeasurableSpace Ω} {μ : Measure Ω} {f : Ω → β} {g : Ω → β'}
 
 theorem indepFun_iff_measure_inter_preimage_eq_mul {mβ : MeasurableSpace β}
     {mβ' : MeasurableSpace β'} :
@@ -539,7 +539,7 @@ theorem indepFun_iff_measure_inter_preimage_eq_mul {mβ : MeasurableSpace β}
     Filter.eventually_pure, kernel.const_apply]
 #align probability_theory.indep_fun_iff_measure_inter_preimage_eq_mul ProbabilityTheory.indepFun_iff_measure_inter_preimage_eq_mul
 
-theorem iIndepFun_iff_measure_inter_preimage_eq_mul {ι : Type _} {β : ι → Type _}
+theorem iIndepFun_iff_measure_inter_preimage_eq_mul {ι : Type*} {β : ι → Type*}
     (m : ∀ x, MeasurableSpace (β x)) (f : ∀ i, Ω → β i) :
     iIndepFun m f μ ↔
       ∀ (S : Finset ι) {sets : ∀ i : ι, Set (β i)} (_H : ∀ i, i ∈ S → MeasurableSet[m i] (sets i)),
@@ -579,7 +579,7 @@ theorem IndepFun.comp {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'}
 /-- If `f` is a family of mutually independent random variables (`iIndepFun m f μ`) and `S, T` are
 two disjoint finite index sets, then the tuple formed by `f i` for `i ∈ S` is independent of the
 tuple `(f i)_i` for `i ∈ T`. -/
-theorem iIndepFun.indepFun_finset [IsProbabilityMeasure μ] {ι : Type _} {β : ι → Type _}
+theorem iIndepFun.indepFun_finset [IsProbabilityMeasure μ] {ι : Type*} {β : ι → Type*}
     {m : ∀ i, MeasurableSpace (β i)} {f : ∀ i, Ω → β i} (S T : Finset ι) (hST : Disjoint S T)
     (hf_Indep : iIndepFun m f μ) (hf_meas : ∀ i, Measurable (f i)) :
     IndepFun (fun a (i : S) => f i a) (fun a (i : T) => f i a) μ :=
@@ -587,7 +587,7 @@ theorem iIndepFun.indepFun_finset [IsProbabilityMeasure μ] {ι : Type _} {β :
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun.indep_fun_finset ProbabilityTheory.iIndepFun.indepFun_finset
 
-theorem iIndepFun.indepFun_prod [IsProbabilityMeasure μ] {ι : Type _} {β : ι → Type _}
+theorem iIndepFun.indepFun_prod [IsProbabilityMeasure μ] {ι : Type*} {β : ι → Type*}
     {m : ∀ i, MeasurableSpace (β i)} {f : ∀ i, Ω → β i} (hf_Indep : iIndepFun m f μ)
     (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
     IndepFun (fun a => (f i a, f j a)) (f k) μ :=
@@ -596,7 +596,7 @@ set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun.indep_fun_prod ProbabilityTheory.iIndepFun.indepFun_prod
 
 @[to_additive]
-theorem iIndepFun.mul [IsProbabilityMeasure μ] {ι : Type _} {β : Type _} {m : MeasurableSpace β}
+theorem iIndepFun.mul [IsProbabilityMeasure μ] {ι : Type*} {β : Type*} {m : MeasurableSpace β}
     [Mul β] [MeasurableMul₂ β] {f : ι → Ω → β} (hf_Indep : iIndepFun (fun _ => m) f μ)
     (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
     IndepFun (f i * f j) (f k) μ :=
@@ -607,7 +607,7 @@ set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun.add ProbabilityTheory.iIndepFun.add
 
 @[to_additive]
-theorem iIndepFun.indepFun_finset_prod_of_not_mem [IsProbabilityMeasure μ] {ι : Type _} {β : Type _}
+theorem iIndepFun.indepFun_finset_prod_of_not_mem [IsProbabilityMeasure μ] {ι : Type*} {β : Type*}
     {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂ β] {f : ι → Ω → β}
     (hf_Indep : iIndepFun (fun _ => m) f μ) (hf_meas : ∀ i, Measurable (f i)) {s : Finset ι} {i : ι}
     (hi : i ∉ s) :
@@ -619,7 +619,7 @@ set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun.indep_fun_finset_sum_of_not_mem ProbabilityTheory.iIndepFun.indepFun_finset_sum_of_not_mem
 
 @[to_additive]
-theorem iIndepFun.indepFun_prod_range_succ [IsProbabilityMeasure μ] {β : Type _}
+theorem iIndepFun.indepFun_prod_range_succ [IsProbabilityMeasure μ] {β : Type*}
     {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂ β] {f : ℕ → Ω → β}
     (hf_Indep : iIndepFun (fun _ => m) f μ) (hf_meas : ∀ i, Measurable (f i)) (n : ℕ) :
     IndepFun (∏ j in Finset.range n, f j) (f n) μ :=

Independence is the special case of independence with respect to a kernel and a measure where the kernel is constant.

@@ -3,7 +3,7 @@ Copyright (c) 2021 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 -/
-import Mathlib.MeasureTheory.Constructions.Pi
+import Mathlib.Probability.Independence.Kernel
 
 #align_import probability.independence.basic from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
 
@@ -33,9 +33,12 @@ import Mathlib.MeasureTheory.Constructions.Pi
 
 ## Implementation notes
 
-We provide one main definition of independence:
-* `iIndepSets`: independence of a family of sets of sets `pi : ι → Set (Set Ω)`.
-Three other independence notions are defined using `iIndepSets`:
+The definitions of independence in this file are a particular case of independence with respect to a
+kernel and a measure, as defined in the file `Kernel.lean`.
+
+We provide four definitions of independence:
+* `iIndepSets`: independence of a family of sets of sets `pi : ι → Set (Set Ω)`. This is meant to
+  be used with π-systems.
 * `iIndep`: independence of a family of measurable space structures `m : ι → MeasurableSpace Ω`,
 * `iIndepSet`: independence of a family of sets `s : ι → Set Ω`,
 * `iIndepFun`: independence of a family of functions. For measurable spaces
@@ -46,11 +49,11 @@ sets of sets, sets, functions) instead of a family. These properties are denoted
 as for a family, but without the starting `i`, for example `IndepFun` is the version of `iIndepFun`
 for two functions.
 
-The definition of independence for `iIndepSets` uses finite sets (`Finset`). An alternative and
-equivalent way of defining independence would have been to use countable sets.
-TODO: prove that equivalence.
+The definition of independence for `iIndepSets` uses finite sets (`Finset`). See
+`ProbabilityTheory.kernel.iIndepSets`. An alternative and equivalent way of defining independence
+would have been to use countable sets.
 
-Most of the definitions and lemma in this file list all variables instead of using the `variables`
+Most of the definitions and lemmas in this file list all variables instead of using the `variable`
 keyword at the beginning of a section, for example
 `lemma Indep.symm {Ω} {m₁ m₂ : MeasurableSpace Ω} [MeasurableSpace Ω] {μ : measure Ω} ...` .
 This is intentional, to be able to control the order of the `MeasurableSpace` variables. Indeed
@@ -78,15 +81,14 @@ for any finite set of indices `s = {i_1, ..., i_n}`, for any sets
 `f i_1 ∈ π i_1, ..., f i_n ∈ π i_n`, then `μ (⋂ i in s, f i) = ∏ i in s, μ (f i) `.
 It will be used for families of pi_systems. -/
 def iIndepSets [MeasurableSpace Ω] (π : ι → Set (Set Ω)) (μ : Measure Ω := by volume_tac) : Prop :=
-    ∀ (s : Finset ι) {f : ι → Set Ω} (_H : ∀ i, i ∈ s → f i ∈ π i),
-    μ (⋂ i ∈ s, f i) = ∏ i in s, μ (f i)
+  kernel.iIndepSets π (kernel.const Unit μ) (Measure.dirac () : Measure Unit)
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_sets ProbabilityTheory.iIndepSets
 
 /-- Two sets of sets `s₁, s₂` are independent with respect to a measure `μ` if for any sets
 `t₁ ∈ p₁, t₂ ∈ s₂`, then `μ (t₁ ∩ t₂) = μ (t₁) * μ (t₂)` -/
-def IndepSets [MeasurableSpace Ω] (s1 s2 : Set (Set Ω)) (μ : Measure Ω := by volume_tac) : Prop
-    := ∀ t1 t2 : Set Ω, t1 ∈ s1 → t2 ∈ s2 → μ (t1 ∩ t2) = μ t1 * μ t2
+def IndepSets [MeasurableSpace Ω] (s1 s2 : Set (Set Ω)) (μ : Measure Ω := by volume_tac) : Prop :=
+  kernel.IndepSets s1 s2 (kernel.const Unit μ) (Measure.dirac () : Measure Unit)
 #align probability_theory.indep_sets ProbabilityTheory.IndepSets
 
 /-- A family of measurable space structures (i.e. of σ-algebras) is independent with respect to a
@@ -95,28 +97,29 @@ define is independent. `m : ι → MeasurableSpace Ω` is independent with respe
 for any finite set of indices `s = {i_1, ..., i_n}`, for any sets
 `f i_1 ∈ m i_1, ..., f i_n ∈ m i_n`, then `μ (⋂ i in s, f i) = ∏ i in s, μ (f i) `. -/
 def iIndep (m : ι → MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Measure Ω := by volume_tac) :
-    Prop := iIndepSets (fun x => { s | MeasurableSet[m x] s }) μ
+    Prop :=
+  kernel.iIndep m (kernel.const Unit μ) (Measure.dirac () : Measure Unit)
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep ProbabilityTheory.iIndep
 
 /-- Two measurable space structures (or σ-algebras) `m₁, m₂` are independent with respect to a
 measure `μ` (defined on a third σ-algebra) if for any sets `t₁ ∈ m₁, t₂ ∈ m₂`,
 `μ (t₁ ∩ t₂) = μ (t₁) * μ (t₂)` -/
-def Indep (m₁ m₂ : MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Measure Ω := by volume_tac) : Prop
-    := IndepSets { s | MeasurableSet[m₁] s } { s | MeasurableSet[m₂] s } μ
+def Indep (m₁ m₂ : MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Measure Ω := by volume_tac) : Prop :=
+  kernel.Indep m₁ m₂ (kernel.const Unit μ) (Measure.dirac () : Measure Unit)
 #align probability_theory.indep ProbabilityTheory.Indep
 
 /-- A family of sets is independent if the family of measurable space structures they generate is
 independent. For a set `s`, the generated measurable space has measurable sets `∅, s, sᶜ, univ`. -/
 def iIndepSet [MeasurableSpace Ω] (s : ι → Set Ω) (μ : Measure Ω := by volume_tac) : Prop :=
-    iIndep (fun i => generateFrom {s i}) μ
+  kernel.iIndepSet s (kernel.const Unit μ) (Measure.dirac () : Measure Unit)
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_set ProbabilityTheory.iIndepSet
 
 /-- Two sets are independent if the two measurable space structures they generate are independent.
 For a set `s`, the generated measurable space structure has measurable sets `∅, s, sᶜ, univ`. -/
 def IndepSet [MeasurableSpace Ω] (s t : Set Ω) (μ : Measure Ω := by volume_tac) : Prop :=
-    Indep (generateFrom {s}) (generateFrom {t}) μ
+  kernel.IndepSet s t (kernel.const Unit μ) (Measure.dirac () : Measure Unit)
 #align probability_theory.indep_set ProbabilityTheory.IndepSet
 
 /-- A family of functions defined on the same space `Ω` and taking values in possibly different
@@ -125,27 +128,99 @@ structures they generate on `Ω` is independent. For a function `g` with codomai
 space structure `m`, the generated measurable space structure is `MeasurableSpace.comap g m`. -/
 def iIndepFun [MeasurableSpace Ω] {β : ι → Type _} (m : ∀ x : ι, MeasurableSpace (β x))
     (f : ∀ x : ι, Ω → β x) (μ : Measure Ω := by volume_tac) : Prop :=
-    iIndep (fun x => MeasurableSpace.comap (f x) (m x)) μ
+  kernel.iIndepFun m f (kernel.const Unit μ) (Measure.dirac () : Measure Unit)
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun ProbabilityTheory.iIndepFun
 
