probability.independence.zero_one ⟷ Mathlib.Probability.Independence.ZeroOne

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

@@ -43,7 +43,7 @@ theorem measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω} (h_indep : I
   · 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 
+  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_indepSet_self
 -/

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

@@ -48,13 +48,13 @@ theorem measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω} (h_indep : I
 #align probability_theory.measure_eq_zero_or_one_or_top_of_indep_set_self ProbabilityTheory.measure_eq_zero_or_one_or_top_of_indepSet_self
 -/
 
-#print ProbabilityTheory.measure_eq_zero_or_one_of_indepSetCat_self /-
-theorem measure_eq_zero_or_one_of_indepSetCat_self [IsFiniteMeasure μ] {t : Set Ω}
+#print ProbabilityTheory.measure_eq_zero_or_one_of_indepSet_self /-
+theorem measure_eq_zero_or_one_of_indepSet_self [IsFiniteMeasure μ] {t : Set Ω}
     (h_indep : IndepSet 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
+#align probability_theory.measure_eq_zero_or_one_of_indep_set_self ProbabilityTheory.measure_eq_zero_or_one_of_indepSet_self
 -/
 
 variable [IsProbabilityMeasure μ] {s : ι → MeasurableSpace Ω}
@@ -145,7 +145,7 @@ theorem measure_zero_or_one_of_measurableSet_limsup (h_le : ∀ n, s n ≤ m0) (
     (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
+  measure_eq_zero_or_one_of_indepSet_self
     ((indep_limsup_self h_le h_indep hf hns hnsp hns_univ).indepSet_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
@@ -182,7 +182,7 @@ 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 : iIndep 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
+  measure_eq_zero_or_one_of_indepSet_self
     ((indep_limsup_atTop_self h_le h_indep).indepSet_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
 -/
@@ -206,7 +206,7 @@ theorem indep_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s 
     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 directed_of_isDirected_ge fun i j hij k hki => hij.trans hki
   · exact fun n => ⟨n, le_rfl⟩
 #align probability_theory.indep_limsup_at_bot_self ProbabilityTheory.indep_limsup_atBot_self
 -/
@@ -217,7 +217,7 @@ 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 : iIndep 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
+  measure_eq_zero_or_one_of_indepSet_self
     ((indep_limsup_atBot_self h_le h_indep).indepSet_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
 -/

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.Probability.Independence.Basic
+import Probability.Independence.Basic
 
 #align_import probability.independence.zero_one from "leanprover-community/mathlib"@"af471b9e3ce868f296626d33189b4ce730fa4c00"
 

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.zero_one
-! 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.Probability.Independence.Basic
 
+#align_import probability.independence.zero_one from "leanprover-community/mathlib"@"af471b9e3ce868f296626d33189b4ce730fa4c00"
+
 /-!
 # Kolmogorov's 0-1 law
 

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

@@ -35,6 +35,7 @@ namespace ProbabilityTheory
 
 variable {Ω ι : Type _} {m m0 : MeasurableSpace Ω} {μ : Measure Ω}
 
+#print ProbabilityTheory.measure_eq_zero_or_one_or_top_of_indepSet_self /-
 theorem measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω} (h_indep : IndepSet t t μ) :
     μ t = 0 ∨ μ t = 1 ∨ μ t = ∞ :=
   by
@@ -48,22 +49,27 @@ theorem measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω} (h_indep : I
   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_indepSet_self
+-/
 
+#print ProbabilityTheory.measure_eq_zero_or_one_of_indepSetCat_self /-
 theorem measure_eq_zero_or_one_of_indepSetCat_self [IsFiniteMeasure μ] {t : Set Ω}
     (h_indep : IndepSet 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 [IsProbabilityMeasure μ] {s : ι → MeasurableSpace Ω}
 
 open Filter
 
+#print ProbabilityTheory.indep_biSup_compl /-
 theorem indep_biSup_compl (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (t : Set ι) :
     Indep (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) μ :=
   indep_iSup_of_disjoint h_le h_indep disjoint_compl_right
 #align probability_theory.indep_bsupr_compl ProbabilityTheory.indep_biSup_compl
+-/
 
 section Abstract
 
@@ -80,6 +86,7 @@ For the example of `f = at_top`, we can take `p = bdd_above` and `ns : ι → se
 
 
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic filter.is_bounded_default -/
+#print ProbabilityTheory.indep_biSup_limsup /-
 theorem indep_biSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
     {t : Set ι} (ht : p t) : Indep (⨆ n ∈ t, s n) (limsup s f) μ :=
   by
@@ -93,7 +100,9 @@ theorem indep_biSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (h
   simp only [Set.mem_compl_iff, eventually_map]
   exact eventually_of_mem (hf t ht) le_iSup₂
 #align probability_theory.indep_bsupr_limsup ProbabilityTheory.indep_biSup_limsup
+-/
 
+#print ProbabilityTheory.indep_iSup_directed_limsup /-
 theorem indep_iSup_directed_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ)
     (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) :
     Indep (⨆ a, ⨆ n ∈ ns a, s n) (limsup s f) μ :=
@@ -108,7 +117,9 @@ theorem indep_iSup_directed_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep
     · exact hc.1 hn
     · exact hc.2 hn
 #align probability_theory.indep_supr_directed_limsup ProbabilityTheory.indep_iSup_directed_limsup
+-/
 
+#print ProbabilityTheory.indep_iSup_limsup /-
 theorem indep_iSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
     (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
     Indep (⨆ n, s n) (limsup s f) μ :=
@@ -122,13 +133,17 @@ theorem indep_iSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf
   haveI : Nonempty (∃ i : α, n ∈ ns i) := ⟨hns_univ n⟩
   rw [this, iSup_const]
 #align probability_theory.indep_supr_limsup ProbabilityTheory.indep_iSup_limsup
+-/
 
+#print ProbabilityTheory.indep_limsup_self /-
 theorem indep_limsup_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
     (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
     Indep (limsup s f) (limsup s f) μ :=
   indep_of_indep_of_le_left (indep_iSup_limsup h_le h_indep hf hns hnsp hns_univ) limsup_le_iSup
 #align probability_theory.indep_limsup_self ProbabilityTheory.indep_limsup_self
+-/
 
+#print ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup /-
 theorem measure_zero_or_one_of_measurableSet_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep 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) :
@@ -137,6 +152,7 @@ theorem measure_zero_or_one_of_measurableSet_limsup (h_le : ∀ n, s n ≤ m0) (
     ((indep_limsup_self h_le h_indep hf hns hnsp hns_univ).indepSet_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
 
@@ -144,6 +160,7 @@ section AtTop
 
 variable [SemilatticeSup ι] [NoMaxOrder ι] [Nonempty ι]
 
