measure_theory.function.ae_eq_of_integral ⟷ Mathlib.MeasureTheory.Function.AEEqOfIntegral

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

@@ -265,9 +265,9 @@ section Real
 
 variable {f : α → ℝ}
 
-#print MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable /-
+#print MeasureTheory.ae_nonneg_of_forall_setIntegral_nonneg_of_stronglyMeasurable /-
 /-- Don't use this lemma. Use `ae_nonneg_of_forall_set_integral_nonneg`. -/
-theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : StronglyMeasurable f)
+theorem ae_nonneg_of_forall_setIntegral_nonneg_of_stronglyMeasurable (hfm : StronglyMeasurable f)
     (hf : Integrable f μ) (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) :
     0 ≤ᵐ[μ] f := by
   simp_rw [eventually_le, Pi.zero_apply]
@@ -305,11 +305,11 @@ theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : Str
   cases' (ENNReal.toReal_eq_zero_iff _).mp h_eq.symm with hμs_eq_zero hμs_eq_top
   · exact hμs_eq_zero
   · exact absurd hμs_eq_top mus.ne
-#align measure_theory.ae_nonneg_of_forall_set_integral_nonneg_of_strongly_measurable MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable
+#align measure_theory.ae_nonneg_of_forall_set_integral_nonneg_of_strongly_measurable MeasureTheory.ae_nonneg_of_forall_setIntegral_nonneg_of_stronglyMeasurable
 -/
 
-#print MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg /-
-theorem ae_nonneg_of_forall_set_integral_nonneg (hf : Integrable f μ)
+#print MeasureTheory.ae_nonneg_of_forall_setIntegral_nonneg /-
+theorem ae_nonneg_of_forall_setIntegral_nonneg (hf : Integrable f μ)
     (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f :=
   by
   rcases hf.1 with ⟨f', hf'_meas, hf_ae⟩
@@ -323,22 +323,22 @@ theorem ae_nonneg_of_forall_set_integral_nonneg (hf : Integrable f μ)
     (ae_nonneg_of_forall_set_integral_nonneg_of_strongly_measurable hf'_meas hf'_integrable
           hf'_zero).trans
       hf_ae.symm.le
-#align measure_theory.ae_nonneg_of_forall_set_integral_nonneg MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg
+#align measure_theory.ae_nonneg_of_forall_set_integral_nonneg MeasureTheory.ae_nonneg_of_forall_setIntegral_nonneg
 -/
 
-#print MeasureTheory.ae_le_of_forall_set_integral_le /-
-theorem ae_le_of_forall_set_integral_le {f g : α → ℝ} (hf : Integrable f μ) (hg : Integrable g μ)
+#print MeasureTheory.ae_le_of_forall_setIntegral_le /-
+theorem ae_le_of_forall_setIntegral_le {f g : α → ℝ} (hf : Integrable f μ) (hg : Integrable g μ)
     (hf_le : ∀ s, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ ≤ ∫ x in s, g x ∂μ) : f ≤ᵐ[μ] g :=
   by
   rw [← eventually_sub_nonneg]
   refine' ae_nonneg_of_forall_set_integral_nonneg (hg.sub hf) fun s hs => _
   rw [integral_sub' hg.integrable_on hf.integrable_on, sub_nonneg]
   exact hf_le s hs
-#align measure_theory.ae_le_of_forall_set_integral_le MeasureTheory.ae_le_of_forall_set_integral_le
+#align measure_theory.ae_le_of_forall_set_integral_le MeasureTheory.ae_le_of_forall_setIntegral_le
 -/
 
-#print MeasureTheory.ae_nonneg_restrict_of_forall_set_integral_nonneg_inter /-
-theorem ae_nonneg_restrict_of_forall_set_integral_nonneg_inter {f : α → ℝ} {t : Set α}
+#print MeasureTheory.ae_nonneg_restrict_of_forall_setIntegral_nonneg_inter /-
+theorem ae_nonneg_restrict_of_forall_setIntegral_nonneg_inter {f : α → ℝ} {t : Set α}
     (hf : IntegrableOn f t μ)
     (hf_zero : ∀ s, MeasurableSet s → μ (s ∩ t) < ∞ → 0 ≤ ∫ x in s ∩ t, f x ∂μ) :
     0 ≤ᵐ[μ.restrict t] f :=
@@ -347,11 +347,11 @@ theorem ae_nonneg_restrict_of_forall_set_integral_nonneg_inter {f : α → ℝ}
   simp_rw [measure.restrict_restrict hs]
   apply hf_zero s hs
   rwa [measure.restrict_apply hs] at h's
-#align measure_theory.ae_nonneg_restrict_of_forall_set_integral_nonneg_inter MeasureTheory.ae_nonneg_restrict_of_forall_set_integral_nonneg_inter
+#align measure_theory.ae_nonneg_restrict_of_forall_set_integral_nonneg_inter MeasureTheory.ae_nonneg_restrict_of_forall_setIntegral_nonneg_inter
 -/
 