 /-- Two functions are independent if the two measurable space structures they generate are
 independent. For a function `f` with codomain having measurable space structure `m`, the generated
 measurable space structure is `MeasurableSpace.comap f m`. -/
-def IndepFun {β γ} [MeasurableSpace Ω] [mβ : MeasurableSpace β] [mγ : MeasurableSpace γ]
+def IndepFun {β γ} [MeasurableSpace Ω] [MeasurableSpace β] [MeasurableSpace γ]
     (f : Ω → β) (g : Ω → γ) (μ : Measure Ω := by volume_tac) : Prop :=
-    Indep (MeasurableSpace.comap f mβ) (MeasurableSpace.comap g mγ) μ
+  kernel.IndepFun f g (kernel.const Unit μ) (Measure.dirac () : Measure Unit)
 #align probability_theory.indep_fun ProbabilityTheory.IndepFun
 
 end Definitions
 
+section Definition_lemmas
+
+lemma iIndepSets_iff [MeasurableSpace Ω] (π : ι → Set (Set Ω)) (μ : Measure Ω) :
+    iIndepSets π μ ↔ ∀ (s : Finset ι) {f : ι → Set Ω} (_H : ∀ i, i ∈ s → f i ∈ π i),
+      μ (⋂ i ∈ s, f i) = ∏ i in s, μ (f i) := by
+  simp only [iIndepSets, kernel.iIndepSets, ae_dirac_eq, Filter.eventually_pure, kernel.const_apply]
+
+lemma IndepSets_iff [MeasurableSpace Ω] (s1 s2 : Set (Set Ω)) (μ : Measure Ω) :
+    IndepSets s1 s2 μ ↔ ∀ t1 t2 : Set Ω, t1 ∈ s1 → t2 ∈ s2 → (μ (t1 ∩ t2) = μ t1 * μ t2) := by
+  simp only [IndepSets, kernel.IndepSets, ae_dirac_eq, Filter.eventually_pure, kernel.const_apply]
+
+lemma iIndep_iff_iIndepSets (m : ι → MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Measure Ω) :
+    iIndep m μ ↔ iIndepSets (fun x ↦ {s | MeasurableSet[m x] s}) μ := by
+  simp only [iIndep, iIndepSets, kernel.iIndep]
+
+lemma iIndep_iff (m : ι → MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Measure Ω) :
+    iIndep m μ ↔ ∀ (s : Finset ι) {f : ι → Set Ω} (_H : ∀ i, i ∈ s → MeasurableSet[m i] (f i)),
+      μ (⋂ i ∈ s, f i) = ∏ i in s, μ (f i) := by
+  simp only [iIndep_iff_iIndepSets, iIndepSets_iff]; rfl
+
+lemma Indep_iff_IndepSets (m₁ m₂ : MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Measure Ω ) :
+    Indep m₁ m₂ μ ↔ IndepSets {s | MeasurableSet[m₁] s} {s | MeasurableSet[m₂] s} μ := by
+  simp only [Indep, IndepSets, kernel.Indep]
+
+lemma Indep_iff (m₁ m₂ : MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Measure Ω ) :
+    Indep m₁ m₂ μ
+      ↔ ∀ t1 t2, MeasurableSet[m₁] t1 → MeasurableSet[m₂] t2 → μ (t1 ∩ t2) = μ t1 * μ t2 := by
+  rw [Indep_iff_IndepSets, IndepSets_iff]; rfl
+
+lemma iIndepSet_iff_iIndep [MeasurableSpace Ω] (s : ι → Set Ω) (μ : Measure Ω) :
+    iIndepSet s μ ↔ iIndep (fun i ↦ generateFrom {s i}) μ := by
+  simp only [iIndepSet, iIndep, kernel.iIndepSet]
+
+lemma iIndepSet_iff [MeasurableSpace Ω] (s : ι → Set Ω) (μ : Measure Ω) :
+    iIndepSet s μ ↔ ∀ (s' : Finset ι) {f : ι → Set Ω}
+      (_H : ∀ i, i ∈ s' → MeasurableSet[generateFrom {s i}] (f i)),
+      μ (⋂ i ∈ s', f i) = ∏ i in s', μ (f i) := by
+  simp only [iIndepSet_iff_iIndep, iIndep_iff]
+
+lemma IndepSet_iff_Indep [MeasurableSpace Ω] (s t : Set Ω) (μ : Measure Ω) :
+    IndepSet s t μ ↔ Indep (generateFrom {s}) (generateFrom {t}) μ := by
+  simp only [IndepSet, Indep, kernel.IndepSet]
+
+lemma IndepSet_iff [MeasurableSpace Ω] (s t : Set Ω) (μ : Measure Ω) :
+    IndepSet s t μ ↔ ∀ t1 t2, MeasurableSet[generateFrom {s}] t1
+      → MeasurableSet[generateFrom {t}] t2 → μ (t1 ∩ t2) = μ t1 * μ t2 := by
+  simp only [IndepSet_iff_Indep, Indep_iff]
+
+lemma iIndepFun_iff_iIndep [MeasurableSpace Ω] {β : ι → Type _}
+    (m : ∀ x : ι, MeasurableSpace (β x)) (f : ∀ x : ι, Ω → β x) (μ : Measure Ω) :
+    iIndepFun m f μ ↔ iIndep (fun x ↦ (m x).comap (f x)) μ := by
+  simp only [iIndepFun, iIndep, kernel.iIndepFun]
+
+lemma iIndepFun_iff [MeasurableSpace Ω] {β : ι → Type _}
+    (m : ∀ x : ι, MeasurableSpace (β x)) (f : ∀ x : ι, Ω → β x) (μ : Measure Ω) :
+    iIndepFun m f μ ↔ ∀ (s : Finset ι) {f' : ι → Set Ω}
+      (_H : ∀ i, i ∈ s → MeasurableSet[(m i).comap (f i)] (f' i)),
+      μ (⋂ i ∈ s, f' i) = ∏ i in s, μ (f' i) := by
+  simp only [iIndepFun_iff_iIndep, iIndep_iff]
+
+lemma IndepFun_iff_Indep [MeasurableSpace Ω] [mβ : MeasurableSpace β]
+    [mγ : MeasurableSpace γ] (f : Ω → β) (g : Ω → γ) (μ : Measure Ω) :
+    IndepFun f g μ ↔ Indep (MeasurableSpace.comap f mβ) (MeasurableSpace.comap g mγ) μ := by
+  simp only [IndepFun, Indep, kernel.IndepFun]
+
+lemma IndepFun_iff {β γ} [MeasurableSpace Ω] [mβ : MeasurableSpace β] [mγ : MeasurableSpace γ]
+    (f : Ω → β) (g : Ω → γ) (μ : Measure Ω) :
+    IndepFun f g μ ↔ ∀ t1 t2, MeasurableSet[MeasurableSpace.comap f mβ] t1
+      → MeasurableSet[MeasurableSpace.comap g mγ] t2 → μ (t1 ∩ t2) = μ t1 * μ t2 := by
+  rw [IndepFun_iff_Indep, Indep_iff]
+
+end Definition_lemmas
+
 section Indep
 
 @[symm]
 theorem IndepSets.symm {s₁ s₂ : Set (Set Ω)} [MeasurableSpace Ω] {μ : Measure Ω}
-    (h : IndepSets s₁ s₂ μ) : IndepSets s₂ s₁ μ := by
-  intro t1 t2 ht1 ht2
-  rw [Set.inter_comm, mul_comm]; exact h t2 t1 ht2 ht1
+    (h : IndepSets s₁ s₂ μ) : IndepSets s₂ s₁ μ :=
+  kernel.IndepSets.symm h
 #align probability_theory.indep_sets.symm ProbabilityTheory.IndepSets.symm
 
 @[symm]
@@ -153,105 +228,94 @@ theorem Indep.symm {m₁ m₂ : MeasurableSpace Ω} [MeasurableSpace Ω] {μ : M
     (h : Indep m₁ m₂ μ) : Indep m₂ m₁ μ := IndepSets.symm h
 #align probability_theory.indep.symm ProbabilityTheory.Indep.symm
 
-theorem indep_bot_right (m' : MeasurableSpace Ω) {m : MeasurableSpace Ω} {μ : Measure Ω}
-    [IsProbabilityMeasure μ] : Indep m' ⊥ μ := by
-  intro s t _hs ht
-  rw [Set.mem_setOf_eq, MeasurableSpace.measurableSet_bot_iff] at ht
-  cases' ht with ht ht
-  · rw [ht, Set.inter_empty, measure_empty, MulZeroClass.mul_zero]
-  · rw [ht, Set.inter_univ, measure_univ, mul_one]
+theorem indep_bot_right (m' : MeasurableSpace Ω) {_m : MeasurableSpace Ω} {μ : Measure Ω}
+    [IsProbabilityMeasure μ] : Indep m' ⊥ μ :=
+  kernel.indep_bot_right m'
 #align probability_theory.indep_bot_right ProbabilityTheory.indep_bot_right
 
 theorem indep_bot_left (m' : MeasurableSpace Ω) {_m : MeasurableSpace Ω} {μ : Measure Ω}
     [IsProbabilityMeasure μ] : Indep ⊥ m' μ := (indep_bot_right m').symm
 #align probability_theory.indep_bot_left ProbabilityTheory.indep_bot_left
 
-theorem indepSet_empty_right {m : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
-    (s : Set Ω) : IndepSet s ∅ μ := by
-  simp only [IndepSet, generateFrom_singleton_empty];
-  exact indep_bot_right _
+theorem indepSet_empty_right {_m : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
+    (s : Set Ω) : IndepSet s ∅ μ :=
+  kernel.indepSet_empty_right s
 #align probability_theory.indep_set_empty_right ProbabilityTheory.indepSet_empty_right
 
 theorem indepSet_empty_left {_m : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
-    (s : Set Ω) : IndepSet ∅ s μ := (indepSet_empty_right s).symm
+    (s : Set Ω) : IndepSet ∅ s μ :=
+  kernel.indepSet_empty_left s
 #align probability_theory.indep_set_empty_left ProbabilityTheory.indepSet_empty_left
 
 theorem indepSets_of_indepSets_of_le_left {s₁ s₂ s₃ : Set (Set Ω)} [MeasurableSpace Ω]
-    {μ : Measure Ω} (h_indep : IndepSets s₁ s₂ μ) (h31 : s₃ ⊆ s₁) : IndepSets s₃ s₂ μ :=
-  fun t1 t2 ht1 ht2 => h_indep t1 t2 (Set.mem_of_subset_of_mem h31 ht1) ht2
+    {μ : Measure Ω} (h_indep : IndepSets s₁ s₂ μ) (h31 : s₃ ⊆ s₁) :
+    IndepSets s₃ s₂ μ :=
+  kernel.indepSets_of_indepSets_of_le_left h_indep h31
 #align probability_theory.indep_sets_of_indep_sets_of_le_left ProbabilityTheory.indepSets_of_indepSets_of_le_left
 
 theorem indepSets_of_indepSets_of_le_right {s₁ s₂ s₃ : Set (Set Ω)} [MeasurableSpace Ω]
-    {μ : Measure Ω} (h_indep : IndepSets s₁ s₂ μ) (h32 : s₃ ⊆ s₂) : IndepSets s₁ s₃ μ :=
-  fun t1 t2 ht1 ht2 => h_indep t1 t2 ht1 (Set.mem_of_subset_of_mem h32 ht2)
+    {μ : Measure Ω} (h_indep : IndepSets s₁ s₂ μ) (h32 : s₃ ⊆ s₂) :
+    IndepSets s₁ s₃ μ :=
+  kernel.indepSets_of_indepSets_of_le_right h_indep h32
 #align probability_theory.indep_sets_of_indep_sets_of_le_right ProbabilityTheory.indepSets_of_indepSets_of_le_right
 
 theorem indep_of_indep_of_le_left {m₁ m₂ m₃ : MeasurableSpace Ω} [MeasurableSpace Ω]
-    {μ : Measure Ω} (h_indep : Indep m₁ m₂ μ) (h31 : m₃ ≤ m₁) : Indep m₃ m₂ μ :=
-  fun t1 t2 ht1 ht2 => h_indep t1 t2 (h31 _ ht1) ht2
+    {μ : Measure Ω} (h_indep : Indep m₁ m₂ μ) (h31 : m₃ ≤ m₁) :
+    Indep m₃ m₂ μ :=
+  kernel.indep_of_indep_of_le_left h_indep h31
 #align probability_theory.indep_of_indep_of_le_left ProbabilityTheory.indep_of_indep_of_le_left
 
 theorem indep_of_indep_of_le_right {m₁ m₂ m₃ : MeasurableSpace Ω} [MeasurableSpace Ω]
-    {μ : Measure Ω} (h_indep : Indep m₁ m₂ μ) (h32 : m₃ ≤ m₂) : Indep m₁ m₃ μ :=
-  fun t1 t2 ht1 ht2 => h_indep t1 t2 ht1 (h32 _ ht2)
+    {μ : Measure Ω} (h_indep : Indep m₁ m₂ μ) (h32 : m₃ ≤ m₂) :
+    Indep m₁ m₃ μ :=
+  kernel.indep_of_indep_of_le_right h_indep h32
 #align probability_theory.indep_of_indep_of_le_right ProbabilityTheory.indep_of_indep_of_le_right
 
 theorem IndepSets.union [MeasurableSpace Ω] {s₁ s₂ s' : Set (Set Ω)} {μ : Measure Ω}
-    (h₁ : IndepSets s₁ s' μ) (h₂ : IndepSets s₂ s' μ) : IndepSets (s₁ ∪ s₂) s' μ := by
-  intro t1 t2 ht1 ht2
-  cases' (Set.mem_union _ _ _).mp ht1 with ht1₁ ht1₂
-  · exact h₁ t1 t2 ht1₁ ht2
-  · exact h₂ t1 t2 ht1₂ ht2
+    (h₁ : IndepSets s₁ s' μ) (h₂ : IndepSets s₂ s' μ) :
+    IndepSets (s₁ ∪ s₂) s' μ :=
+  kernel.IndepSets.union h₁ h₂
 #align probability_theory.indep_sets.union ProbabilityTheory.IndepSets.union
 
 @[simp]
 theorem IndepSets.union_iff [MeasurableSpace Ω] {s₁ s₂ s' : Set (Set Ω)} {μ : Measure Ω} :
     IndepSets (s₁ ∪ s₂) s' μ ↔ IndepSets s₁ s' μ ∧ IndepSets s₂ s' μ :=
-  ⟨fun h =>
-    ⟨indepSets_of_indepSets_of_le_left h (Set.subset_union_left s₁ s₂),
-      indepSets_of_indepSets_of_le_left h (Set.subset_union_right s₁ s₂)⟩,
-    fun h => IndepSets.union h.left h.right⟩
+  kernel.IndepSets.union_iff
 #align probability_theory.indep_sets.union_iff ProbabilityTheory.IndepSets.union_iff
 
 theorem IndepSets.iUnion [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
-    {μ : Measure Ω} (hyp : ∀ n, IndepSets (s n) s' μ) : IndepSets (⋃ n, s n) s' μ := by
-  intro t1 t2 ht1 ht2
-  rw [Set.mem_iUnion] at ht1
-  cases' ht1 with n ht1
-  exact hyp n t1 t2 ht1 ht2
+    {μ : Measure Ω} (hyp : ∀ n, IndepSets (s n) s' μ) :
+    IndepSets (⋃ n, s n) s' μ :=
+  kernel.IndepSets.iUnion hyp
 #align probability_theory.indep_sets.Union ProbabilityTheory.IndepSets.iUnion
 
 theorem IndepSets.bUnion [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
     {μ : Measure Ω} {u : Set ι} (hyp : ∀ n ∈ u, IndepSets (s n) s' μ) :
-    IndepSets (⋃ n ∈ u, s n) s' μ := by
-  intro t1 t2 ht1 ht2
-  simp_rw [Set.mem_iUnion] at ht1
-  rcases ht1 with ⟨n, hpn, ht1⟩
-  exact hyp n hpn t1 t2 ht1 ht2
+    IndepSets (⋃ n ∈ u, s n) s' μ :=
+  kernel.IndepSets.bUnion hyp
 #align probability_theory.indep_sets.bUnion ProbabilityTheory.IndepSets.bUnion
 