+#print ProbabilityTheory.indep_limsup_atTop_self /-
 theorem indep_limsup_atTop_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) :
     Indep (limsup s atTop) (limsup s atTop) μ :=
   by
@@ -159,7 +176,9 @@ theorem indep_limsup_atTop_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s 
   · 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.indep_limsup_atTop_self
+-/
 
+#print ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup_atTop /-
 /-- **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`. -/
@@ -169,6 +188,7 @@ theorem measure_zero_or_one_of_measurableSet_limsup_atTop (h_le : ∀ n, s n ≤
   measure_eq_zero_or_one_of_indepSetCat_self
     ((indep_limsup_atTop_self h_le h_indep).indepSet_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
 
@@ -176,6 +196,7 @@ section AtBot
 
 variable [SemilatticeInf ι] [NoMinOrder ι] [Nonempty ι]
 
+#print ProbabilityTheory.indep_limsup_atBot_self /-
 theorem indep_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) :
     Indep (limsup s atBot) (limsup s atBot) μ :=
   by
@@ -191,7 +212,9 @@ theorem indep_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s 
   · 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.indep_limsup_atBot_self
+-/
 
+#print ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup_atBot /-
 /-- **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)
@@ -200,6 +223,7 @@ theorem measure_zero_or_one_of_measurableSet_limsup_atBot (h_le : ∀ n, s n ≤
   measure_eq_zero_or_one_of_indepSetCat_self
     ((indep_limsup_atBot_self h_le h_indep).indepSet_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
 

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.zero_one
-! 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.Probability.Independence.Basic
 /-!
 # Kolmogorov's 0-1 law
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 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.
 
@@ -32,8 +35,8 @@ namespace ProbabilityTheory
 
 variable {Ω ι : Type _} {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 = ∞ :=
+theorem measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω} (h_indep : IndepSet 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))
@@ -44,23 +47,23 @@ theorem measure_eq_zero_or_one_or_top_of_indepSetCat_self {t : Set Ω}
   · 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
+#align probability_theory.measure_eq_zero_or_one_or_top_of_indep_set_self ProbabilityTheory.measure_eq_zero_or_one_or_top_of_indepSet_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 :=
+theorem measure_eq_zero_or_one_of_indepSetCat_self [IsFiniteMeasure μ] {t : Set Ω}
+    (h_indep : IndepSet 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}
+variable [IsProbabilityMeasure 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_iSup_of_disjoint h_le h_indep disjoint_compl_right
-#align probability_theory.indep_bsupr_compl ProbabilityTheory.indepCat_bsupr_compl
+theorem indep_biSup_compl (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (t : Set ι) :
+    Indep (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) μ :=
+  indep_iSup_of_disjoint h_le h_indep disjoint_compl_right
+#align probability_theory.indep_bsupr_compl ProbabilityTheory.indep_biSup_compl
 
 section Abstract
 
@@ -77,8 +80,8 @@ For the example of `f = at_top`, we can take `p = bdd_above` and `ns : ι → se
 
 
 /- ./././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 :=
+theorem indep_biSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
+    {t : Set ι} (ht : p t) : Indep (⨆ 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'
@@ -89,11 +92,11 @@ theorem indepCat_bsupr_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ)
       _
   simp only [Set.mem_compl_iff, eventually_map]
   exact eventually_of_mem (hf t ht) le_iSup₂
-#align probability_theory.indep_bsupr_limsup ProbabilityTheory.indepCat_bsupr_limsup
+#align probability_theory.indep_bsupr_limsup ProbabilityTheory.indep_biSup_limsup
 
-theorem indepCat_iSup_directed_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ)
+theorem indep_iSup_directed_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep 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 :=
+    Indep (⨆ 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)
@@ -104,11 +107,11 @@ theorem indepCat_iSup_directed_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : Inde
     refine' ⟨c, _, _⟩ <;> refine' iSup_mono fun n => iSup_mono' fun hn => ⟨_, le_rfl⟩
     · exact hc.1 hn
     · exact hc.2 hn
-#align probability_theory.indep_supr_directed_limsup ProbabilityTheory.indepCat_iSup_directed_limsup
+#align probability_theory.indep_supr_directed_limsup ProbabilityTheory.indep_iSup_directed_limsup
 
-theorem indepCat_iSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
+theorem indep_iSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep 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 :=
+    Indep (⨆ n, s n) (limsup s f) μ :=
   by
   suffices (⨆ a, ⨆ n ∈ ns a, s n) = ⨆ n, s n by
     rw [← this]
@@ -118,21 +121,20 @@ theorem indepCat_iSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) (
   have : (⨆ (i : α) (H : n ∈ ns i), s n) = ⨆ h : ∃ i, n ∈ ns i, s n := by rw [iSup_exists]
   haveI : Nonempty (∃ i : α, n ∈ ns i) := ⟨hns_univ n⟩
   rw [this, iSup_const]
-#align probability_theory.indep_supr_limsup ProbabilityTheory.indepCat_iSup_limsup
+#align probability_theory.indep_supr_limsup ProbabilityTheory.indep_iSup_limsup
 
-theorem indepCat_limsup_self (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
+theorem indep_limsup_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep 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_iSup_limsup h_le h_indep hf hns hnsp hns_univ)
-    limsup_le_iSup
-#align probability_theory.indep_limsup_self ProbabilityTheory.indepCat_limsup_self
+    Indep (limsup s f) (limsup s f) μ :=
+  indep_of_indep_of_le_left (indep_iSup_limsup h_le h_indep hf hns hnsp hns_univ) limsup_le_iSup
+#align probability_theory.indep_limsup_self ProbabilityTheory.indep_limsup_self
 
-theorem measure_zero_or_one_of_measurableSet_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ)
+theorem measure_zero_or_one_of_measurableSet_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep 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
+    ((indep_limsup_self h_le h_indep hf hns hnsp hns_univ).indepSet_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
 
@@ -142,8 +144,8 @@ 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 :=
+theorem indep_limsup_atTop_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) :
+    Indep (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
@@ -156,16 +158,16 @@ theorem indepCat_limsup_atTop_self (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s
     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
+#align probability_theory.indep_limsup_at_top_self ProbabilityTheory.indep_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) :
+    (h_indep : iIndep 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)
+    ((indep_limsup_atTop_self h_le h_indep).indepSet_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
@@ -174,8 +176,8 @@ 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 :=
+theorem indep_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) :
+    Indep (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
@@ -188,15 +190,15 @@ theorem indepCat_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s
     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
+#align probability_theory.indep_limsup_at_bot_self ProbabilityTheory.indep_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) :
+    (h_indep : iIndep 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)
+    ((indep_limsup_atBot_self h_le h_indep).indepSet_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