-#print MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite /-
-theorem ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite [SigmaFinite μ] {f : α → ℝ}
+#print MeasureTheory.ae_nonneg_of_forall_setIntegral_nonneg_of_sigmaFinite /-
+theorem ae_nonneg_of_forall_setIntegral_nonneg_of_sigmaFinite [SigmaFinite μ] {f : α → ℝ}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f :=
   by
@@ -362,11 +362,11 @@ theorem ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite [SigmaFinite μ]
   exact
     hf_zero _ (s_meas.inter t_meas)
       (lt_of_le_of_lt (measure_mono (Set.inter_subset_right _ _)) t_lt_top)
-#align measure_theory.ae_nonneg_of_forall_set_integral_nonneg_of_sigma_finite MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite
+#align measure_theory.ae_nonneg_of_forall_set_integral_nonneg_of_sigma_finite MeasureTheory.ae_nonneg_of_forall_setIntegral_nonneg_of_sigmaFinite
 -/
 
-#print MeasureTheory.AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg /-
-theorem AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg {f : α → ℝ}
+#print MeasureTheory.AEFinStronglyMeasurable.ae_nonneg_of_forall_setIntegral_nonneg /-
+theorem AEFinStronglyMeasurable.ae_nonneg_of_forall_setIntegral_nonneg {f : α → ℝ}
     (hf : AEFinStronglyMeasurable f μ)
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f :=
@@ -383,11 +383,11 @@ theorem AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg {f : α
   · rw [measure.restrict_restrict hs]
     rw [measure.restrict_apply hs] at hμts
     exact hf_zero (s ∩ t) (hs.inter hf.measurable_set) hμts
-#align measure_theory.ae_fin_strongly_measurable.ae_nonneg_of_forall_set_integral_nonneg MeasureTheory.AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg
+#align measure_theory.ae_fin_strongly_measurable.ae_nonneg_of_forall_set_integral_nonneg MeasureTheory.AEFinStronglyMeasurable.ae_nonneg_of_forall_setIntegral_nonneg
 -/
 
-#print MeasureTheory.ae_nonneg_restrict_of_forall_set_integral_nonneg /-
-theorem ae_nonneg_restrict_of_forall_set_integral_nonneg {f : α → ℝ}
+#print MeasureTheory.ae_nonneg_restrict_of_forall_setIntegral_nonneg /-
+theorem ae_nonneg_restrict_of_forall_setIntegral_nonneg {f : α → ℝ}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) {t : Set α}
     (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : 0 ≤ᵐ[μ.restrict t] f :=
@@ -397,11 +397,11 @@ theorem ae_nonneg_restrict_of_forall_set_integral_nonneg {f : α → ℝ}
       (hf_int_finite t ht (lt_top_iff_ne_top.mpr hμt)) fun s hs h's => _
   refine' hf_zero (s ∩ t) (hs.inter ht) _
   exact (measure_mono (Set.inter_subset_right s t)).trans_lt (lt_top_iff_ne_top.mpr hμt)
-#align measure_theory.ae_nonneg_restrict_of_forall_set_integral_nonneg MeasureTheory.ae_nonneg_restrict_of_forall_set_integral_nonneg
+#align measure_theory.ae_nonneg_restrict_of_forall_set_integral_nonneg MeasureTheory.ae_nonneg_restrict_of_forall_setIntegral_nonneg
 -/
 