 theorem IndepSets.inter [MeasurableSpace Ω] {s₁ s' : Set (Set Ω)} (s₂ : Set (Set Ω))
     {μ : Measure Ω} (h₁ : IndepSets s₁ s' μ) : IndepSets (s₁ ∩ s₂) s' μ :=
-  fun t1 t2 ht1 ht2 => h₁ t1 t2 ((Set.mem_inter_iff _ _ _).mp ht1).left ht2
+  kernel.IndepSets.inter s₂ h₁
 #align probability_theory.indep_sets.inter ProbabilityTheory.IndepSets.inter
 
 theorem IndepSets.iInter [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
-    {μ : Measure Ω} (h : ∃ n, IndepSets (s n) s' μ) : IndepSets (⋂ n, s n) s' μ := by
-  intro t1 t2 ht1 ht2; cases' h with n h; exact h t1 t2 (Set.mem_iInter.mp ht1 n) ht2
+    {μ : Measure Ω} (h : ∃ n, IndepSets (s n) s' μ) :
+    IndepSets (⋂ n, s n) s' μ :=
+  kernel.IndepSets.iInter h
 #align probability_theory.indep_sets.Inter ProbabilityTheory.IndepSets.iInter
 
 theorem IndepSets.bInter [MeasurableSpace Ω] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
     {μ : Measure Ω} {u : Set ι} (h : ∃ n ∈ u, IndepSets (s n) s' μ) :
-    IndepSets (⋂ n ∈ u, s n) s' μ := by
-  intro t1 t2 ht1 ht2
-  rcases h with ⟨n, hn, h⟩
-  exact h t1 t2 (Set.biInter_subset_of_mem hn ht1) ht2
+    IndepSets (⋂ n ∈ u, s n) s' μ :=
+  kernel.IndepSets.bInter h
 #align probability_theory.indep_sets.bInter ProbabilityTheory.IndepSets.bInter
 
 theorem indepSets_singleton_iff [MeasurableSpace Ω] {s t : Set Ω} {μ : Measure Ω} :
-    IndepSets {s} {t} μ ↔ μ (s ∩ t) = μ s * μ t :=
-  ⟨fun h => h s t rfl rfl, fun h s1 t1 hs1 ht1 => by
-    rwa [Set.mem_singleton_iff.mp hs1, Set.mem_singleton_iff.mp ht1]⟩
+    IndepSets {s} {t} μ ↔ μ (s ∩ t) = μ s * μ t := by
+  simp only [IndepSets, kernel.indepSets_singleton_iff, ae_dirac_eq, Filter.eventually_pure,
+    kernel.const_apply]
 #align probability_theory.indep_sets_singleton_iff ProbabilityTheory.indepSets_singleton_iff
 
 end Indep
@@ -262,43 +326,22 @@ end Indep
 section FromIndepToIndep
 
 theorem iIndepSets.indepSets {s : ι → Set (Set Ω)} [MeasurableSpace Ω] {μ : Measure Ω}
-    (h_indep : iIndepSets s μ) {i j : ι} (hij : i ≠ j) : IndepSets (s i) (s j) μ := by
-  classical
-    intro t₁ t₂ ht₁ ht₂
-    have hf_m : ∀ x : ι, x ∈ ({i, j} : Finset ι) → ite (x = i) t₁ t₂ ∈ s x := by
-      intro x hx
-      cases' Finset.mem_insert.mp hx with hx hx
-      · simp [hx, ht₁]
-      · simp [Finset.mem_singleton.mp hx, hij.symm, ht₂]
-    have h1 : t₁ = ite (i = i) t₁ t₂ := by simp only [if_true, eq_self_iff_true]
-    have h2 : t₂ = ite (j = i) t₁ t₂ := by simp only [hij.symm, if_false]
-    have h_inter :
-      ⋂ (t : ι) (_ : t ∈ ({i, j} : Finset ι)), ite (t = i) t₁ t₂ =
-        ite (i = i) t₁ t₂ ∩ ite (j = i) t₁ t₂ :=
-      by simp only [Finset.set_biInter_singleton, Finset.set_biInter_insert]
-    have h_prod :
-      (∏ t : ι in ({i, j} : Finset ι), μ (ite (t = i) t₁ t₂)) =
-        μ (ite (i = i) t₁ t₂) * μ (ite (j = i) t₁ t₂) := by
-      simp only [hij, Finset.prod_singleton, Finset.prod_insert, not_false_iff,
-        Finset.mem_singleton]
-    rw [h1]
-    nth_rw 2 [h2]
-    nth_rw 4 [h2]
-    rw [← h_inter, ← h_prod, h_indep {i, j} hf_m]
+    (h_indep : iIndepSets s μ) {i j : ι} (hij : i ≠ j) : IndepSets (s i) (s j) μ :=
+  kernel.iIndepSets.indepSets h_indep hij
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_sets.indep_sets ProbabilityTheory.iIndepSets.indepSets
 
 theorem iIndep.indep {m : ι → MeasurableSpace Ω} [MeasurableSpace Ω] {μ : Measure Ω}
-    (h_indep : iIndep m μ) {i j : ι} (hij : i ≠ j) : Indep (m i) (m j) μ := by
-  change IndepSets ((fun x => MeasurableSet[m x]) i) ((fun x => MeasurableSet[m x]) j) μ
-  exact iIndepSets.indepSets h_indep hij
+    (h_indep : iIndep m μ) {i j : ι} (hij : i ≠ j) : Indep (m i) (m j) μ :=
+  kernel.iIndep.indep h_indep hij
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep.indep ProbabilityTheory.iIndep.indep
 
 theorem iIndepFun.indepFun {_m₀ : MeasurableSpace Ω} {μ : Measure Ω} {β : ι → Type _}
     {m : ∀ x, MeasurableSpace (β x)} {f : ∀ i, Ω → β i} (hf_Indep : iIndepFun m f μ) {i j : ι}
-    (hij : i ≠ j) : IndepFun (f i) (f j) μ :=
-  hf_Indep.indep hij
+    (hij : i ≠ j) :
+    IndepFun (f i) (f j) μ :=
+  kernel.iIndepFun.indepFun hf_Indep hij
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun.indep_fun ProbabilityTheory.iIndepFun.indepFun
 
@@ -318,16 +361,15 @@ section FromMeasurableSpacesToSetsOfSets
 
 theorem iIndep.iIndepSets [MeasurableSpace Ω] {μ : Measure Ω} {m : ι → MeasurableSpace Ω}
     {s : ι → Set (Set Ω)} (hms : ∀ n, m n = generateFrom (s n)) (h_indep : iIndep m μ) :
-    iIndepSets s μ := fun S f hfs =>
-  h_indep S fun x hxS =>
-    ((hms x).symm ▸ measurableSet_generateFrom (hfs x hxS) : MeasurableSet[m x] (f x))
+    iIndepSets s μ :=
+  kernel.iIndep.iIndepSets hms h_indep
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep.Indep_sets ProbabilityTheory.iIndep.iIndepSets
 
 theorem Indep.indepSets [MeasurableSpace Ω] {μ : Measure Ω} {s1 s2 : Set (Set Ω)}
-    (h_indep : Indep (generateFrom s1) (generateFrom s2) μ) : IndepSets s1 s2 μ :=
-  fun t1 t2 ht1 ht2 =>
-  h_indep t1 t2 (measurableSet_generateFrom ht1) (measurableSet_generateFrom ht2)
+    (h_indep : Indep (generateFrom s1) (generateFrom s2) μ) :
+    IndepSets s1 s2 μ :=
+  kernel.Indep.indepSets h_indep
 #align probability_theory.indep.indep_sets ProbabilityTheory.Indep.indepSets
 
 end FromMeasurableSpacesToSetsOfSets
@@ -336,246 +378,97 @@ section FromPiSystemsToMeasurableSpaces
 
 /-! ### Independence of generating π-systems implies independence of measurable space structures -/
 
-
-private theorem IndepSets.indep_aux {m2 : MeasurableSpace Ω} {m : MeasurableSpace Ω}
-    {μ : Measure Ω} [IsProbabilityMeasure μ] {p1 p2 : Set (Set Ω)} (h2 : m2 ≤ m)
-    (hp2 : IsPiSystem p2) (hpm2 : m2 = generateFrom p2) (hyp : IndepSets p1 p2 μ) {t1 t2 : Set Ω}
-    (ht1 : t1 ∈ p1) (ht2m : MeasurableSet[m2] t2) : μ (t1 ∩ t2) = μ t1 * μ t2 := by
-  let μ_inter := μ.restrict t1
-  let ν := μ t1 • μ
-  have h_univ : μ_inter Set.univ = ν Set.univ := by
-    rw [Measure.restrict_apply_univ, Measure.smul_apply, smul_eq_mul, measure_univ, mul_one]
-  haveI : IsFiniteMeasure μ_inter := @Restrict.isFiniteMeasure Ω _ t1 μ ⟨measure_lt_top μ t1⟩
-  rw [Set.inter_comm, ← Measure.restrict_apply (h2 t2 ht2m)]
-  refine' ext_on_measurableSpace_of_generate_finite m p2 (fun t ht => _) h2 hpm2 hp2 h_univ ht2m
-  have ht2 : MeasurableSet[m] t := by
-    refine' h2 _ _
-    rw [hpm2]
-    exact measurableSet_generateFrom ht
-  rw [Measure.restrict_apply ht2, Measure.smul_apply, Set.inter_comm]
-  exact hyp t1 t ht1 ht
-
 theorem IndepSets.indep {m1 m2 : MeasurableSpace Ω} {m : MeasurableSpace Ω} {μ : Measure Ω}
     [IsProbabilityMeasure μ] {p1 p2 : Set (Set Ω)} (h1 : m1 ≤ m) (h2 : m2 ≤ m) (hp1 : IsPiSystem p1)
     (hp2 : IsPiSystem p2) (hpm1 : m1 = generateFrom p1) (hpm2 : m2 = generateFrom p2)
-    (hyp : IndepSets p1 p2 μ) : Indep m1 m2 μ := by
-  intro t1 t2 ht1 ht2
-  let μ_inter := μ.restrict t2
-  let ν := μ t2 • μ
-  have h_univ : μ_inter Set.univ = ν Set.univ := by
-    rw [Measure.restrict_apply_univ, Measure.smul_apply, smul_eq_mul, measure_univ, mul_one]
-  haveI : IsFiniteMeasure μ_inter := @Restrict.isFiniteMeasure Ω _ t2 μ ⟨measure_lt_top μ t2⟩
-  rw [mul_comm, ← Measure.restrict_apply (h1 t1 ht1)]
-  refine' ext_on_measurableSpace_of_generate_finite m p1 (fun t ht => _) h1 hpm1 hp1 h_univ ht1
-  have ht1 : MeasurableSet[m] t := by
-    refine' h1 _ _
-    rw [hpm1]
-    exact measurableSet_generateFrom ht
-  rw [Measure.restrict_apply ht1, Measure.smul_apply, smul_eq_mul, mul_comm]
-  exact IndepSets.indep_aux h2 hp2 hpm2 hyp ht ht2
+    (hyp : IndepSets p1 p2 μ) :
+    Indep m1 m2 μ :=
+  kernel.IndepSets.indep h1 h2 hp1 hp2 hpm1 hpm2 hyp
 #align probability_theory.indep_sets.indep ProbabilityTheory.IndepSets.indep
 
 theorem IndepSets.indep' {_m : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
     {p1 p2 : Set (Set Ω)} (hp1m : ∀ s ∈ p1, MeasurableSet s) (hp2m : ∀ s ∈ p2, MeasurableSet s)
     (hp1 : IsPiSystem p1) (hp2 : IsPiSystem p2) (hyp : IndepSets p1 p2 μ) :
     Indep (generateFrom p1) (generateFrom p2) μ :=
-  hyp.indep (generateFrom_le hp1m) (generateFrom_le hp2m) hp1 hp2 rfl rfl
+  kernel.IndepSets.indep' hp1m hp2m hp1 hp2 hyp
 #align probability_theory.indep_sets.indep' ProbabilityTheory.IndepSets.indep'
 
 variable {m0 : MeasurableSpace Ω} {μ : Measure Ω}
 
 theorem indepSets_piiUnionInter_of_disjoint [IsProbabilityMeasure μ] {s : ι → Set (Set Ω)}
     {S T : Set ι} (h_indep : iIndepSets s μ) (hST : Disjoint S T) :
-    IndepSets (piiUnionInter s S) (piiUnionInter s T) μ := by
-  rintro t1 t2 ⟨p1, hp1, f1, ht1_m, ht1_eq⟩ ⟨p2, hp2, f2, ht2_m, ht2_eq⟩
-  classical
-    let g i := ite (i ∈ p1) (f1 i) Set.univ ∩ ite (i ∈ p2) (f2 i) Set.univ
-    have h_P_inter : μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, μ (g n) := by
-      have hgm : ∀ i ∈ p1 ∪ p2, g i ∈ s i := by
-        intro i hi_mem_union
-        rw [Finset.mem_union] at hi_mem_union
-        cases' hi_mem_union with hi1 hi2
-        · have hi2 : i ∉ p2 := fun hip2 => Set.disjoint_left.mp hST (hp1 hi1) (hp2 hip2)
-          simp_rw [if_pos hi1, if_neg hi2, Set.inter_univ]
-          exact ht1_m i hi1
-        · have hi1 : i ∉ p1 := fun hip1 => Set.disjoint_right.mp hST (hp2 hi2) (hp1 hip1)
-          simp_rw [if_neg hi1, if_pos hi2, Set.univ_inter]
-          exact ht2_m i hi2
-      have h_p1_inter_p2 :
-        ((⋂ x ∈ p1, f1 x) ∩ ⋂ x ∈ p2, f2 x) =
-          ⋂ i ∈ p1 ∪ p2, ite (i ∈ p1) (f1 i) Set.univ ∩ ite (i ∈ p2) (f2 i) Set.univ := by
-        ext1 x
-        simp only [Set.mem_ite_univ_right, Set.mem_inter_iff, Set.mem_iInter, Finset.mem_union]
-        exact
-          ⟨fun h i _ => ⟨h.1 i, h.2 i⟩, fun h =>
-            ⟨fun i hi => (h i (Or.inl hi)).1 hi, fun i hi => (h i (Or.inr hi)).2 hi⟩⟩
-      rw [ht1_eq, ht2_eq, h_p1_inter_p2, ← h_indep _ hgm]
-    have h_μg : ∀ n, μ (g n) = ite (n ∈ p1) (μ (f1 n)) 1 * ite (n ∈ p2) (μ (f2 n)) 1 := by
-      intro n
-      dsimp only
-      split_ifs with h1 h2
-      · exact absurd rfl (Set.disjoint_iff_forall_ne.mp hST (hp1 h1) (hp2 h2))
-      all_goals simp only [measure_univ, one_mul, mul_one, Set.inter_univ, Set.univ_inter]
-    simp_rw [h_P_inter, h_μg, Finset.prod_mul_distrib,
-      Finset.prod_ite_mem (p1 ∪ p2) p1 fun x => μ (f1 x), Finset.union_inter_cancel_left,
-      Finset.prod_ite_mem (p1 ∪ p2) p2 fun x => μ (f2 x), Finset.union_inter_cancel_right, ht1_eq, ←
-      h_indep p1 ht1_m, ht2_eq, ← h_indep p2 ht2_m]
+    IndepSets (piiUnionInter s S) (piiUnionInter s T) μ :=
+  kernel.indepSets_piiUnionInter_of_disjoint h_indep hST
 #align probability_theory.indep_sets_pi_Union_Inter_of_disjoint ProbabilityTheory.indepSets_piiUnionInter_of_disjoint
 