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

@@ -42,7 +42,7 @@ theorem measure_eq_zero_or_one_or_top_of_indepSetCat_self {t : Set Ω}
   · 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
+  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
 

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

@@ -26,7 +26,7 @@ is measurable with respect to the tail σ-algebra `limsup s at_top` has probabil
 
 open MeasureTheory MeasurableSpace
 
-open MeasureTheory ENNReal
+open scoped MeasureTheory ENNReal
 
 namespace ProbabilityTheory
 

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

@@ -152,8 +152,7 @@ theorem indepCat_limsup_atTop_self (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s
     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)
+    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⟩
@@ -185,8 +184,7 @@ theorem indepCat_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s
     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)
+    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⟩

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

@@ -59,7 +59,7 @@ 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
+  indepCat_iSup_of_disjoint h_le h_indep disjoint_compl_right
 #align probability_theory.indep_bsupr_compl ProbabilityTheory.indepCat_bsupr_compl
 
 section Abstract
@@ -88,43 +88,43 @@ theorem indepCat_bsupr_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ)
           is_bounded_default)
       _
   simp only [Set.mem_compl_iff, eventually_map]
-  exact eventually_of_mem (hf t ht) le_supᵢ₂
+  exact eventually_of_mem (hf t ht) le_iSup₂
 #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)
+theorem indepCat_iSup_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)
+  · exact fun a => iSup₂_le fun n hn => h_le n
+  · exact limsup_le_supr.trans (iSup_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⟩
+    refine' ⟨c, _, _⟩ <;> refine' iSup_mono fun n => iSup_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
+#align probability_theory.indep_supr_directed_limsup ProbabilityTheory.indepCat_iSup_directed_limsup
 
-theorem indepCat_supᵢ_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
+theorem indepCat_iSup_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]
+  rw [iSup_comm]
+  refine' iSup_congr fun n => _
+  have : (⨆ (i : α) (H : n ∈ ns i), s n) = ⨆ h : ∃ i, n ∈ ns i, s n := by rw [iSup_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
+  rw [this, iSup_const]
+#align probability_theory.indep_supr_limsup ProbabilityTheory.indepCat_iSup_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
+  indepCat_of_indepCat_of_le_left (indepCat_iSup_limsup h_le h_indep hf hns hnsp hns_univ)
+    limsup_le_iSup
 #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)

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

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

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -247,7 +247,7 @@ theorem kernel.indep_limsup_atTop_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIn
     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)
+    suffices ∀ i ∈ t, i < c from 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⟩
@@ -301,7 +301,7 @@ theorem kernel.indep_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIn
     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)
+    suffices ∀ i ∈ t, c < i from lt_irrefl c (this c hct)
     exact fun i hi => hc.trans_lt (hb.trans_le (ha hi))
   · exact Antitone.directed_le fun _ _ ↦ Set.Ici_subset_Ici.2
   · exact fun n => ⟨n, le_rfl⟩

Also replace some Type _ with Type*.

@@ -28,7 +28,7 @@ open scoped MeasureTheory ENNReal
 
 namespace ProbabilityTheory
 
-variable {α Ω ι : Type _} {_mα : MeasurableSpace α} {m m0 : MeasurableSpace Ω}
+variable {α Ω ι : Type*} {_mα : MeasurableSpace α} {m m0 : MeasurableSpace Ω}
   {κ : kernel α Ω} {μα : Measure α} {μ : Measure Ω}
 
 theorem kernel.measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω}
@@ -63,7 +63,7 @@ theorem measure_eq_zero_or_one_of_indepSet_self [IsFiniteMeasure μ] {t : Set Ω
 #align probability_theory.measure_eq_zero_or_one_of_indep_set_self ProbabilityTheory.measure_eq_zero_or_one_of_indepSet_self
 
 theorem condexp_eq_zero_or_one_of_condIndepSet_self
-    [TopologicalSpace Ω] [BorelSpace Ω] [PolishSpace Ω] [Nonempty Ω]
+    [StandardBorelSpace Ω] [Nonempty Ω]
     (hm : m ≤ m0) [hμ : IsFiniteMeasure μ] {t : Set Ω} (ht : MeasurableSet t)
     (h_indep : CondIndepSet m hm t t μ) :
     ∀ᵐ ω ∂μ, (μ⟦t | m⟧) ω = 0 ∨ (μ⟦t | m⟧) ω = 1 := by
@@ -87,7 +87,7 @@ theorem indep_biSup_compl (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (t
   kernel.indep_biSup_compl h_le h_indep t
 #align probability_theory.indep_bsupr_compl ProbabilityTheory.indep_biSup_compl
 
-theorem condIndep_biSup_compl [TopologicalSpace Ω] [BorelSpace Ω] [PolishSpace Ω] [Nonempty Ω]
+theorem condIndep_biSup_compl [StandardBorelSpace Ω] [Nonempty Ω]
     (hm : m ≤ m0) [IsFiniteMeasure μ]
     (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) (t : Set ι) :
     CondIndep m (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) hm μ :=
@@ -121,7 +121,7 @@ theorem indep_biSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (h
   kernel.indep_biSup_limsup h_le h_indep hf ht
 #align probability_theory.indep_bsupr_limsup ProbabilityTheory.indep_biSup_limsup
 
-theorem condIndep_biSup_limsup [TopologicalSpace Ω] [BorelSpace Ω] [PolishSpace Ω] [Nonempty Ω]
+theorem condIndep_biSup_limsup [StandardBorelSpace Ω] [Nonempty Ω]
     (hm : m ≤ m0) [IsFiniteMeasure μ]
     (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) (hf : ∀ t, p t → tᶜ ∈ f)
     {t : Set ι} (ht : p t) :
@@ -147,7 +147,7 @@ theorem indep_iSup_directed_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep
   kernel.indep_iSup_directed_limsup h_le h_indep hf hns hnsp
 #align probability_theory.indep_supr_directed_limsup ProbabilityTheory.indep_iSup_directed_limsup
 
-theorem condIndep_iSup_directed_limsup [TopologicalSpace Ω] [BorelSpace Ω] [PolishSpace Ω]
+theorem condIndep_iSup_directed_limsup [StandardBorelSpace Ω]
     [Nonempty Ω] (hm : m ≤ m0) [IsFiniteMeasure μ]
     (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ)
     (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) :
@@ -173,7 +173,7 @@ theorem indep_iSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf
   kernel.indep_iSup_limsup h_le h_indep hf hns hnsp hns_univ
 #align probability_theory.indep_supr_limsup ProbabilityTheory.indep_iSup_limsup
 