-#print MeasureTheory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real /-
-theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real {f : α → ℝ}
+#print MeasureTheory.ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real /-
+theorem ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real {f : α → ℝ}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) {t : Set α}
     (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] 0 :=
@@ -422,13 +422,13 @@ theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real {f : α → ℝ}
   simp_rw [Pi.neg_apply]
   rw [integral_neg, neg_nonneg]
   exact (hf_zero s hs hμs).le
-#align measure_theory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real MeasureTheory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real
+#align measure_theory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real MeasureTheory.ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real
 -/
 
 end Real
 
-#print MeasureTheory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero /-
-theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero {f : α → E}
+#print MeasureTheory.ae_eq_zero_restrict_of_forall_setIntegral_eq_zero /-
+theorem ae_eq_zero_restrict_of_forall_setIntegral_eq_zero {f : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) {t : Set α}
     (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] 0 :=
@@ -442,11 +442,11 @@ theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero {f : α → E}
   · intro s hs hμs
     rw [ContinuousLinearMap.integral_comp_comm c (hf_int_finite s hs hμs), hf_zero s hs hμs]
     exact ContinuousLinearMap.map_zero _
-#align measure_theory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero MeasureTheory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero
+#align measure_theory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero MeasureTheory.ae_eq_zero_restrict_of_forall_setIntegral_eq_zero
 -/
 
-#print MeasureTheory.ae_eq_restrict_of_forall_set_integral_eq /-
-theorem ae_eq_restrict_of_forall_set_integral_eq {f g : α → E}
+#print MeasureTheory.ae_eq_restrict_of_forall_setIntegral_eq /-
+theorem ae_eq_restrict_of_forall_setIntegral_eq {f g : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
     (hfg_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ)
@@ -461,11 +461,11 @@ theorem ae_eq_restrict_of_forall_set_integral_eq {f g : α → E}
   have hfg_int : ∀ s, MeasurableSet s → μ s < ∞ → integrable_on (f - g) s μ := fun s hs hμs =>
     (hf_int_finite s hs hμs).sub (hg_int_finite s hs hμs)
   exact ae_eq_zero_restrict_of_forall_set_integral_eq_zero hfg_int hfg' ht hμt
-#align measure_theory.ae_eq_restrict_of_forall_set_integral_eq MeasureTheory.ae_eq_restrict_of_forall_set_integral_eq
+#align measure_theory.ae_eq_restrict_of_forall_set_integral_eq MeasureTheory.ae_eq_restrict_of_forall_setIntegral_eq
 -/
 
-#print MeasureTheory.ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite /-
-theorem ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f : α → E}
+#print MeasureTheory.ae_eq_zero_of_forall_setIntegral_eq_of_sigmaFinite /-
+theorem ae_eq_zero_of_forall_setIntegral_eq_of_sigmaFinite [SigmaFinite μ] {f : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 :=
   by
@@ -478,11 +478,11 @@ theorem ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f
   have hμn : μ (S n) < ∞ := measure_spanning_sets_lt_top μ n
   rw [← measure.restrict_apply' h_meas_n]
   exact ae_eq_zero_restrict_of_forall_set_integral_eq_zero hf_int_finite hf_zero h_meas_n hμn.ne
-#align measure_theory.ae_eq_zero_of_forall_set_integral_eq_of_sigma_finite MeasureTheory.ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite
+#align measure_theory.ae_eq_zero_of_forall_set_integral_eq_of_sigma_finite MeasureTheory.ae_eq_zero_of_forall_setIntegral_eq_of_sigmaFinite
 -/
 
-#print MeasureTheory.ae_eq_of_forall_set_integral_eq_of_sigmaFinite /-
-theorem ae_eq_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f g : α → E}
+#print MeasureTheory.ae_eq_of_forall_setIntegral_eq_of_sigmaFinite /-
+theorem ae_eq_of_forall_setIntegral_eq_of_sigmaFinite [SigmaFinite μ] {f g : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
     (hfg_eq : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) :
@@ -496,11 +496,11 @@ theorem ae_eq_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f g : 
   have hfg_int : ∀ s, MeasurableSet s → μ s < ∞ → integrable_on (f - g) s μ := fun s hs hμs =>
     (hf_int_finite s hs hμs).sub (hg_int_finite s hs hμs)
   exact ae_eq_zero_of_forall_set_integral_eq_of_sigma_finite hfg_int hfg
-#align measure_theory.ae_eq_of_forall_set_integral_eq_of_sigma_finite MeasureTheory.ae_eq_of_forall_set_integral_eq_of_sigmaFinite
+#align measure_theory.ae_eq_of_forall_set_integral_eq_of_sigma_finite MeasureTheory.ae_eq_of_forall_setIntegral_eq_of_sigmaFinite
 -/
 
-#print MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero /-
-theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero {f : α → E}
+#print MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_of_forall_setIntegral_eq_zero /-
+theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_setIntegral_eq_zero {f : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
     (hf : AEFinStronglyMeasurable f μ) : f =ᵐ[μ] 0 :=
@@ -518,11 +518,11 @@ theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero {f : 
     rw [measure.restrict_restrict hs]
     rw [measure.restrict_apply hs] at hμs
     exact hf_zero _ (hs.inter hf.measurable_set) hμs
-#align measure_theory.ae_fin_strongly_measurable.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero
+#align measure_theory.ae_fin_strongly_measurable.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_of_forall_setIntegral_eq_zero
 -/
 