 theorem iIndepSet.indep_generateFrom_of_disjoint [IsProbabilityMeasure μ] {s : ι → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (S T : Set ι) (hST : Disjoint S T) :
-    Indep (generateFrom { t | ∃ n ∈ S, s n = t }) (generateFrom { t | ∃ k ∈ T, s k = t }) μ := by
-  rw [← generateFrom_piiUnionInter_singleton_left, ← generateFrom_piiUnionInter_singleton_left]
-  refine'
-    IndepSets.indep'
-      (fun t ht => generateFrom_piiUnionInter_le _ _ _ _ (measurableSet_generateFrom ht))
-      (fun t ht => generateFrom_piiUnionInter_le _ _ _ _ (measurableSet_generateFrom ht)) _ _ _
-  · exact fun k => generateFrom_le fun t ht => (Set.mem_singleton_iff.1 ht).symm ▸ hsm k
-  · exact fun k => generateFrom_le fun t ht => (Set.mem_singleton_iff.1 ht).symm ▸ hsm k
-  · exact isPiSystem_piiUnionInter _ (fun k => IsPiSystem.singleton _) _
-  · exact isPiSystem_piiUnionInter _ (fun k => IsPiSystem.singleton _) _
-  · classical exact indepSets_piiUnionInter_of_disjoint (iIndep.iIndepSets (fun n => rfl) hs) hST
+    Indep (generateFrom { t | ∃ n ∈ S, s n = t }) (generateFrom { t | ∃ k ∈ T, s k = t }) μ :=
+  kernel.iIndepSet.indep_generateFrom_of_disjoint hsm hs S T hST
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_set.indep_generate_from_of_disjoint ProbabilityTheory.iIndepSet.indep_generateFrom_of_disjoint
 
 theorem indep_iSup_of_disjoint [IsProbabilityMeasure μ] {m : ι → MeasurableSpace Ω}
     (h_le : ∀ i, m i ≤ m0) (h_indep : iIndep m μ) {S T : Set ι} (hST : Disjoint S T) :
-    Indep (⨆ i ∈ S, m i) (⨆ i ∈ T, m i) μ := by
-  refine'
-    IndepSets.indep (iSup₂_le fun i _ => h_le i) (iSup₂_le fun i _ => h_le i) _ _
-      (generateFrom_piiUnionInter_measurableSet m S).symm
-      (generateFrom_piiUnionInter_measurableSet m T).symm _
-  · exact isPiSystem_piiUnionInter _ (fun n => @isPiSystem_measurableSet Ω (m n)) _
-  · exact isPiSystem_piiUnionInter _ (fun n => @isPiSystem_measurableSet Ω (m n)) _
-  · classical exact indepSets_piiUnionInter_of_disjoint h_indep hST
+    Indep (⨆ i ∈ S, m i) (⨆ i ∈ T, m i) μ :=
+  kernel.indep_iSup_of_disjoint h_le h_indep hST
 #align probability_theory.indep_supr_of_disjoint ProbabilityTheory.indep_iSup_of_disjoint
 
 theorem indep_iSup_of_directed_le {Ω} {m : ι → MeasurableSpace Ω} {m' m0 : MeasurableSpace Ω}
     {μ : Measure Ω} [IsProbabilityMeasure μ] (h_indep : ∀ i, Indep (m i) m' μ)
     (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0) (hm : Directed (· ≤ ·) m) :
-    Indep (⨆ i, m i) m' μ := by
-  let p : ι → Set (Set Ω) := fun n => { t | MeasurableSet[m n] t }
-  have hp : ∀ n, IsPiSystem (p n) := fun n => @isPiSystem_measurableSet Ω (m n)
-  have h_gen_n : ∀ n, m n = generateFrom (p n) := fun n =>
-    (@generateFrom_measurableSet Ω (m n)).symm
-  have hp_supr_pi : IsPiSystem (⋃ n, p n) := isPiSystem_iUnion_of_directed_le p hp hm
-  let p' := { t : Set Ω | MeasurableSet[m'] t }
-  have hp'_pi : IsPiSystem p' := @isPiSystem_measurableSet Ω m'
-  have h_gen' : m' = generateFrom p' := (@generateFrom_measurableSet Ω m').symm
-  -- the π-systems defined are independent
-  have h_pi_system_indep : IndepSets (⋃ n, p n) p' μ := by
-    refine IndepSets.iUnion ?_
-    conv at h_indep =>
-      intro i
-      rw [h_gen_n i, h_gen']
-    exact fun n => (h_indep n).indepSets
-  -- now go from π-systems to σ-algebras
-  refine' IndepSets.indep (iSup_le h_le) h_le' hp_supr_pi hp'_pi _ h_gen' h_pi_system_indep
-  exact (generateFrom_iUnion_measurableSet _).symm
+    Indep (⨆ i, m i) m' μ :=
+  kernel.indep_iSup_of_directed_le h_indep h_le h_le' hm
 #align probability_theory.indep_supr_of_directed_le ProbabilityTheory.indep_iSup_of_directed_le
 
 theorem iIndepSet.indep_generateFrom_lt [Preorder ι] [IsProbabilityMeasure μ] {s : ι → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (i : ι) :
-    Indep (generateFrom {s i}) (generateFrom { t | ∃ j < i, s j = t }) μ := by
-  convert iIndepSet.indep_generateFrom_of_disjoint hsm hs {i} { j | j < i }
-    (Set.disjoint_singleton_left.mpr (lt_irrefl _))
-  simp only [Set.mem_singleton_iff, exists_prop, exists_eq_left, Set.setOf_eq_eq_singleton']
+    Indep (generateFrom {s i}) (generateFrom { t | ∃ j < i, s j = t }) μ :=
+  kernel.iIndepSet.indep_generateFrom_lt hsm hs i
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_set.indep_generate_from_lt ProbabilityTheory.iIndepSet.indep_generateFrom_lt
 
 theorem iIndepSet.indep_generateFrom_le [LinearOrder ι] [IsProbabilityMeasure μ] {s : ι → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (i : ι) {k : ι} (hk : i < k) :
-    Indep (generateFrom {s k}) (generateFrom { t | ∃ j ≤ i, s j = t }) μ := by
-  convert iIndepSet.indep_generateFrom_of_disjoint hsm hs {k} { j | j ≤ i }
-      (Set.disjoint_singleton_left.mpr hk.not_le)
-  simp only [Set.mem_singleton_iff, exists_prop, exists_eq_left, Set.setOf_eq_eq_singleton']
+    Indep (generateFrom {s k}) (generateFrom { t | ∃ j ≤ i, s j = t }) μ :=
+  kernel.iIndepSet.indep_generateFrom_le hsm hs i hk
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_set.indep_generate_from_le ProbabilityTheory.iIndepSet.indep_generateFrom_le
 
 theorem iIndepSet.indep_generateFrom_le_nat [IsProbabilityMeasure μ] {s : ℕ → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (n : ℕ) :
     Indep (generateFrom {s (n + 1)}) (generateFrom { t | ∃ k ≤ n, s k = t }) μ :=
-  iIndepSet.indep_generateFrom_le hsm hs _ n.lt_succ_self
+  kernel.iIndepSet.indep_generateFrom_le_nat hsm hs n
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_set.indep_generate_from_le_nat ProbabilityTheory.iIndepSet.indep_generateFrom_le_nat
 
 theorem indep_iSup_of_monotone [SemilatticeSup ι] {Ω} {m : ι → MeasurableSpace Ω}
     {m' m0 : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
-    (h_indep : ∀ i, Indep (m i) m' μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0)
-    (hm : Monotone m) : Indep (⨆ i, m i) m' μ :=
-  indep_iSup_of_directed_le h_indep h_le h_le' (Monotone.directed_le hm)
+    (h_indep : ∀ i, Indep (m i) m' μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0) (hm : Monotone m) :
+    Indep (⨆ i, m i) m' μ :=
+  kernel.indep_iSup_of_monotone h_indep h_le h_le' hm
 #align probability_theory.indep_supr_of_monotone ProbabilityTheory.indep_iSup_of_monotone
 
 theorem indep_iSup_of_antitone [SemilatticeInf ι] {Ω} {m : ι → MeasurableSpace Ω}
     {m' m0 : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
-    (h_indep : ∀ i, Indep (m i) m' μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0)
-    (hm : Antitone m) : Indep (⨆ i, m i) m' μ :=
-  indep_iSup_of_directed_le h_indep h_le h_le' (directed_of_inf hm)
+    (h_indep : ∀ i, Indep (m i) m' μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0) (hm : Antitone m) :
+    Indep (⨆ i, m i) m' μ :=
+  kernel.indep_iSup_of_antitone h_indep h_le h_le' hm
 #align probability_theory.indep_supr_of_antitone ProbabilityTheory.indep_iSup_of_antitone
 
 theorem iIndepSets.piiUnionInter_of_not_mem {π : ι → Set (Set Ω)} {a : ι} {S : Finset ι}
-    (hp_ind : iIndepSets π μ) (haS : a ∉ S) : IndepSets (piiUnionInter π S) (π a) μ := by
-  rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
-  rw [Finset.coe_subset] at hs_mem
-  classical
-    let f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
-    have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n := by
-      intro n hn_mem_insert
-      dsimp only
-      cases' Finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
-      · simp [hn_mem, ht2_mem_pia]
-      · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
-        simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
-    have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
-    have h_t1 : t1 = ⋂ n ∈ s, f n := by
-      suffices h_forall : ∀ n ∈ s, f n = ft1 n
-      · rw [ht1_eq]
-        ext x
-        simp_rw [Set.mem_iInter]
-        conv => lhs; intro i hns; rw [← h_forall i hns]
-      intro n hnS
-      have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hnS)
-      simp_rw [if_pos hnS, if_neg hn_ne_a]
-    have h_μ_t1 : μ t1 = ∏ n in s, μ (f n) := by rw [h_t1, ← hp_ind s h_f_mem_pi]
-    have h_t2 : t2 = f a := by simp
-    have h_μ_inter : μ (t1 ∩ t2) = ∏ n in insert a s, μ (f n) := by
-      have h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n := by
-        rw [h_t1, h_t2, Finset.set_biInter_insert, Set.inter_comm]
-      rw [h_t1_inter_t2, ← hp_ind (insert a s) h_f_mem]
-    have has : a ∉ s := fun has_mem => haS (hs_mem has_mem)
-    rw [h_μ_inter, Finset.prod_insert has, h_t2, mul_comm, h_μ_t1]
+    (hp_ind : iIndepSets π μ) (haS : a ∉ S) :
+    IndepSets (piiUnionInter π S) (π a) μ :=
+  kernel.iIndepSets.piiUnionInter_of_not_mem hp_ind haS
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_sets.pi_Union_Inter_of_not_mem ProbabilityTheory.iIndepSets.piiUnionInter_of_not_mem
 
 /-- The measurable space structures generated by independent pi-systems are independent. -/
 theorem iIndepSets.iIndep [IsProbabilityMeasure μ] (m : ι → MeasurableSpace Ω)
     (h_le : ∀ i, m i ≤ m0) (π : ι → Set (Set Ω)) (h_pi : ∀ n, IsPiSystem (π n))
-    (h_generate : ∀ i, m i = generateFrom (π i)) (h_ind : iIndepSets π μ) : iIndep m μ := by
-  classical
-    intro s f
-    refine Finset.induction ?_ ?_ s
-    · simp only [Finset.not_mem_empty, Set.mem_setOf_eq, IsEmpty.forall_iff, implies_true,
-      Set.iInter_of_empty, Set.iInter_univ, measure_univ, Finset.prod_empty, forall_true_left]
-    · intro a S ha_notin_S h_rec hf_m
-      have hf_m_S : ∀ x ∈ S, MeasurableSet[m x] (f x) := fun x hx => hf_m x (by simp [hx])
-      rw [Finset.set_biInter_insert, Finset.prod_insert ha_notin_S, ← h_rec hf_m_S]
-      let p := piiUnionInter π S
-      set m_p := generateFrom p with hS_eq_generate
-      have h_indep : @Indep Ω m_p (m a) m0 μ := by
-        have hp : IsPiSystem p := isPiSystem_piiUnionInter π h_pi S
-        have h_le' : ∀ i, generateFrom (π i) ≤ m0 := fun i => (h_generate i).symm.trans_le (h_le i)
-        have hm_p : m_p ≤ m0 := generateFrom_piiUnionInter_le π h_le' S
-        exact IndepSets.indep hm_p (h_le a) hp (h_pi a) hS_eq_generate (h_generate a)
-            (iIndepSets.piiUnionInter_of_not_mem h_ind ha_notin_S)
-      refine @Indep.symm Ω _ _ m0 μ h_indep (f a) (⋂ n ∈ S, f n)
-        (hf_m a (Finset.mem_insert_self a S)) ?_
-      have h_le_p : ∀ i ∈ S, m i ≤ m_p := by
-        intro n hn
-        rw [hS_eq_generate, h_generate n]
-        exact le_generateFrom_piiUnionInter _ hn
-      have h_S_f : ∀ i ∈ S, MeasurableSet[m_p] (f i) :=
-        fun i hi => (h_le_p i hi) (f i) (hf_m_S i hi)
-      exact S.measurableSet_biInter h_S_f
+    (h_generate : ∀ i, m i = generateFrom (π i)) (h_ind : iIndepSets π μ) :
+    iIndep m μ :=
+  kernel.iIndepSets.iIndep m h_le π h_pi h_generate h_ind
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_sets.Indep ProbabilityTheory.IndepSets.indep
 