-theorem condIndep_iSup_limsup [TopologicalSpace Ω] [BorelSpace Ω] [PolishSpace Ω] [Nonempty Ω]
+theorem condIndep_iSup_limsup [StandardBorelSpace Ω] [Nonempty Ω]
     (hm : m ≤ m0) [IsFiniteMeasure μ]
     (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) (hf : ∀ t, p t → tᶜ ∈ f)
     (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
@@ -192,7 +192,7 @@ theorem indep_limsup_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf
   kernel.indep_limsup_self h_le h_indep hf hns hnsp hns_univ
 #align probability_theory.indep_limsup_self ProbabilityTheory.indep_limsup_self
 
-theorem condIndep_limsup_self [TopologicalSpace Ω] [BorelSpace Ω] [PolishSpace Ω] [Nonempty Ω]
+theorem condIndep_limsup_self [StandardBorelSpace Ω] [Nonempty Ω]
     (hm : m ≤ m0) [IsFiniteMeasure μ]
     (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) (hf : ∀ t, p t → tᶜ ∈ f)
     (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
@@ -217,8 +217,8 @@ theorem measure_zero_or_one_of_measurableSet_limsup (h_le : ∀ n, s n ≤ m0) (
       ht_tail
 #align probability_theory.measure_zero_or_one_of_measurable_set_limsup ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup
 
-theorem condexp_zero_or_one_of_measurableSet_limsup [TopologicalSpace Ω] [BorelSpace Ω]
-    [PolishSpace Ω] [Nonempty Ω] (hm : m ≤ m0) [IsFiniteMeasure μ]
+theorem condexp_zero_or_one_of_measurableSet_limsup [StandardBorelSpace Ω] [Nonempty Ω]
+    (hm : m ≤ m0) [IsFiniteMeasure μ]
     (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm 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 : MeasurableSet[limsup s f] t) :
@@ -257,7 +257,7 @@ theorem indep_limsup_atTop_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s 
   kernel.indep_limsup_atTop_self h_le h_indep
 #align probability_theory.indep_limsup_at_top_self ProbabilityTheory.indep_limsup_atTop_self
 
-theorem condIndep_limsup_atTop_self [TopologicalSpace Ω] [BorelSpace Ω] [PolishSpace Ω] [Nonempty Ω]
+theorem condIndep_limsup_atTop_self [StandardBorelSpace Ω] [Nonempty Ω]
     (hm : m ≤ m0) [IsFiniteMeasure μ]
     (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) :
     CondIndep m (limsup s atTop) (limsup s atTop) hm μ :=
@@ -279,9 +279,8 @@ theorem measure_zero_or_one_of_measurableSet_limsup_atTop (h_le : ∀ n, s n ≤
     using kernel.measure_zero_or_one_of_measurableSet_limsup_atTop h_le h_indep ht_tail
 #align probability_theory.measure_zero_or_one_of_measurable_set_limsup_at_top ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup_atTop
 
-theorem condexp_zero_or_one_of_measurableSet_limsup_atTop [TopologicalSpace Ω] [BorelSpace Ω]
-    [PolishSpace Ω] [Nonempty Ω] (hm : m ≤ m0) [IsFiniteMeasure μ]
-    (h_le : ∀ n, s n ≤ m0)
+theorem condexp_zero_or_one_of_measurableSet_limsup_atTop [StandardBorelSpace Ω] [Nonempty Ω]
+    (hm : m ≤ m0) [IsFiniteMeasure μ] (h_le : ∀ n, s n ≤ m0)
     (h_indep : iCondIndep m hm s μ) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atTop] t) :
     ∀ᵐ ω ∂μ, (μ⟦t | m⟧) ω = 0 ∨ (μ⟦t | m⟧) ω = 1 :=
   condexp_eq_zero_or_one_of_condIndepSet_self hm (limsup_le_iSup.trans (iSup_le h_le) t ht_tail)
@@ -312,7 +311,7 @@ theorem indep_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s 
   kernel.indep_limsup_atBot_self h_le h_indep
 #align probability_theory.indep_limsup_at_bot_self ProbabilityTheory.indep_limsup_atBot_self
 
-theorem condIndep_limsup_atBot_self [TopologicalSpace Ω] [BorelSpace Ω] [PolishSpace Ω] [Nonempty Ω]
+theorem condIndep_limsup_atBot_self [StandardBorelSpace Ω] [Nonempty Ω]
     (hm : m ≤ m0) [IsFiniteMeasure μ]
     (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) :
     CondIndep m (limsup s atBot) (limsup s atBot) hm μ :=
@@ -337,9 +336,8 @@ theorem measure_zero_or_one_of_measurableSet_limsup_atBot (h_le : ∀ n, s n ≤
 
 /-- **Kolmogorov's 0-1 law**, conditional version: any event in the tail σ-algebra of a
 conditinoally independent sequence of sub-σ-algebras has conditional probability 0 or 1. -/
-theorem condexp_zero_or_one_of_measurableSet_limsup_atBot [TopologicalSpace Ω] [BorelSpace Ω]
-    [PolishSpace Ω] [Nonempty Ω] (hm : m ≤ m0) [IsFiniteMeasure μ]
-    (h_le : ∀ n, s n ≤ m0)
+theorem condexp_zero_or_one_of_measurableSet_limsup_atBot [StandardBorelSpace Ω] [Nonempty Ω]
+    (hm : m ≤ m0) [IsFiniteMeasure μ] (h_le : ∀ n, s n ≤ m0)
     (h_indep : iCondIndep m hm s μ) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atBot] t) :
     ∀ᵐ ω ∂μ, (μ⟦t | m⟧) ω = 0 ∨ (μ⟦t | m⟧) ω = 1 :=
   condexp_eq_zero_or_one_of_condIndepSet_self hm (limsup_le_iSup.trans (iSup_le h_le) t ht_tail)

Define conditional independence of sigma-algebras, sets and functions. This is a special case of independence with respect to a kernel and a measure. Conditional independence is obtained by using the conditional expectation kernel condexpKernel.

Co-authored-by: Rémy Degenne <remydegenne@gmail.com> Co-authored-by: RemyDegenne <remydegenne@gmail.com>

@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 -/
 import Mathlib.Probability.Independence.Basic
+import Mathlib.Probability.Independence.Conditional
 
 #align_import probability.independence.zero_one from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
 
@@ -27,36 +28,71 @@ open scoped MeasureTheory ENNReal
 
 namespace ProbabilityTheory
 
-variable {Ω ι : Type*} {m m0 : MeasurableSpace Ω} {μ : Measure Ω}
+variable {α Ω ι : Type _} {_mα : MeasurableSpace α} {m m0 : MeasurableSpace Ω}
+  {κ : kernel α Ω} {μα : Measure α} {μ : Measure Ω}
 