@@ -593,13 +486,10 @@ We prove the following equivalences on `IndepSet`, for measurable sets `s, t`.
 
 variable {s t : Set Ω} (S T : Set (Set Ω))
 
-theorem indepSet_iff_indepSets_singleton {m0 : MeasurableSpace Ω} (hs_meas : MeasurableSet s)
+theorem indepSet_iff_indepSets_singleton {_m0 : MeasurableSpace Ω} (hs_meas : MeasurableSet s)
     (ht_meas : MeasurableSet t) (μ : Measure Ω := by volume_tac)
     [IsProbabilityMeasure μ] : IndepSet s t μ ↔ IndepSets {s} {t} μ :=
-  ⟨Indep.indepSets, fun h =>
-    IndepSets.indep (generateFrom_le fun u hu => by rwa [Set.mem_singleton_iff.mp hu])
-      (generateFrom_le fun u hu => by rwa [Set.mem_singleton_iff.mp hu]) (IsPiSystem.singleton s)
-      (IsPiSystem.singleton t) rfl rfl h⟩
+  kernel.indepSet_iff_indepSets_singleton hs_meas ht_meas _ _
 #align probability_theory.indep_set_iff_indep_sets_singleton ProbabilityTheory.indepSet_iff_indepSets_singleton
 
 theorem indepSet_iff_measure_inter_eq_mul {_m0 : MeasurableSpace Ω} (hs_meas : MeasurableSet s)
@@ -611,32 +501,22 @@ theorem indepSet_iff_measure_inter_eq_mul {_m0 : MeasurableSpace Ω} (hs_meas :
 theorem IndepSets.indepSet_of_mem {_m0 : MeasurableSpace Ω} (hs : s ∈ S) (ht : t ∈ T)
     (hs_meas : MeasurableSet s) (ht_meas : MeasurableSet t)
     (μ : Measure Ω := by volume_tac) [IsProbabilityMeasure μ]
-    (h_indep : IndepSets S T μ) : IndepSet s t μ :=
-  (indepSet_iff_measure_inter_eq_mul hs_meas ht_meas μ).mpr (h_indep s t hs ht)
+    (h_indep : IndepSets S T μ) :
+    IndepSet s t μ :=
+  kernel.IndepSets.indepSet_of_mem _ _ hs ht hs_meas ht_meas _ _ h_indep
 #align probability_theory.indep_sets.indep_set_of_mem ProbabilityTheory.IndepSets.indepSet_of_mem
 
-theorem Indep.indepSet_of_measurableSet {m₁ m₂ m0 : MeasurableSpace Ω} {μ : Measure Ω}
+theorem Indep.indepSet_of_measurableSet {m₁ m₂ _m0 : MeasurableSpace Ω} {μ : Measure Ω}
     (h_indep : Indep m₁ m₂ μ) {s t : Set Ω} (hs : MeasurableSet[m₁] s)
-    (ht : MeasurableSet[m₂] t) : IndepSet s t μ := by
-  refine fun s' t' hs' ht' => h_indep s' t' ?_ ?_
-  · refine @generateFrom_induction _ (fun u => MeasurableSet[m₁] u) {s} ?_ ?_ ?_ ?_ _ hs'
-    · simp only [Set.mem_singleton_iff, forall_eq, hs]
-    · exact @MeasurableSet.empty _ m₁
-    · exact fun u hu => hu.compl
-    · exact fun f hf => MeasurableSet.iUnion hf
-  · refine @generateFrom_induction _ (fun u => MeasurableSet[m₂] u) {t} ?_ ?_ ?_ ?_ _ ht'
-    · simp only [Set.mem_singleton_iff, forall_eq, ht]
-    · exact @MeasurableSet.empty _ m₂
-    · exact fun u hu => hu.compl
-    · exact fun f hf => MeasurableSet.iUnion hf
+    (ht : MeasurableSet[m₂] t) :
+    IndepSet s t μ :=
+  kernel.Indep.indepSet_of_measurableSet h_indep hs ht
 #align probability_theory.indep.indep_set_of_measurable_set ProbabilityTheory.Indep.indepSet_of_measurableSet
 
 theorem indep_iff_forall_indepSet (m₁ m₂ : MeasurableSpace Ω) {_m0 : MeasurableSpace Ω}
     (μ : Measure Ω) :
     Indep m₁ m₂ μ ↔ ∀ s t, MeasurableSet[m₁] s → MeasurableSet[m₂] t → IndepSet s t μ :=
-  ⟨fun h => fun _s _t hs ht => h.indepSet_of_measurableSet hs ht, fun h s t hs ht =>
-    h s t hs ht s t (measurableSet_generateFrom (Set.mem_singleton s))
-      (measurableSet_generateFrom (Set.mem_singleton t))⟩
+  kernel.indep_iff_forall_indepSet m₁ m₂ _ _
 #align probability_theory.indep_iff_forall_indep_set ProbabilityTheory.indep_iff_forall_indepSet
 
 end IndepSet
@@ -653,11 +533,10 @@ variable {β β' γ γ' : Type _} {mΩ : MeasurableSpace Ω} {μ : Measure Ω} {
 theorem indepFun_iff_measure_inter_preimage_eq_mul {mβ : MeasurableSpace β}
     {mβ' : MeasurableSpace β'} :
     IndepFun f g μ ↔
-      ∀ s t,
-        MeasurableSet s → MeasurableSet t → μ (f ⁻¹' s ∩ g ⁻¹' t) = μ (f ⁻¹' s) * μ (g ⁻¹' t) := by
-  constructor <;> intro h
-  · refine' fun s t hs ht => h (f ⁻¹' s) (g ⁻¹' t) ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩
-  · rintro _ _ ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩; exact h s t hs ht
+      ∀ s t, MeasurableSet s → MeasurableSet t
+        → μ (f ⁻¹' s ∩ g ⁻¹' t) = μ (f ⁻¹' s) * μ (g ⁻¹' t) := by
+  simp only [IndepFun, kernel.indepFun_iff_measure_inter_preimage_eq_mul, ae_dirac_eq,
+    Filter.eventually_pure, kernel.const_apply]
 #align probability_theory.indep_fun_iff_measure_inter_preimage_eq_mul ProbabilityTheory.indepFun_iff_measure_inter_preimage_eq_mul
 
 theorem iIndepFun_iff_measure_inter_preimage_eq_mul {ι : Type _} {β : ι → Type _}
@@ -665,30 +544,8 @@ theorem iIndepFun_iff_measure_inter_preimage_eq_mul {ι : Type _} {β : ι → T
     iIndepFun m f μ ↔
       ∀ (S : Finset ι) {sets : ∀ i : ι, Set (β i)} (_H : ∀ i, i ∈ S → MeasurableSet[m i] (sets i)),
         μ (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, μ (f i ⁻¹' sets i) := by
-  refine' ⟨fun h S sets h_meas => h _ fun i hi_mem => ⟨sets i, h_meas i hi_mem, rfl⟩, _⟩
-  intro h S setsΩ h_meas
-  classical
-    let setsβ : ∀ i : ι, Set (β i) := fun i =>
-      dite (i ∈ S) (fun hi_mem => (h_meas i hi_mem).choose) fun _ => Set.univ
-    have h_measβ : ∀ i ∈ S, MeasurableSet[m i] (setsβ i) := by
-      intro i hi_mem
-      simp_rw [dif_pos hi_mem]
-      exact (h_meas i hi_mem).choose_spec.1
-    have h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i := by
-      intro i hi_mem
-      simp_rw [dif_pos hi_mem]
-      exact (h_meas i hi_mem).choose_spec.2.symm
-    have h_left_eq : μ (⋂ i ∈ S, setsΩ i) = μ (⋂ i ∈ S, f i ⁻¹' setsβ i) := by
-      congr with x
-      simp_rw [Set.mem_iInter]
-      constructor <;> intro h i hi_mem <;> specialize h i hi_mem
-      · rwa [h_preim i hi_mem] at h
-      · rwa [h_preim i hi_mem]
-    have h_right_eq : (∏ i in S, μ (setsΩ i)) = ∏ i in S, μ (f i ⁻¹' setsβ i) := by
-      refine' Finset.prod_congr rfl fun i hi_mem => _
-      rw [h_preim i hi_mem]
-    rw [h_left_eq, h_right_eq]
-    exact h S h_measβ
+  simp only [iIndepFun, kernel.iIndepFun_iff_measure_inter_preimage_eq_mul, ae_dirac_eq,
+    Filter.eventually_pure, kernel.const_apply]
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun_iff_measure_inter_preimage_eq_mul ProbabilityTheory.iIndepFun_iff_measure_inter_preimage_eq_mul
 
@@ -696,10 +553,8 @@ theorem indepFun_iff_indepSet_preimage {mβ : MeasurableSpace β} {mβ' : Measur
     [IsProbabilityMeasure μ] (hf : Measurable f) (hg : Measurable g) :
     IndepFun f g μ ↔
       ∀ s t, MeasurableSet s → MeasurableSet t → IndepSet (f ⁻¹' s) (g ⁻¹' t) μ := by
-  refine' indepFun_iff_measure_inter_preimage_eq_mul.trans _
-  constructor <;> intro h s t hs ht <;> specialize h s t hs ht
-  · rwa [indepSet_iff_measure_inter_eq_mul (hf hs) (hg ht) μ]
-  · rwa [← indepSet_iff_measure_inter_eq_mul (hf hs) (hg ht) μ]
+  simp only [IndepFun, IndepSet, kernel.indepFun_iff_indepSet_preimage hf hg, ae_dirac_eq,
+    Filter.eventually_pure, kernel.const_apply]
 #align probability_theory.indep_fun_iff_indep_set_preimage ProbabilityTheory.indepFun_iff_indepSet_preimage
 
 @[symm]
@@ -709,21 +564,16 @@ nonrec theorem IndepFun.symm {_ : MeasurableSpace β} {f g : Ω → β} (hfg : I
 
 theorem IndepFun.ae_eq {mβ : MeasurableSpace β} {f g f' g' : Ω → β} (hfg : IndepFun f g μ)
     (hf : f =ᵐ[μ] f') (hg : g =ᵐ[μ] g') : IndepFun f' g' μ := by
-  rintro _ _ ⟨A, hA, rfl⟩ ⟨B, hB, rfl⟩
-  have h1 : f ⁻¹' A =ᵐ[μ] f' ⁻¹' A := hf.fun_comp A
-  have h2 : g ⁻¹' B =ᵐ[μ] g' ⁻¹' B := hg.fun_comp B
-  rw [← measure_congr h1, ← measure_congr h2, ← measure_congr (h1.inter h2)]
-  exact hfg _ _ ⟨_, hA, rfl⟩ ⟨_, hB, rfl⟩
+  refine kernel.IndepFun.ae_eq hfg ?_ ?_ <;>
+    simp only [ae_dirac_eq, Filter.eventually_pure, kernel.const_apply]
+  exacts [hf, hg]
 #align probability_theory.indep_fun.ae_eq ProbabilityTheory.IndepFun.ae_eq
 
-theorem IndepFun.comp {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
-    {mγ : MeasurableSpace γ} {mγ' : MeasurableSpace γ'} {φ : β → γ} {ψ : β' → γ'}
+theorem IndepFun.comp {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'}
+    {_mγ : MeasurableSpace γ} {_mγ' : MeasurableSpace γ'} {φ : β → γ} {ψ : β' → γ'}
     (hfg : IndepFun f g μ) (hφ : Measurable φ) (hψ : Measurable ψ) :
-    IndepFun (φ ∘ f) (ψ ∘ g) μ := by
-  rintro _ _ ⟨A, hA, rfl⟩ ⟨B, hB, rfl⟩
-  apply hfg
-  · exact ⟨φ ⁻¹' A, hφ hA, Set.preimage_comp.symm⟩
-  · exact ⟨ψ ⁻¹' B, hψ hB, Set.preimage_comp.symm⟩
+    IndepFun (φ ∘ f) (ψ ∘ g) μ :=
+  kernel.IndepFun.comp hfg hφ hψ
 #align probability_theory.indep_fun.comp ProbabilityTheory.IndepFun.comp
 
 /-- If `f` is a family of mutually independent random variables (`iIndepFun m f μ`) and `S, T` are
@@ -732,117 +582,16 @@ tuple `(f i)_i` for `i ∈ T`. -/
 theorem iIndepFun.indepFun_finset [IsProbabilityMeasure μ] {ι : Type _} {β : ι → Type _}
     {m : ∀ i, MeasurableSpace (β i)} {f : ∀ i, Ω → β i} (S T : Finset ι) (hST : Disjoint S T)
     (hf_Indep : iIndepFun m f μ) (hf_meas : ∀ i, Measurable (f i)) :
-    IndepFun (fun a (i : S) => f i a) (fun a (i : T) => f i a) μ := by
-  -- We introduce π-systems, build from the π-system of boxes which generates `MeasurableSpace.pi`.
-  let πSβ := Set.pi (Set.univ : Set S) ''
-    Set.pi (Set.univ : Set S) fun i => { s : Set (β i) | MeasurableSet[m i] s }
-  let πS := { s : Set Ω | ∃ t ∈ πSβ, (fun a (i : S) => f i a) ⁻¹' t = s }
-  have hπS_pi : IsPiSystem πS := by exact IsPiSystem.comap (@isPiSystem_pi _ _ ?_) _
-  have hπS_gen : (MeasurableSpace.pi.comap fun a (i : S) => f i a) = generateFrom πS := by
-    rw [generateFrom_pi.symm, comap_generateFrom]
-    congr
-  let πTβ := Set.pi (Set.univ : Set T) ''
-      Set.pi (Set.univ : Set T) fun i => { s : Set (β i) | MeasurableSet[m i] s }
-  let πT := { s : Set Ω | ∃ t ∈ πTβ, (fun a (i : T) => f i a) ⁻¹' t = s }
-  have hπT_pi : IsPiSystem πT := by exact IsPiSystem.comap (@isPiSystem_pi _ _ ?_) _
-  have hπT_gen : (MeasurableSpace.pi.comap fun a (i : T) => f i a) = generateFrom πT := by
-    rw [generateFrom_pi.symm, comap_generateFrom]
-    congr
-  -- To prove independence, we prove independence of the generating π-systems.
-  refine IndepSets.indep (Measurable.comap_le (measurable_pi_iff.mpr fun i => hf_meas i))
-    (Measurable.comap_le (measurable_pi_iff.mpr fun i => hf_meas i)) hπS_pi hπT_pi hπS_gen hπT_gen
-    ?_
-  rintro _ _ ⟨s, ⟨sets_s, hs1, hs2⟩, rfl⟩ ⟨t, ⟨sets_t, ht1, ht2⟩, rfl⟩
-  simp only [Set.mem_univ_pi, Set.mem_setOf_eq] at hs1 ht1
-  rw [← hs2, ← ht2]
-  classical
-    let sets_s' : ∀ i : ι, Set (β i) := fun i =>
-      dite (i ∈ S) (fun hi => sets_s ⟨i, hi⟩) fun _ => Set.univ
-    have h_sets_s'_eq : ∀ {i} (hi : i ∈ S), sets_s' i = sets_s ⟨i, hi⟩ := by
-      intro i hi; simp_rw [dif_pos hi]
-    have h_sets_s'_univ : ∀ {i} (_hi : i ∈ T), sets_s' i = Set.univ := by
-      intro i hi; simp_rw [dif_neg (Finset.disjoint_right.mp hST hi)]
-    let sets_t' : ∀ i : ι, Set (β i) := fun i =>
-      dite (i ∈ T) (fun hi => sets_t ⟨i, hi⟩) fun _ => Set.univ
-    have h_sets_t'_univ : ∀ {i} (_hi : i ∈ S), sets_t' i = Set.univ := by
-      intro i hi; simp_rw [dif_neg (Finset.disjoint_left.mp hST hi)]
-    have h_meas_s' : ∀ i ∈ S, MeasurableSet (sets_s' i) := by
-      intro i hi; rw [h_sets_s'_eq hi]; exact hs1 _
-    have h_meas_t' : ∀ i ∈ T, MeasurableSet (sets_t' i) := by
-      intro i hi; simp_rw [dif_pos hi]; exact ht1 _
-    have h_eq_inter_S : (fun (ω : Ω) (i : ↥S) =>
-      f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i := by
-      ext1 x
-      simp_rw [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
-      constructor <;> intro h
-      · intro i hi; simp only [h_sets_s'_eq hi, Set.mem_preimage]; exact h ⟨i, hi⟩
-      · rintro ⟨i, hi⟩; specialize h i hi; rwa [dif_pos hi] at h
-    have h_eq_inter_T : (fun (ω : Ω) (i : ↥T) => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t
-      = ⋂ i ∈ T, f i ⁻¹' sets_t' i := by
-      ext1 x
-      simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
-      constructor <;> intro h
-      · intro i hi; simp_rw [dif_pos hi]; exact h ⟨i, hi⟩
-      · rintro ⟨i, hi⟩; specialize h i hi; simp_rw [dif_pos hi] at h; exact h
-    rw [iIndepFun_iff_measure_inter_preimage_eq_mul] at hf_Indep
-    rw [h_eq_inter_S, h_eq_inter_T, hf_Indep S h_meas_s', hf_Indep T h_meas_t']
-    have h_Inter_inter :
-      ((⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i) =
-        ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) := by
-      ext1 x
-      simp_rw [Set.mem_inter_iff, Set.mem_iInter, Set.mem_preimage, Finset.mem_union]
-      constructor <;> intro h
-      · intro i hi
-        cases' hi with hiS hiT
-        · replace h := h.1 i hiS
-          simp_rw [dif_pos hiS, dif_neg (Finset.disjoint_left.mp hST hiS)]
-          exact ⟨by rwa [dif_pos hiS] at h, Set.mem_univ _⟩
-        · replace h := h.2 i hiT
-          simp_rw [dif_pos hiT, dif_neg (Finset.disjoint_right.mp hST hiT)]
-          exact ⟨Set.mem_univ _, by rwa [dif_pos hiT] at h⟩
-      · exact ⟨fun i hi => (h i (Or.inl hi)).1, fun i hi => (h i (Or.inr hi)).2⟩
-    rw [h_Inter_inter, hf_Indep (S ∪ T)]
-    swap
-    · intro i hi_mem
-      rw [Finset.mem_union] at hi_mem
-      cases' hi_mem with hi_mem hi_mem
-      · rw [h_sets_t'_univ hi_mem, Set.inter_univ]; exact h_meas_s' i hi_mem
-      · rw [h_sets_s'_univ hi_mem, Set.univ_inter]; exact h_meas_t' i hi_mem
-    rw [Finset.prod_union hST]
-    congr 1
-    · refine' Finset.prod_congr rfl fun i hi => _
-      rw [h_sets_t'_univ hi, Set.inter_univ]
-    · refine' Finset.prod_congr rfl fun i hi => _
-      rw [h_sets_s'_univ hi, Set.univ_inter]
+    IndepFun (fun a (i : S) => f i a) (fun a (i : T) => f i a) μ :=
+  kernel.iIndepFun.indepFun_finset S T hST hf_Indep hf_meas
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun.indep_fun_finset ProbabilityTheory.iIndepFun.indepFun_finset
 
 theorem iIndepFun.indepFun_prod [IsProbabilityMeasure μ] {ι : Type _} {β : ι → Type _}
     {m : ∀ i, MeasurableSpace (β i)} {f : ∀ i, Ω → β i} (hf_Indep : iIndepFun m f μ)
     (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
-    IndepFun (fun a => (f i a, f j a)) (f k) μ := by
-  classical
-    have h_right : f k =
-      (fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩) ∘
-      fun a (j : ({k} : Finset ι)) => f j a := rfl
-    have h_meas_right :  Measurable fun p : ∀ j : ({k} : Finset ι),
-      β j => p ⟨k, Finset.mem_singleton_self k⟩ := measurable_pi_apply _
-    let s : Finset ι := {i, j}
-    have h_left : (fun ω => (f i ω, f j ω)) = (fun p : ∀ l : s, β l =>
-      (p ⟨i, Finset.mem_insert_self i _⟩,
-      p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩)) ∘ fun a (j : s) => f j a := by
-      ext1 a
-      simp only [Prod.mk.inj_iff]
-      constructor
-    have h_meas_left : Measurable fun p : ∀ l : s, β l =>
-      (p ⟨i, Finset.mem_insert_self i _⟩,
-      p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩) :=
-        Measurable.prod (measurable_pi_apply _) (measurable_pi_apply _)
-    rw [h_left, h_right]
-    refine' (hf_Indep.indepFun_finset s {k} _ hf_meas).comp h_meas_left h_meas_right
-    rw [Finset.disjoint_singleton_right]
-    simp only [Finset.mem_insert, Finset.mem_singleton, not_or]
-    exact ⟨hik.symm, hjk.symm⟩
+    IndepFun (fun a => (f i a, f j a)) (f k) μ :=
+  kernel.iIndepFun.indepFun_prod hf_Indep hf_meas i j k hik hjk
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun.indep_fun_prod ProbabilityTheory.iIndepFun.indepFun_prod
 
@@ -850,11 +599,8 @@ set_option linter.uppercaseLean3 false in
 theorem iIndepFun.mul [IsProbabilityMeasure μ] {ι : Type _} {β : Type _} {m : MeasurableSpace β}
     [Mul β] [MeasurableMul₂ β] {f : ι → Ω → β} (hf_Indep : iIndepFun (fun _ => m) f μ)
     (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
-    IndepFun (f i * f j) (f k) μ := by
-  have : IndepFun (fun ω => (f i ω, f j ω)) (f k) μ :=
-    hf_Indep.indepFun_prod hf_meas i j k hik hjk
-  change IndepFun ((fun p : β × β => p.fst * p.snd) ∘ fun ω => (f i ω, f j ω)) (id ∘ f k) μ
-  exact IndepFun.comp this (measurable_fst.mul measurable_snd) measurable_id
+    IndepFun (f i * f j) (f k) μ :=
+  kernel.iIndepFun.mul hf_Indep hf_meas i j k hik hjk
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun.mul ProbabilityTheory.iIndepFun.mul
 set_option linter.uppercaseLean3 false in
@@ -864,24 +610,9 @@ set_option linter.uppercaseLean3 false in
 theorem iIndepFun.indepFun_finset_prod_of_not_mem [IsProbabilityMeasure μ] {ι : Type _} {β : Type _}
     {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂ β] {f : ι → Ω → β}
     (hf_Indep : iIndepFun (fun _ => m) f μ) (hf_meas : ∀ i, Measurable (f i)) {s : Finset ι} {i : ι}
-    (hi : i ∉ s) : IndepFun (∏ j in s, f j) (f i) μ := by
-  classical
-    have h_right : f i =
-      (fun p : ∀ _j : ({i} : Finset ι), β => p ⟨i, Finset.mem_singleton_self i⟩) ∘
-      fun a (j : ({i} : Finset ι)) => f j a := rfl
-    have h_meas_right : Measurable fun p : ∀ _j : ({i} : Finset ι), β
-      => p ⟨i, Finset.mem_singleton_self i⟩ := measurable_pi_apply ⟨i, Finset.mem_singleton_self i⟩
-    have h_left : ∏ j in s, f j = (fun p : ∀ _j : s, β => ∏ j, p j) ∘ fun a (j : s) => f j a := by
-      ext1 a
-      simp only [Function.comp_apply]
-      have : (∏ j : ↥s, f (↑j) a) = (∏ j : ↥s, f ↑j) a := by rw [Finset.prod_apply]
-      rw [this, Finset.prod_coe_sort]
-    have h_meas_left : Measurable fun p : ∀ _j : s, β => ∏ j, p j :=
-      Finset.univ.measurable_prod fun (j : ↥s) (_H : j ∈ Finset.univ) => measurable_pi_apply j
-    rw [h_left, h_right]
-    exact
-      (hf_Indep.indepFun_finset s {i} (Finset.disjoint_singleton_left.mpr hi).symm hf_meas).comp
-        h_meas_left h_meas_right
+    (hi : i ∉ s) :
+    IndepFun (∏ j in s, f j) (f i) μ :=
+  kernel.iIndepFun.indepFun_finset_prod_of_not_mem hf_Indep hf_meas hi
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun.indep_fun_finset_prod_of_not_mem ProbabilityTheory.iIndepFun.indepFun_finset_prod_of_not_mem
 set_option linter.uppercaseLean3 false in
@@ -892,26 +623,16 @@ theorem iIndepFun.indepFun_prod_range_succ [IsProbabilityMeasure μ] {β : Type
     {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂ β] {f : ℕ → Ω → β}
     (hf_Indep : iIndepFun (fun _ => m) f μ) (hf_meas : ∀ i, Measurable (f i)) (n : ℕ) :
     IndepFun (∏ j in Finset.range n, f j) (f n) μ :=
-  hf_Indep.indepFun_finset_prod_of_not_mem hf_meas Finset.not_mem_range_self
+  kernel.iIndepFun.indepFun_prod_range_succ hf_Indep hf_meas n
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun.indep_fun_prod_range_succ ProbabilityTheory.iIndepFun.indepFun_prod_range_succ
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun.indep_fun_sum_range_succ ProbabilityTheory.iIndepFun.indepFun_sum_range_succ
 
 theorem iIndepSet.iIndepFun_indicator [Zero β] [One β] {m : MeasurableSpace β} {s : ι → Set Ω}
-    (hs : iIndepSet s μ) : iIndepFun (fun _n => m) (fun n => (s n).indicator fun _ω => 1) μ := by
-  classical
-    rw [iIndepFun_iff_measure_inter_preimage_eq_mul]
-    rintro S π _hπ
-    simp_rw [Set.indicator_const_preimage_eq_union]
-    refine' @hs S (fun i => ite (1 ∈ π i) (s i) ∅ ∪ ite ((0 : β) ∈ π i) (s i)ᶜ ∅) fun i _hi => _
-    have hsi : MeasurableSet[generateFrom {s i}] (s i) :=
-      measurableSet_generateFrom (Set.mem_singleton _)
-    refine'
-      MeasurableSet.union (MeasurableSet.ite' (fun _ => hsi) fun _ => _)
-        (MeasurableSet.ite' (fun _ => hsi.compl) fun _ => _)
-    · exact @MeasurableSet.empty _ (generateFrom {s i})
-    · exact @MeasurableSet.empty _ (generateFrom {s i})
+    (hs : iIndepSet s μ) :
+    iIndepFun (fun _n => m) (fun n => (s n).indicator fun _ω => 1) μ :=
+  kernel.iIndepSet.iIndepFun_indicator hs
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_set.Indep_fun_indicator ProbabilityTheory.iIndepSet.iIndepFun_indicator
 

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,14 +2,11 @@
 Copyright (c) 2021 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
-
-! This file was ported from Lean 3 source module probability.independence.basic
-! leanprover-community/mathlib commit 2f8347015b12b0864dfaf366ec4909eb70c78740
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.MeasureTheory.Constructions.Pi
 
+#align_import probability.independence.basic from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
+
 /-!
 # Independence of sets of sets and measure spaces (σ-algebras)
 

• Also remove most superfluous parentheses around big operators (∑, ∏ and variants).
• roughly the used regex: ([^a-zA-Zα-ωΑ-Ω'𝓝ℳ₀𝕂ₛ)]) \(([∑∏][^()∑∏]*,[^()∑∏:]*)\) ([⊂⊆=<≤]) replaced by $1 $2 $3

@@ -874,7 +874,7 @@ theorem iIndepFun.indepFun_finset_prod_of_not_mem [IsProbabilityMeasure μ] {ι
       fun a (j : ({i} : Finset ι)) => f j a := rfl
     have h_meas_right : Measurable fun p : ∀ _j : ({i} : Finset ι), β
       => p ⟨i, Finset.mem_singleton_self i⟩ := measurable_pi_apply ⟨i, Finset.mem_singleton_self i⟩
-    have h_left : (∏ j in s, f j) = (fun p : ∀ _j : s, β => ∏ j, p j) ∘ fun a (j : s) => f j a := by
+    have h_left : ∏ j in s, f j = (fun p : ∀ _j : s, β => ∏ j, p j) ∘ fun a (j : s) => f j a := by
       ext1 a
       simp only [Function.comp_apply]
       have : (∏ j : ↥s, f (↑j) a) = (∏ j : ↥s, f ↑j) a := by rw [Finset.prod_apply]