-theorem measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω}
-    (h_indep : IndepSet t t μ) : μ t = 0 ∨ μ t = 1 ∨ μ t = ∞ := by
-  rw [IndepSet_iff] at h_indep
+theorem kernel.measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω}
+    (h_indep : kernel.IndepSet t t κ μα) :
+    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 ∨ κ a t = ∞ := by
   specialize h_indep t t (measurableSet_generateFrom (Set.mem_singleton t))
     (measurableSet_generateFrom (Set.mem_singleton t))
-  by_cases h0 : μ t = 0
+  filter_upwards [h_indep] with a ha
+  by_cases h0 : κ a t = 0
   · exact Or.inl h0
-  by_cases h_top : μ t = ∞
+  by_cases h_top : κ a 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)
+  rw [← one_mul (κ a (t ∩ t)), Set.inter_self, ENNReal.mul_eq_mul_right h0 h_top] at ha
+  exact Or.inr (Or.inl ha.symm)
+
+theorem measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω}
+    (h_indep : IndepSet t t μ) : μ t = 0 ∨ μ t = 1 ∨ μ t = ∞ := by
+  simpa only [ae_dirac_eq, Filter.eventually_pure]
+    using kernel.measure_eq_zero_or_one_or_top_of_indepSet_self h_indep
 #align probability_theory.measure_eq_zero_or_one_or_top_of_indep_set_self ProbabilityTheory.measure_eq_zero_or_one_or_top_of_indepSet_self
 
+theorem kernel.measure_eq_zero_or_one_of_indepSet_self [∀ a, IsFiniteMeasure (κ a)] {t : Set Ω}
+    (h_indep : IndepSet t t κ μα) :
+    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 := by
+  filter_upwards [measure_eq_zero_or_one_or_top_of_indepSet_self h_indep] with a h_0_1_top
+  simpa only [measure_ne_top (κ a), or_false] using h_0_1_top
+
 theorem measure_eq_zero_or_one_of_indepSet_self [IsFiniteMeasure μ] {t : Set Ω}
     (h_indep : IndepSet t t μ) : μ t = 0 ∨ μ t = 1 := by
-  have h_0_1_top := measure_eq_zero_or_one_or_top_of_indepSet_self h_indep
-  simpa [measure_ne_top μ] using h_0_1_top
+  simpa only [ae_dirac_eq, Filter.eventually_pure]
+    using kernel.measure_eq_zero_or_one_of_indepSet_self h_indep
 #align probability_theory.measure_eq_zero_or_one_of_indep_set_self ProbabilityTheory.measure_eq_zero_or_one_of_indepSet_self
 
-variable [IsProbabilityMeasure μ] {s : ι → MeasurableSpace Ω}
+theorem condexp_eq_zero_or_one_of_condIndepSet_self
+    [TopologicalSpace Ω] [BorelSpace Ω] [PolishSpace Ω] [Nonempty Ω]
+    (hm : m ≤ m0) [hμ : IsFiniteMeasure μ] {t : Set Ω} (ht : MeasurableSet t)
+    (h_indep : CondIndepSet m hm t t μ) :
+    ∀ᵐ ω ∂μ, (μ⟦t | m⟧) ω = 0 ∨ (μ⟦t | m⟧) ω = 1 := by
+  have h := ae_of_ae_trim hm (kernel.measure_eq_zero_or_one_of_indepSet_self h_indep)
+  filter_upwards [condexpKernel_ae_eq_condexp hm ht, h] with ω hω_eq hω
+  rw [← hω_eq, ENNReal.toReal_eq_zero_iff, ENNReal.toReal_eq_one_iff]
+  cases hω with
+  | inl h => exact Or.inl (Or.inl h)
+  | inr h => exact Or.inr h
+
+variable [IsMarkovKernel κ] [IsProbabilityMeasure μ] {s : ι → MeasurableSpace Ω}
 
 open Filter
 
+theorem kernel.indep_biSup_compl (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα) (t : Set ι) :
+    Indep (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) κ μα :=
+  indep_iSup_of_disjoint h_le h_indep disjoint_compl_right
+
 theorem indep_biSup_compl (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (t : Set ι) :
     Indep (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) μ :=
-  indep_iSup_of_disjoint h_le h_indep disjoint_compl_right
+  kernel.indep_biSup_compl h_le h_indep t
 #align probability_theory.indep_bsupr_compl ProbabilityTheory.indep_biSup_compl
 
+theorem condIndep_biSup_compl [TopologicalSpace Ω] [BorelSpace Ω] [PolishSpace Ω] [Nonempty Ω]
+    (hm : m ≤ m0) [IsFiniteMeasure μ]
+    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) (t : Set ι) :
+    CondIndep m (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) hm μ :=
+  kernel.indep_biSup_compl h_le h_indep t
+
 section Abstract
 
 variable {α : Type*} {p : Set ι → Prop} {f : Filter ι} {ns : α → Set ι}
@@ -71,18 +107,30 @@ For the example of `f = atTop`, we can take
 `p = bddAbove` and `ns : ι → Set ι := fun i => Set.Iic i`.
 -/
 
-
-theorem indep_biSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
-    {t : Set ι} (ht : p t) : Indep (⨆ n ∈ t, s n) (limsup s f) μ := by
+theorem kernel.indep_biSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα)
+    (hf : ∀ t, p t → tᶜ ∈ f) {t : Set ι} (ht : p t) :
+    Indep (⨆ n ∈ t, s n) (limsup s f) κ μα := by
   refine' indep_of_indep_of_le_right (indep_biSup_compl h_le h_indep t) _
   refine' limsSup_le_of_le (by isBoundedDefault) _
   simp only [Set.mem_compl_iff, eventually_map]
   exact eventually_of_mem (hf t ht) le_iSup₂
+
+theorem indep_biSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
+    {t : Set ι} (ht : p t) :
+    Indep (⨆ n ∈ t, s n) (limsup s f) μ :=
+  kernel.indep_biSup_limsup h_le h_indep hf ht
 #align probability_theory.indep_bsupr_limsup ProbabilityTheory.indep_biSup_limsup
 
-theorem indep_iSup_directed_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ)
+theorem condIndep_biSup_limsup [TopologicalSpace Ω] [BorelSpace Ω] [PolishSpace Ω] [Nonempty Ω]
+    (hm : m ≤ m0) [IsFiniteMeasure μ]
+    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) (hf : ∀ t, p t → tᶜ ∈ f)
+    {t : Set ι} (ht : p t) :
+    CondIndep m (⨆ n ∈ t, s n) (limsup s f) hm μ :=
+  kernel.indep_biSup_limsup h_le h_indep hf ht
+
+theorem kernel.indep_iSup_directed_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα)
     (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) :
-    Indep (⨆ a, ⨆ n ∈ ns a, s n) (limsup s f) μ := by
+    Indep (⨆ a, ⨆ n ∈ ns a, s n) (limsup s f) κ μα := by
   refine' indep_iSup_of_directed_le _ _ _ _
   · exact fun a => indep_biSup_limsup h_le h_indep hf (hnsp a)
   · exact fun a => iSup₂_le fun n _ => h_le n
@@ -92,11 +140,24 @@ theorem indep_iSup_directed_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep
     refine' ⟨c, _, _⟩ <;> refine' iSup_mono fun n => iSup_mono' fun hn => ⟨_, le_rfl⟩
     · exact hc.1 hn
     · exact hc.2 hn
+
+theorem indep_iSup_directed_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ)
+    (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) :
+    Indep (⨆ a, ⨆ n ∈ ns a, s n) (limsup s f) μ :=
+  kernel.indep_iSup_directed_limsup h_le h_indep hf hns hnsp
 #align probability_theory.indep_supr_directed_limsup ProbabilityTheory.indep_iSup_directed_limsup
 
-theorem indep_iSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
+theorem condIndep_iSup_directed_limsup [TopologicalSpace Ω] [BorelSpace Ω] [PolishSpace Ω]
+    [Nonempty Ω] (hm : m ≤ m0) [IsFiniteMeasure μ]
+    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ)
+    (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) :
+    CondIndep m (⨆ a, ⨆ n ∈ ns a, s n) (limsup s f) hm μ :=
+  kernel.indep_iSup_directed_limsup h_le h_indep hf hns hnsp
+
+theorem kernel.indep_iSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα)
+    (hf : ∀ t, p t → tᶜ ∈ f)
     (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
-    Indep (⨆ n, s n) (limsup s f) μ := by
+    Indep (⨆ n, s n) (limsup s f) κ μα := by
   suffices (⨆ a, ⨆ n ∈ ns a, s n) = ⨆ n, s n by
     rw [← this]
     exact indep_iSup_directed_limsup h_le h_indep hf hns hnsp
@@ -105,31 +166,80 @@ theorem indep_iSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf
   have h : ⨆ (i : α) (_ : n ∈ ns i), s n = ⨆ _ : ∃ i, n ∈ ns i, s n := by rw [iSup_exists]
   haveI : Nonempty (∃ i : α, n ∈ ns i) := ⟨hns_univ n⟩
   rw [h, iSup_const]
+
+theorem indep_iSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
+    (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
+    Indep (⨆ n, s n) (limsup s f) μ :=
+  kernel.indep_iSup_limsup h_le h_indep hf hns hnsp hns_univ
 #align probability_theory.indep_supr_limsup ProbabilityTheory.indep_iSup_limsup
 
+theorem condIndep_iSup_limsup [TopologicalSpace Ω] [BorelSpace Ω] [PolishSpace Ω] [Nonempty Ω]
+    (hm : m ≤ m0) [IsFiniteMeasure μ]
+    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) (hf : ∀ t, p t → tᶜ ∈ f)
+    (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
+    CondIndep m (⨆ n, s n) (limsup s f) hm μ :=
+  kernel.indep_iSup_limsup h_le h_indep hf hns hnsp hns_univ
+
+theorem kernel.indep_limsup_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα)
+    (hf : ∀ t, p t → tᶜ ∈ f)
+    (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
+    Indep (limsup s f) (limsup s f) κ μα :=
+  indep_of_indep_of_le_left (indep_iSup_limsup h_le h_indep hf hns hnsp hns_univ) limsup_le_iSup
+
 theorem indep_limsup_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
     (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
     Indep (limsup s f) (limsup s f) μ :=
-  indep_of_indep_of_le_left (indep_iSup_limsup h_le h_indep hf hns hnsp hns_univ) limsup_le_iSup
+  kernel.indep_limsup_self h_le h_indep hf hns hnsp hns_univ
 #align probability_theory.indep_limsup_self ProbabilityTheory.indep_limsup_self
 
-theorem measure_zero_or_one_of_measurableSet_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ)
+theorem condIndep_limsup_self [TopologicalSpace Ω] [BorelSpace Ω] [PolishSpace Ω] [Nonempty Ω]
+    (hm : m ≤ m0) [IsFiniteMeasure μ]
+    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) (hf : ∀ t, p t → tᶜ ∈ f)
+    (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
+    CondIndep m (limsup s f) (limsup s f) hm μ :=
+  kernel.indep_limsup_self h_le h_indep hf hns hnsp hns_univ
+
+theorem kernel.measure_zero_or_one_of_measurableSet_limsup (h_le : ∀ n, s n ≤ m0)
+    (h_indep : iIndep s κ μα)
     (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 : MeasurableSet[limsup s f] t) :
-    μ t = 0 ∨ μ t = 1 :=
+    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 :=
   measure_eq_zero_or_one_of_indepSet_self
     ((indep_limsup_self h_le h_indep hf hns hnsp hns_univ).indepSet_of_measurableSet ht_tail
       ht_tail)
+
+theorem measure_zero_or_one_of_measurableSet_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep 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 : MeasurableSet[limsup s f] t) :
+    μ t = 0 ∨ μ t = 1 :=
+  by simpa only [ae_dirac_eq, Filter.eventually_pure]
+    using kernel.measure_zero_or_one_of_measurableSet_limsup h_le h_indep hf hns hnsp hns_univ
+      ht_tail
 #align probability_theory.measure_zero_or_one_of_measurable_set_limsup ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup
 
+theorem condexp_zero_or_one_of_measurableSet_limsup [TopologicalSpace Ω] [BorelSpace Ω]
+    [PolishSpace Ω] [Nonempty Ω] (hm : m ≤ m0) [IsFiniteMeasure μ]
+    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm 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 : MeasurableSet[limsup s f] t) :
+    ∀ᵐ ω ∂μ, (μ⟦t | m⟧) ω = 0 ∨ (μ⟦t | m⟧) ω = 1 := by
+  have h := ae_of_ae_trim hm
+    (kernel.measure_zero_or_one_of_measurableSet_limsup h_le h_indep hf hns hnsp hns_univ ht_tail)
+  have ht : MeasurableSet t := limsup_le_iSup.trans (iSup_le h_le) t ht_tail
+  filter_upwards [condexpKernel_ae_eq_condexp hm ht, h] with ω hω_eq hω
+  rw [← hω_eq, ENNReal.toReal_eq_zero_iff, ENNReal.toReal_eq_one_iff]
+  cases hω with
+  | inl h => exact Or.inl (Or.inl h)
+  | inr h => exact Or.inr h
+
 end Abstract
 
 section AtTop
 
 variable [SemilatticeSup ι] [NoMaxOrder ι] [Nonempty ι]
 