@@ -276,7 +276,7 @@ theorem iIndepSets.indepSets {s : ι → Set (Set Ω)} [MeasurableSpace Ω] {μ
     have h1 : t₁ = ite (i = i) t₁ t₂ := by simp only [if_true, eq_self_iff_true]
     have h2 : t₂ = ite (j = i) t₁ t₂ := by simp only [hij.symm, if_false]
     have h_inter :
-      (⋂ (t : ι) (_ : t ∈ ({i, j} : Finset ι)), ite (t = i) t₁ t₂) =
+      ⋂ (t : ι) (_ : t ∈ ({i, j} : Finset ι)), ite (t = i) t₁ t₂ =
         ite (i = i) t₁ t₂ ∩ ite (j = i) t₁ t₂ :=
       by simp only [Finset.set_biInter_singleton, Finset.set_biInter_insert]
     have h_prod :

Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>

@@ -907,7 +907,7 @@ theorem iIndepSet.iIndepFun_indicator [Zero β] [One β] {m : MeasurableSpace β
     rw [iIndepFun_iff_measure_inter_preimage_eq_mul]
     rintro S π _hπ
     simp_rw [Set.indicator_const_preimage_eq_union]
-    refine' @hs S (fun i => ite (1 ∈ π i) (s i) ∅ ∪ ite ((0 : β) ∈ π i) (s iᶜ) ∅) fun i _hi => _
+    refine' @hs S (fun i => ite (1 ∈ π i) (s i) ∅ ∪ ite ((0 : β) ∈ π i) (s i)ᶜ ∅) fun i _hi => _
     have hsi : MeasurableSet[generateFrom {s i}] (s i) :=
       measurableSet_generateFrom (Set.mem_singleton _)
     refine'

@@ -147,7 +147,7 @@ section Indep
 @[symm]
 theorem IndepSets.symm {s₁ s₂ : Set (Set Ω)} [MeasurableSpace Ω] {μ : Measure Ω}
     (h : IndepSets s₁ s₂ μ) : IndepSets s₂ s₁ μ := by
-  intro t1 t2 ht1 ht2;
+  intro t1 t2 ht1 ht2
   rw [Set.inter_comm, mul_comm]; exact h t2 t1 ht2 ht1
 #align probability_theory.indep_sets.symm ProbabilityTheory.IndepSets.symm
 
@@ -259,7 +259,7 @@ theorem indepSets_singleton_iff [MeasurableSpace Ω] {s t : Set Ω} {μ : Measur
 
 end Indep
 
-/-! ### Deducing `indep` from `Indep` -/
+/-! ### Deducing `Indep` from `iIndep` -/
 
 
 section FromIndepToIndep
@@ -340,7 +340,7 @@ section FromPiSystemsToMeasurableSpaces
 /-! ### Independence of generating π-systems implies independence of measurable space structures -/
 
 
-private theorem indep_sets.indep_aux {m2 : MeasurableSpace Ω} {m : MeasurableSpace Ω}
+private theorem IndepSets.indep_aux {m2 : MeasurableSpace Ω} {m : MeasurableSpace Ω}
     {μ : Measure Ω} [IsProbabilityMeasure μ] {p1 p2 : Set (Set Ω)} (h2 : m2 ≤ m)
     (hp2 : IsPiSystem p2) (hpm2 : m2 = generateFrom p2) (hyp : IndepSets p1 p2 μ) {t1 t2 : Set Ω}
     (ht1 : t1 ∈ p1) (ht2m : MeasurableSet[m2] t2) : μ (t1 ∩ t2) = μ t1 * μ t2 := by
@@ -375,7 +375,7 @@ theorem IndepSets.indep {m1 m2 : MeasurableSpace Ω} {m : MeasurableSpace Ω} {
     rw [hpm1]
     exact measurableSet_generateFrom ht
   rw [Measure.restrict_apply ht1, Measure.smul_apply, smul_eq_mul, mul_comm]
-  exact indep_sets.indep_aux h2 hp2 hpm2 hyp ht ht2
+  exact IndepSets.indep_aux h2 hp2 hpm2 hyp ht ht2
 #align probability_theory.indep_sets.indep ProbabilityTheory.IndepSets.indep
 
 theorem IndepSets.indep' {_m : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
@@ -588,9 +588,9 @@ section IndepSet
 
 /-! ### Independence of measurable sets
 
-We prove the following equivalences on `indep_set`, for measurable sets `s, t`.
-* `indep_set s t μ ↔ μ (s ∩ t) = μ s * μ t`,
-* `indep_set s t μ ↔ indep_sets {s} {t} μ`.
+We prove the following equivalences on `IndepSet`, for measurable sets `s, t`.
+* `IndepSet s t μ ↔ μ (s ∩ t) = μ s * μ t`,
+* `IndepSet s t μ ↔ IndepSets {s} {t} μ`.
 -/
 
 

@@ -276,7 +276,7 @@ theorem iIndepSets.indepSets {s : ι → Set (Set Ω)} [MeasurableSpace Ω] {μ
     have h1 : t₁ = ite (i = i) t₁ t₂ := by simp only [if_true, eq_self_iff_true]
     have h2 : t₂ = ite (j = i) t₁ t₂ := by simp only [hij.symm, if_false]
     have h_inter :
-      (⋂ (t : ι) (_H : t ∈ ({i, j} : Finset ι)), ite (t = i) t₁ t₂) =
+      (⋂ (t : ι) (_ : t ∈ ({i, j} : Finset ι)), ite (t = i) t₁ t₂) =
         ite (i = i) t₁ t₂ ∩ ite (j = i) t₁ t₂ :=
       by simp only [Finset.set_biInter_singleton, Finset.set_biInter_insert]
     have h_prod :

I have misported is_foo to Foo because I misunderstood the rule for IsLawfulFoo. This PR recover Is of Foo which is ported from is_foo. This PR also renames some misported theorems.

@@ -157,7 +157,7 @@ theorem Indep.symm {m₁ m₂ : MeasurableSpace Ω} [MeasurableSpace Ω] {μ : M
 #align probability_theory.indep.symm ProbabilityTheory.Indep.symm
 
 theorem indep_bot_right (m' : MeasurableSpace Ω) {m : MeasurableSpace Ω} {μ : Measure Ω}
-    [ProbabilityMeasure μ] : Indep m' ⊥ μ := by
+    [IsProbabilityMeasure μ] : Indep m' ⊥ μ := by
   intro s t _hs ht
   rw [Set.mem_setOf_eq, MeasurableSpace.measurableSet_bot_iff] at ht
   cases' ht with ht ht
@@ -166,16 +166,16 @@ theorem indep_bot_right (m' : MeasurableSpace Ω) {m : MeasurableSpace Ω} {μ :
 #align probability_theory.indep_bot_right ProbabilityTheory.indep_bot_right
 
 theorem indep_bot_left (m' : MeasurableSpace Ω) {_m : MeasurableSpace Ω} {μ : Measure Ω}
-    [ProbabilityMeasure μ] : Indep ⊥ m' μ := (indep_bot_right m').symm
+    [IsProbabilityMeasure μ] : Indep ⊥ m' μ := (indep_bot_right m').symm
 #align probability_theory.indep_bot_left ProbabilityTheory.indep_bot_left
 
-theorem indepSet_empty_right {m : MeasurableSpace Ω} {μ : Measure Ω} [ProbabilityMeasure μ]
+theorem indepSet_empty_right {m : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
     (s : Set Ω) : IndepSet s ∅ μ := by
   simp only [IndepSet, generateFrom_singleton_empty];
   exact indep_bot_right _
 #align probability_theory.indep_set_empty_right ProbabilityTheory.indepSet_empty_right
 
-theorem indepSet_empty_left {_m : MeasurableSpace Ω} {μ : Measure Ω} [ProbabilityMeasure μ]
+theorem indepSet_empty_left {_m : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
     (s : Set Ω) : IndepSet ∅ s μ := (indepSet_empty_right s).symm
 #align probability_theory.indep_set_empty_left ProbabilityTheory.indepSet_empty_left
 
@@ -341,14 +341,14 @@ section FromPiSystemsToMeasurableSpaces
 
 
 private theorem indep_sets.indep_aux {m2 : MeasurableSpace Ω} {m : MeasurableSpace Ω}
-    {μ : Measure Ω} [ProbabilityMeasure μ] {p1 p2 : Set (Set Ω)} (h2 : m2 ≤ m) (hp2 : IsPiSystem p2)
-    (hpm2 : m2 = generateFrom p2) (hyp : IndepSets p1 p2 μ) {t1 t2 : Set Ω} (ht1 : t1 ∈ p1)
-    (ht2m : MeasurableSet[m2] t2) : μ (t1 ∩ t2) = μ t1 * μ t2 := by
+    {μ : Measure Ω} [IsProbabilityMeasure μ] {p1 p2 : Set (Set Ω)} (h2 : m2 ≤ m)
+    (hp2 : IsPiSystem p2) (hpm2 : m2 = generateFrom p2) (hyp : IndepSets p1 p2 μ) {t1 t2 : Set Ω}
+    (ht1 : t1 ∈ p1) (ht2m : MeasurableSet[m2] t2) : μ (t1 ∩ t2) = μ t1 * μ t2 := by
   let μ_inter := μ.restrict t1
   let ν := μ t1 • μ
   have h_univ : μ_inter Set.univ = ν Set.univ := by
     rw [Measure.restrict_apply_univ, Measure.smul_apply, smul_eq_mul, measure_univ, mul_one]
-  haveI : FiniteMeasure μ_inter := @Restrict.finiteMeasure Ω _ t1 μ ⟨measure_lt_top μ t1⟩
+  haveI : IsFiniteMeasure μ_inter := @Restrict.isFiniteMeasure Ω _ t1 μ ⟨measure_lt_top μ t1⟩
   rw [Set.inter_comm, ← Measure.restrict_apply (h2 t2 ht2m)]
   refine' ext_on_measurableSpace_of_generate_finite m p2 (fun t ht => _) h2 hpm2 hp2 h_univ ht2m
   have ht2 : MeasurableSet[m] t := by
@@ -359,7 +359,7 @@ private theorem indep_sets.indep_aux {m2 : MeasurableSpace Ω} {m : MeasurableSp
   exact hyp t1 t ht1 ht
 
 theorem IndepSets.indep {m1 m2 : MeasurableSpace Ω} {m : MeasurableSpace Ω} {μ : Measure Ω}
-    [ProbabilityMeasure μ] {p1 p2 : Set (Set Ω)} (h1 : m1 ≤ m) (h2 : m2 ≤ m) (hp1 : IsPiSystem p1)
+    [IsProbabilityMeasure μ] {p1 p2 : Set (Set Ω)} (h1 : m1 ≤ m) (h2 : m2 ≤ m) (hp1 : IsPiSystem p1)
     (hp2 : IsPiSystem p2) (hpm1 : m1 = generateFrom p1) (hpm2 : m2 = generateFrom p2)
     (hyp : IndepSets p1 p2 μ) : Indep m1 m2 μ := by
   intro t1 t2 ht1 ht2
@@ -367,7 +367,7 @@ theorem IndepSets.indep {m1 m2 : MeasurableSpace Ω} {m : MeasurableSpace Ω} {
   let ν := μ t2 • μ
   have h_univ : μ_inter Set.univ = ν Set.univ := by
     rw [Measure.restrict_apply_univ, Measure.smul_apply, smul_eq_mul, measure_univ, mul_one]
-  haveI : FiniteMeasure μ_inter := @Restrict.finiteMeasure Ω _ t2 μ ⟨measure_lt_top μ t2⟩
+  haveI : IsFiniteMeasure μ_inter := @Restrict.isFiniteMeasure Ω _ t2 μ ⟨measure_lt_top μ t2⟩
   rw [mul_comm, ← Measure.restrict_apply (h1 t1 ht1)]
   refine' ext_on_measurableSpace_of_generate_finite m p1 (fun t ht => _) h1 hpm1 hp1 h_univ ht1
   have ht1 : MeasurableSet[m] t := by
@@ -378,7 +378,7 @@ theorem IndepSets.indep {m1 m2 : MeasurableSpace Ω} {m : MeasurableSpace Ω} {
   exact indep_sets.indep_aux h2 hp2 hpm2 hyp ht ht2
 #align probability_theory.indep_sets.indep ProbabilityTheory.IndepSets.indep
 
-theorem IndepSets.indep' {_m : MeasurableSpace Ω} {μ : Measure Ω} [ProbabilityMeasure μ]
+theorem IndepSets.indep' {_m : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
     {p1 p2 : Set (Set Ω)} (hp1m : ∀ s ∈ p1, MeasurableSet s) (hp2m : ∀ s ∈ p2, MeasurableSet s)
     (hp1 : IsPiSystem p1) (hp2 : IsPiSystem p2) (hyp : IndepSets p1 p2 μ) :
     Indep (generateFrom p1) (generateFrom p2) μ :=
@@ -387,7 +387,7 @@ theorem IndepSets.indep' {_m : MeasurableSpace Ω} {μ : Measure Ω} [Probabilit
 
 variable {m0 : MeasurableSpace Ω} {μ : Measure Ω}
 
-theorem indepSets_piiUnionInter_of_disjoint [ProbabilityMeasure μ] {s : ι → Set (Set Ω)}
+theorem indepSets_piiUnionInter_of_disjoint [IsProbabilityMeasure μ] {s : ι → Set (Set Ω)}
     {S T : Set ι} (h_indep : iIndepSets s μ) (hST : Disjoint S T) :
     IndepSets (piiUnionInter s S) (piiUnionInter s T) μ := by
   rintro t1 t2 ⟨p1, hp1, f1, ht1_m, ht1_eq⟩ ⟨p2, hp2, f2, ht2_m, ht2_eq⟩
@@ -425,7 +425,7 @@ theorem indepSets_piiUnionInter_of_disjoint [ProbabilityMeasure μ] {s : ι →
       h_indep p1 ht1_m, ht2_eq, ← h_indep p2 ht2_m]
 #align probability_theory.indep_sets_pi_Union_Inter_of_disjoint ProbabilityTheory.indepSets_piiUnionInter_of_disjoint
 
-theorem iIndepSet.indep_generateFrom_of_disjoint [ProbabilityMeasure μ] {s : ι → Set Ω}
+theorem iIndepSet.indep_generateFrom_of_disjoint [IsProbabilityMeasure μ] {s : ι → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (S T : Set ι) (hST : Disjoint S T) :
     Indep (generateFrom { t | ∃ n ∈ S, s n = t }) (generateFrom { t | ∃ k ∈ T, s k = t }) μ := by
   rw [← generateFrom_piiUnionInter_singleton_left, ← generateFrom_piiUnionInter_singleton_left]
@@ -441,7 +441,7 @@ theorem iIndepSet.indep_generateFrom_of_disjoint [ProbabilityMeasure μ] {s : ι
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_set.indep_generate_from_of_disjoint ProbabilityTheory.iIndepSet.indep_generateFrom_of_disjoint
 