-theorem indep_limsup_atTop_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) :
-    Indep (limsup s atTop) (limsup s atTop) μ := by
+theorem kernel.indep_limsup_atTop_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα) :
+    Indep (limsup s atTop) (limsup s atTop) κ μα := 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 _
@@ -141,26 +251,50 @@ theorem indep_limsup_atTop_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s 
     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⟩
+
+theorem indep_limsup_atTop_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) :
+    Indep (limsup s atTop) (limsup s atTop) μ :=
+  kernel.indep_limsup_atTop_self h_le h_indep
 #align probability_theory.indep_limsup_at_top_self ProbabilityTheory.indep_limsup_atTop_self
 
+theorem condIndep_limsup_atTop_self [TopologicalSpace Ω] [BorelSpace Ω] [PolishSpace Ω] [Nonempty Ω]
+    (hm : m ≤ m0) [IsFiniteMeasure μ]
+    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) :
+    CondIndep m (limsup s atTop) (limsup s atTop) hm μ :=
+  kernel.indep_limsup_atTop_self h_le h_indep
+
+theorem kernel.measure_zero_or_one_of_measurableSet_limsup_atTop (h_le : ∀ n, s n ≤ m0)
+    (h_indep : iIndep s κ μα) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atTop] t) :
+    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 :=
+  measure_eq_zero_or_one_of_indepSet_self
+    ((indep_limsup_atTop_self h_le h_indep).indepSet_of_measurableSet ht_tail ht_tail)
+
 /-- **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 atTop` 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 : iIndep s μ) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atTop] t) :
-    μ t = 0 ∨ μ t = 1 :=
-  measure_eq_zero_or_one_of_indepSet_self
-    ((indep_limsup_atTop_self h_le h_indep).indepSet_of_measurableSet ht_tail ht_tail)
+    μ t = 0 ∨ μ t = 1 := by
+  simpa only [ae_dirac_eq, Filter.eventually_pure]
+    using kernel.measure_zero_or_one_of_measurableSet_limsup_atTop h_le h_indep ht_tail
 #align probability_theory.measure_zero_or_one_of_measurable_set_limsup_at_top ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup_atTop
 
+theorem condexp_zero_or_one_of_measurableSet_limsup_atTop [TopologicalSpace Ω] [BorelSpace Ω]
+    [PolishSpace Ω] [Nonempty Ω] (hm : m ≤ m0) [IsFiniteMeasure μ]
+    (h_le : ∀ n, s n ≤ m0)
+    (h_indep : iCondIndep m hm s μ) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atTop] t) :
+    ∀ᵐ ω ∂μ, (μ⟦t | m⟧) ω = 0 ∨ (μ⟦t | m⟧) ω = 1 :=
+  condexp_eq_zero_or_one_of_condIndepSet_self hm (limsup_le_iSup.trans (iSup_le h_le) t ht_tail)
+    ((condIndep_limsup_atTop_self hm h_le h_indep).condIndepSet_of_measurableSet ht_tail ht_tail)
+
 end AtTop
 
 section AtBot
 
 variable [SemilatticeInf ι] [NoMinOrder ι] [Nonempty ι]
 
-theorem indep_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) :
-    Indep (limsup s atBot) (limsup s atBot) μ := by
+theorem kernel.indep_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα) :
+    Indep (limsup s atBot) (limsup s atBot) κ μα := 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 _
@@ -172,17 +306,45 @@ theorem indep_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s 
     exact fun i hi => hc.trans_lt (hb.trans_le (ha hi))
   · exact Antitone.directed_le fun _ _ ↦ Set.Ici_subset_Ici.2
   · exact fun n => ⟨n, le_rfl⟩
+
+theorem indep_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) :
+    Indep (limsup s atBot) (limsup s atBot) μ :=
+  kernel.indep_limsup_atBot_self h_le h_indep
 #align probability_theory.indep_limsup_at_bot_self ProbabilityTheory.indep_limsup_atBot_self
 
+theorem condIndep_limsup_atBot_self [TopologicalSpace Ω] [BorelSpace Ω] [PolishSpace Ω] [Nonempty Ω]
+    (hm : m ≤ m0) [IsFiniteMeasure μ]
+    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) :
+    CondIndep m (limsup s atBot) (limsup s atBot) hm μ :=
+  kernel.indep_limsup_atBot_self h_le h_indep
+
+/-- **Kolmogorov's 0-1 law**, kernel version: any event in the tail σ-algebra of an independent
+sequence of sub-σ-algebras has probability 0 or 1 almost surely. -/
+theorem kernel.measure_zero_or_one_of_measurableSet_limsup_atBot (h_le : ∀ n, s n ≤ m0)
+    (h_indep : iIndep s κ μα) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atBot] t) :
+    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 :=
+  measure_eq_zero_or_one_of_indepSet_self
+    ((indep_limsup_atBot_self h_le h_indep).indepSet_of_measurableSet ht_tail ht_tail)
+
 /-- **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 : iIndep s μ) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atBot] t) :
-    μ t = 0 ∨ μ t = 1 :=
-  measure_eq_zero_or_one_of_indepSet_self
-    ((indep_limsup_atBot_self h_le h_indep).indepSet_of_measurableSet ht_tail ht_tail)
+    μ t = 0 ∨ μ t = 1 := by
+  simpa only [ae_dirac_eq, Filter.eventually_pure]
+    using kernel.measure_zero_or_one_of_measurableSet_limsup_atBot h_le h_indep ht_tail
 #align probability_theory.measure_zero_or_one_of_measurable_set_limsup_at_bot ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup_atBot
 
+/-- **Kolmogorov's 0-1 law**, conditional version: any event in the tail σ-algebra of a
+conditinoally independent sequence of sub-σ-algebras has conditional probability 0 or 1. -/
+theorem condexp_zero_or_one_of_measurableSet_limsup_atBot [TopologicalSpace Ω] [BorelSpace Ω]
+    [PolishSpace Ω] [Nonempty Ω] (hm : m ≤ m0) [IsFiniteMeasure μ]
+    (h_le : ∀ n, s n ≤ m0)
+    (h_indep : iCondIndep m hm s μ) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atBot] t) :
+    ∀ᵐ ω ∂μ, (μ⟦t | m⟧) ω = 0 ∨ (μ⟦t | m⟧) ω = 1 :=
+  condexp_eq_zero_or_one_of_condIndepSet_self hm (limsup_le_iSup.trans (iSup_le h_le) t ht_tail)
+    ((condIndep_limsup_atBot_self hm h_le h_indep).condIndepSet_of_measurableSet ht_tail ht_tail)
+
 end AtBot
 
 end ProbabilityTheory

New lemmas / instances

• An archimedean ordered semiring is directed upwards.
• Filter.hasAntitoneBasis_atTop;
• Filter.HasAntitoneBasis.iInf_principal;

Fix typos

• Docstrings: "if the agree" -> "if they agree".
• ProbabilityTheory.measure_eq_zero_or_one_of_indepSetCat_self -> ProbabilityTheory.measure_eq_zero_or_one_of_indepSet_self.