-theorem indep_iSup_of_disjoint [ProbabilityMeasure μ] {m : ι → MeasurableSpace Ω}
+theorem indep_iSup_of_disjoint [IsProbabilityMeasure μ] {m : ι → MeasurableSpace Ω}
     (h_le : ∀ i, m i ≤ m0) (h_indep : iIndep m μ) {S T : Set ι} (hST : Disjoint S T) :
     Indep (⨆ i ∈ S, m i) (⨆ i ∈ T, m i) μ := by
   refine'
@@ -454,7 +454,7 @@ theorem indep_iSup_of_disjoint [ProbabilityMeasure μ] {m : ι → MeasurableSpa
 #align probability_theory.indep_supr_of_disjoint ProbabilityTheory.indep_iSup_of_disjoint
 
 theorem indep_iSup_of_directed_le {Ω} {m : ι → MeasurableSpace Ω} {m' m0 : MeasurableSpace Ω}
-    {μ : Measure Ω} [ProbabilityMeasure μ] (h_indep : ∀ i, Indep (m i) m' μ)
+    {μ : Measure Ω} [IsProbabilityMeasure μ] (h_indep : ∀ i, Indep (m i) m' μ)
     (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0) (hm : Directed (· ≤ ·) m) :
     Indep (⨆ i, m i) m' μ := by
   let p : ι → Set (Set Ω) := fun n => { t | MeasurableSet[m n] t }
@@ -477,7 +477,7 @@ theorem indep_iSup_of_directed_le {Ω} {m : ι → MeasurableSpace Ω} {m' m0 :
   exact (generateFrom_iUnion_measurableSet _).symm
 #align probability_theory.indep_supr_of_directed_le ProbabilityTheory.indep_iSup_of_directed_le
 
-theorem iIndepSet.indep_generateFrom_lt [Preorder ι] [ProbabilityMeasure μ] {s : ι → Set Ω}
+theorem iIndepSet.indep_generateFrom_lt [Preorder ι] [IsProbabilityMeasure μ] {s : ι → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (i : ι) :
     Indep (generateFrom {s i}) (generateFrom { t | ∃ j < i, s j = t }) μ := by
   convert iIndepSet.indep_generateFrom_of_disjoint hsm hs {i} { j | j < i }
@@ -486,7 +486,7 @@ theorem iIndepSet.indep_generateFrom_lt [Preorder ι] [ProbabilityMeasure μ] {s
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_set.indep_generate_from_lt ProbabilityTheory.iIndepSet.indep_generateFrom_lt
 
-theorem iIndepSet.indep_generateFrom_le [LinearOrder ι] [ProbabilityMeasure μ] {s : ι → Set Ω}
+theorem iIndepSet.indep_generateFrom_le [LinearOrder ι] [IsProbabilityMeasure μ] {s : ι → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (i : ι) {k : ι} (hk : i < k) :
     Indep (generateFrom {s k}) (generateFrom { t | ∃ j ≤ i, s j = t }) μ := by
   convert iIndepSet.indep_generateFrom_of_disjoint hsm hs {k} { j | j ≤ i }
@@ -495,7 +495,7 @@ theorem iIndepSet.indep_generateFrom_le [LinearOrder ι] [ProbabilityMeasure μ]
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_set.indep_generate_from_le ProbabilityTheory.iIndepSet.indep_generateFrom_le
 
-theorem iIndepSet.indep_generateFrom_le_nat [ProbabilityMeasure μ] {s : ℕ → Set Ω}
+theorem iIndepSet.indep_generateFrom_le_nat [IsProbabilityMeasure μ] {s : ℕ → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (n : ℕ) :
     Indep (generateFrom {s (n + 1)}) (generateFrom { t | ∃ k ≤ n, s k = t }) μ :=
   iIndepSet.indep_generateFrom_le hsm hs _ n.lt_succ_self
@@ -503,14 +503,14 @@ set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_set.indep_generate_from_le_nat ProbabilityTheory.iIndepSet.indep_generateFrom_le_nat
 
 theorem indep_iSup_of_monotone [SemilatticeSup ι] {Ω} {m : ι → MeasurableSpace Ω}
-    {m' m0 : MeasurableSpace Ω} {μ : Measure Ω} [ProbabilityMeasure μ]
+    {m' m0 : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
     (h_indep : ∀ i, Indep (m i) m' μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0)
     (hm : Monotone m) : Indep (⨆ i, m i) m' μ :=
   indep_iSup_of_directed_le h_indep h_le h_le' (Monotone.directed_le hm)
 #align probability_theory.indep_supr_of_monotone ProbabilityTheory.indep_iSup_of_monotone
 
 theorem indep_iSup_of_antitone [SemilatticeInf ι] {Ω} {m : ι → MeasurableSpace Ω}
-    {m' m0 : MeasurableSpace Ω} {μ : Measure Ω} [ProbabilityMeasure μ]
+    {m' m0 : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
     (h_indep : ∀ i, Indep (m i) m' μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0)
     (hm : Antitone m) : Indep (⨆ i, m i) m' μ :=
   indep_iSup_of_directed_le h_indep h_le h_le' (directed_of_inf hm)
@@ -551,8 +551,8 @@ set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_sets.pi_Union_Inter_of_not_mem ProbabilityTheory.iIndepSets.piiUnionInter_of_not_mem
 
 /-- The measurable space structures generated by independent pi-systems are independent. -/
-theorem iIndepSets.iIndep [ProbabilityMeasure μ] (m : ι → MeasurableSpace Ω) (h_le : ∀ i, m i ≤ m0)
-    (π : ι → Set (Set Ω)) (h_pi : ∀ n, IsPiSystem (π n))
+theorem iIndepSets.iIndep [IsProbabilityMeasure μ] (m : ι → MeasurableSpace Ω)
+    (h_le : ∀ i, m i ≤ m0) (π : ι → Set (Set Ω)) (h_pi : ∀ n, IsPiSystem (π n))
     (h_generate : ∀ i, m i = generateFrom (π i)) (h_ind : iIndepSets π μ) : iIndep m μ := by
   classical
     intro s f
@@ -598,7 +598,7 @@ variable {s t : Set Ω} (S T : Set (Set Ω))
 
 theorem indepSet_iff_indepSets_singleton {m0 : MeasurableSpace Ω} (hs_meas : MeasurableSet s)
     (ht_meas : MeasurableSet t) (μ : Measure Ω := by volume_tac)
-    [ProbabilityMeasure μ] : IndepSet s t μ ↔ IndepSets {s} {t} μ :=
+    [IsProbabilityMeasure μ] : IndepSet s t μ ↔ IndepSets {s} {t} μ :=
   ⟨Indep.indepSets, fun h =>
     IndepSets.indep (generateFrom_le fun u hu => by rwa [Set.mem_singleton_iff.mp hu])
       (generateFrom_le fun u hu => by rwa [Set.mem_singleton_iff.mp hu]) (IsPiSystem.singleton s)
@@ -607,13 +607,13 @@ theorem indepSet_iff_indepSets_singleton {m0 : MeasurableSpace Ω} (hs_meas : Me
 
 theorem indepSet_iff_measure_inter_eq_mul {_m0 : MeasurableSpace Ω} (hs_meas : MeasurableSet s)
     (ht_meas : MeasurableSet t) (μ : Measure Ω := by volume_tac)
-    [ProbabilityMeasure μ] : IndepSet s t μ ↔ μ (s ∩ t) = μ s * μ t :=
+    [IsProbabilityMeasure μ] : IndepSet s t μ ↔ μ (s ∩ t) = μ s * μ t :=
   (indepSet_iff_indepSets_singleton hs_meas ht_meas μ).trans indepSets_singleton_iff
 #align probability_theory.indep_set_iff_measure_inter_eq_mul ProbabilityTheory.indepSet_iff_measure_inter_eq_mul
 
 theorem IndepSets.indepSet_of_mem {_m0 : MeasurableSpace Ω} (hs : s ∈ S) (ht : t ∈ T)
     (hs_meas : MeasurableSet s) (ht_meas : MeasurableSet t)
-    (μ : Measure Ω := by volume_tac) [ProbabilityMeasure μ]
+    (μ : Measure Ω := by volume_tac) [IsProbabilityMeasure μ]
     (h_indep : IndepSets S T μ) : IndepSet s t μ :=
   (indepSet_iff_measure_inter_eq_mul hs_meas ht_meas μ).mpr (h_indep s t hs ht)
 #align probability_theory.indep_sets.indep_set_of_mem ProbabilityTheory.IndepSets.indepSet_of_mem
@@ -696,7 +696,7 @@ set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun_iff_measure_inter_preimage_eq_mul ProbabilityTheory.iIndepFun_iff_measure_inter_preimage_eq_mul
 
 theorem indepFun_iff_indepSet_preimage {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
-    [ProbabilityMeasure μ] (hf : Measurable f) (hg : Measurable g) :
+    [IsProbabilityMeasure μ] (hf : Measurable f) (hg : Measurable g) :
     IndepFun f g μ ↔
       ∀ s t, MeasurableSet s → MeasurableSet t → IndepSet (f ⁻¹' s) (g ⁻¹' t) μ := by
   refine' indepFun_iff_measure_inter_preimage_eq_mul.trans _
@@ -732,7 +732,7 @@ theorem IndepFun.comp {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
 /-- If `f` is a family of mutually independent random variables (`iIndepFun m f μ`) and `S, T` are
 two disjoint finite index sets, then the tuple formed by `f i` for `i ∈ S` is independent of the
 tuple `(f i)_i` for `i ∈ T`. -/
-theorem iIndepFun.indepFun_finset [ProbabilityMeasure μ] {ι : Type _} {β : ι → Type _}
+theorem iIndepFun.indepFun_finset [IsProbabilityMeasure μ] {ι : Type _} {β : ι → Type _}
     {m : ∀ i, MeasurableSpace (β i)} {f : ∀ i, Ω → β i} (S T : Finset ι) (hST : Disjoint S T)
     (hf_Indep : iIndepFun m f μ) (hf_meas : ∀ i, Measurable (f i)) :
     IndepFun (fun a (i : S) => f i a) (fun a (i : T) => f i a) μ := by
@@ -820,7 +820,7 @@ theorem iIndepFun.indepFun_finset [ProbabilityMeasure μ] {ι : Type _} {β : ι
 set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun.indep_fun_finset ProbabilityTheory.iIndepFun.indepFun_finset
 
-theorem iIndepFun.indepFun_prod [ProbabilityMeasure μ] {ι : Type _} {β : ι → Type _}
+theorem iIndepFun.indepFun_prod [IsProbabilityMeasure μ] {ι : Type _} {β : ι → Type _}
     {m : ∀ i, MeasurableSpace (β i)} {f : ∀ i, Ω → β i} (hf_Indep : iIndepFun m f μ)
     (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
     IndepFun (fun a => (f i a, f j a)) (f k) μ := by
@@ -850,7 +850,7 @@ set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun.indep_fun_prod ProbabilityTheory.iIndepFun.indepFun_prod
 
 @[to_additive]
-theorem iIndepFun.mul [ProbabilityMeasure μ] {ι : Type _} {β : Type _} {m : MeasurableSpace β}
+theorem iIndepFun.mul [IsProbabilityMeasure μ] {ι : Type _} {β : Type _} {m : MeasurableSpace β}
     [Mul β] [MeasurableMul₂ β] {f : ι → Ω → β} (hf_Indep : iIndepFun (fun _ => m) f μ)
     (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
     IndepFun (f i * f j) (f k) μ := by
@@ -864,7 +864,7 @@ set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun.add ProbabilityTheory.iIndepFun.add
 
 @[to_additive]
-theorem iIndepFun.indepFun_finset_prod_of_not_mem [ProbabilityMeasure μ] {ι : Type _} {β : Type _}
+theorem iIndepFun.indepFun_finset_prod_of_not_mem [IsProbabilityMeasure μ] {ι : Type _} {β : Type _}
     {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂ β] {f : ι → Ω → β}
     (hf_Indep : iIndepFun (fun _ => m) f μ) (hf_meas : ∀ i, Measurable (f i)) {s : Finset ι} {i : ι}
     (hi : i ∉ s) : IndepFun (∏ j in s, f j) (f i) μ := by
@@ -891,7 +891,7 @@ set_option linter.uppercaseLean3 false in
 #align probability_theory.Indep_fun.indep_fun_finset_sum_of_not_mem ProbabilityTheory.iIndepFun.indepFun_finset_sum_of_not_mem
 
 @[to_additive]
-theorem iIndepFun.indepFun_prod_range_succ [ProbabilityMeasure μ] {β : Type _}
+theorem iIndepFun.indepFun_prod_range_succ [IsProbabilityMeasure μ] {β : Type _}
     {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂ β] {f : ℕ → Ω → β}
     (hf_Indep : iIndepFun (fun _ => m) f μ) (hf_meas : ∀ i, Measurable (f i)) (n : ℕ) :
     IndepFun (∏ j in Finset.range n, f j) (f n) μ :=

@@ -90,7 +90,7 @@ set_option linter.uppercaseLean3 false in
 `t₁ ∈ p₁, t₂ ∈ s₂`, then `μ (t₁ ∩ t₂) = μ (t₁) * μ (t₂)` -/
 def IndepSets [MeasurableSpace Ω] (s1 s2 : Set (Set Ω)) (μ : Measure Ω := by volume_tac) : Prop
     := ∀ t1 t2 : Set Ω, t1 ∈ s1 → t2 ∈ s2 → μ (t1 ∩ t2) = μ t1 * μ t2
-#align probability_theory.indep_sets ProbabilityTheory.iIndepSets
+#align probability_theory.indep_sets ProbabilityTheory.IndepSets
 
 /-- A family of measurable space structures (i.e. of σ-algebras) is independent with respect to a
 measure `μ` (typically defined on a finer σ-algebra) if the family of sets of measurable sets they

The names of many definitions and lemmas in the Mathlib3 file contained Indep so Mathport added Cat at a bunch of places that had to be removed. The Mathlib4 equivalent is iIndep see this thread