Weaken typeclass assumptions

From a semilattice to a directed type

• MeasureTheory.tendsto_measure_iUnion;
• MeasureTheory.tendsto_measure_iInter;
• Monotone.directed_le, Monotone.directed_ge;
• Antitone.directed_le, Antitone.directed_ge;
• directed_of_sup, renamed to directed_of_isDirected_le;
• directed_of_inf, renamed to directed_of_isDirected_ge;

From a strict ordered semiring to an ordered semiring

• tendsto_nat_cast_atTop_atTop;
• Filter.Eventually.nat_cast_atTop;
• atTop_hasAntitoneBasis_of_archimedean;

@@ -42,11 +42,11 @@ theorem measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω}
   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_indepSet_self
 
-theorem measure_eq_zero_or_one_of_indepSetCat_self [IsFiniteMeasure μ] {t : Set Ω}
+theorem measure_eq_zero_or_one_of_indepSet_self [IsFiniteMeasure μ] {t : Set Ω}
     (h_indep : IndepSet t t μ) : μ t = 0 ∨ μ t = 1 := by
   have h_0_1_top := measure_eq_zero_or_one_or_top_of_indepSet_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
+#align probability_theory.measure_eq_zero_or_one_of_indep_set_self ProbabilityTheory.measure_eq_zero_or_one_of_indepSet_self
 
 variable [IsProbabilityMeasure μ] {s : ι → MeasurableSpace Ω}
 
@@ -117,7 +117,7 @@ theorem measure_zero_or_one_of_measurableSet_limsup (h_le : ∀ n, s n ≤ m0) (
     (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 : MeasurableSet[limsup s f] t) :
     μ t = 0 ∨ μ t = 1 :=
-  measure_eq_zero_or_one_of_indepSetCat_self
+  measure_eq_zero_or_one_of_indepSet_self
     ((indep_limsup_self h_le h_indep hf hns hnsp hns_univ).indepSet_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
@@ -149,7 +149,7 @@ The tail σ-algebra `limsup s atTop` 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 : iIndep s μ) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atTop] t) :
     μ t = 0 ∨ μ t = 1 :=
-  measure_eq_zero_or_one_of_indepSetCat_self
+  measure_eq_zero_or_one_of_indepSet_self
     ((indep_limsup_atTop_self h_le h_indep).indepSet_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
 
@@ -170,7 +170,7 @@ theorem indep_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s 
     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 Antitone.directed_le fun _ _ ↦ Set.Ici_subset_Ici.2
   · exact fun n => ⟨n, le_rfl⟩
 #align probability_theory.indep_limsup_at_bot_self ProbabilityTheory.indep_limsup_atBot_self
 
@@ -179,7 +179,7 @@ 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 : iIndep s μ) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atBot] t) :
     μ t = 0 ∨ μ t = 1 :=
-  measure_eq_zero_or_one_of_indepSetCat_self
+  measure_eq_zero_or_one_of_indepSet_self
     ((indep_limsup_atBot_self h_le h_indep).indepSet_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
 

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

This has nice performance benefits.

@@ -27,7 +27,7 @@ open scoped MeasureTheory ENNReal
 
 namespace ProbabilityTheory
 
-variable {Ω ι : Type _} {m m0 : MeasurableSpace Ω} {μ : Measure Ω}
+variable {Ω ι : Type*} {m m0 : MeasurableSpace Ω} {μ : Measure Ω}
 
 theorem measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω}
     (h_indep : IndepSet t t μ) : μ t = 0 ∨ μ t = 1 ∨ μ t = ∞ := by
@@ -59,7 +59,7 @@ theorem indep_biSup_compl (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (t
 
 section Abstract
 
-variable {α : Type _} {p : Set ι → Prop} {f : Filter ι} {ns : α → Set ι}
+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:

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

@@ -31,6 +31,7 @@ variable {Ω ι : Type _} {m m0 : MeasurableSpace Ω} {μ : Measure Ω}
 
 theorem measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω}
     (h_indep : IndepSet t t μ) : μ t = 0 ∨ μ t = 1 ∨ μ t = ∞ := by
+  rw [IndepSet_iff] at h_indep
   specialize h_indep t t (measurableSet_generateFrom (Set.mem_singleton t))
     (measurableSet_generateFrom (Set.mem_singleton t))
   by_cases h0 : μ t = 0

• 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.zero_one
-! 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.Probability.Independence.Basic
 
+#align_import probability.independence.zero_one from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
+
 /-!
 # Kolmogorov's 0-1 law
 

@@ -104,7 +104,7 @@ theorem indep_iSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf
     exact indep_iSup_directed_limsup h_le h_indep hf hns hnsp
   rw [iSup_comm]
   refine' iSup_congr fun n => _
-  have h : (⨆ (i : α) (_ : n ∈ ns i), s n) = ⨆ _ : ∃ i, n ∈ ns i, s n := by rw [iSup_exists]
+  have h : ⨆ (i : α) (_ : n ∈ ns i), s n = ⨆ _ : ∃ i, n ∈ ns i, s n := by rw [iSup_exists]
   haveI : Nonempty (∃ i : α, n ∈ ns i) := ⟨hns_univ n⟩
   rw [h, iSup_const]
 #align probability_theory.indep_supr_limsup ProbabilityTheory.indep_iSup_limsup

@@ -104,7 +104,7 @@ theorem indep_iSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf
     exact indep_iSup_directed_limsup h_le h_indep hf hns hnsp
   rw [iSup_comm]
   refine' iSup_congr fun n => _
-  have h : (⨆ (i : α) (_H : n ∈ ns i), s n) = ⨆ _h : ∃ i, n ∈ ns i, s n := by rw [iSup_exists]
+  have h : (⨆ (i : α) (_ : n ∈ ns i), s n) = ⨆ _ : ∃ i, n ∈ ns i, s n := by rw [iSup_exists]
   haveI : Nonempty (∃ i : α, n ∈ ns i) := ⟨hns_univ n⟩
   rw [h, iSup_const]
 #align probability_theory.indep_supr_limsup ProbabilityTheory.indep_iSup_limsup

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.

@@ -44,13 +44,13 @@ theorem measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω}
   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_indepSet_self
 
-theorem measure_eq_zero_or_one_of_indepSetCat_self [FiniteMeasure μ] {t : Set Ω}
+theorem measure_eq_zero_or_one_of_indepSetCat_self [IsFiniteMeasure μ] {t : Set Ω}
     (h_indep : IndepSet t t μ) : μ t = 0 ∨ μ t = 1 := by
   have h_0_1_top := measure_eq_zero_or_one_or_top_of_indepSet_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}
+variable [IsProbabilityMeasure μ] {s : ι → MeasurableSpace Ω}
 
 open Filter