-#print MeasureTheory.AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq /-
-theorem AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq {f g : α → E}
+#print MeasureTheory.AEFinStronglyMeasurable.ae_eq_of_forall_setIntegral_eq /-
+theorem AEFinStronglyMeasurable.ae_eq_of_forall_setIntegral_eq {f g : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
     (hfg_eq : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ)
@@ -536,33 +536,33 @@ theorem AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq {f g : α → E}
       sub_eq_zero.mpr (hfg_eq s hs hμs)]
   have hfg_int : ∀ s, MeasurableSet s → μ s < ∞ → integrable_on (f - g) s μ := fun s hs hμs =>
     (hf_int_finite s hs hμs).sub (hg_int_finite s hs hμs)
-  exact (hf.sub hg).ae_eq_zero_of_forall_set_integral_eq_zero hfg_int hfg
-#align measure_theory.ae_fin_strongly_measurable.ae_eq_of_forall_set_integral_eq MeasureTheory.AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq
+  exact (hf.sub hg).ae_eq_zero_of_forall_setIntegral_eq_zero hfg_int hfg
+#align measure_theory.ae_fin_strongly_measurable.ae_eq_of_forall_set_integral_eq MeasureTheory.AEFinStronglyMeasurable.ae_eq_of_forall_setIntegral_eq
 -/
 
-#print MeasureTheory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero /-
-theorem Lp.ae_eq_zero_of_forall_set_integral_eq_zero (f : Lp E p μ) (hp_ne_zero : p ≠ 0)
+#print MeasureTheory.Lp.ae_eq_zero_of_forall_setIntegral_eq_zero /-
+theorem Lp.ae_eq_zero_of_forall_setIntegral_eq_zero (f : Lp E p μ) (hp_ne_zero : p ≠ 0)
     (hp_ne_top : p ≠ ∞) (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 :=
-  AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero hf_int_finite hf_zero
+  AEFinStronglyMeasurable.ae_eq_zero_of_forall_setIntegral_eq_zero hf_int_finite hf_zero
     (Lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).AEFinStronglyMeasurable
-#align measure_theory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero
+#align measure_theory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.Lp.ae_eq_zero_of_forall_setIntegral_eq_zero
 -/
 
-#print MeasureTheory.Lp.ae_eq_of_forall_set_integral_eq /-
-theorem Lp.ae_eq_of_forall_set_integral_eq (f g : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
+#print MeasureTheory.Lp.ae_eq_of_forall_setIntegral_eq /-
+theorem Lp.ae_eq_of_forall_setIntegral_eq (f g : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
     (hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) :
     f =ᵐ[μ] g :=
-  AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq hf_int_finite hg_int_finite hfg
+  AEFinStronglyMeasurable.ae_eq_of_forall_setIntegral_eq hf_int_finite hg_int_finite hfg
     (Lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).AEFinStronglyMeasurable
     (Lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).AEFinStronglyMeasurable
-#align measure_theory.Lp.ae_eq_of_forall_set_integral_eq MeasureTheory.Lp.ae_eq_of_forall_set_integral_eq
+#align measure_theory.Lp.ae_eq_of_forall_set_integral_eq MeasureTheory.Lp.ae_eq_of_forall_setIntegral_eq
 -/
 
-#print MeasureTheory.ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim /-
-theorem ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim (hm : m ≤ m0) {f : α → E}
+#print MeasureTheory.ae_eq_zero_of_forall_setIntegral_eq_of_finStronglyMeasurable_trim /-
+theorem ae_eq_zero_of_forall_setIntegral_eq_of_finStronglyMeasurable_trim (hm : m ≤ m0) {f : α → E}
     (hf_int_finite : ∀ s, measurable_set[m] s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s : Set α, measurable_set[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
     (hf : FinStronglyMeasurable f (μ.trim hm)) : f =ᵐ[μ] 0 :=
@@ -590,11 +590,11 @@ theorem ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim (hm :
       trim_measurable_set_eq hm (hs.inter ht_meas)] at hμs
     rw [← integral_trim hm hf_meas_m]
     exact hf_zero _ (hs.inter ht_meas) hμs
-#align measure_theory.ae_eq_zero_of_forall_set_integral_eq_of_fin_strongly_measurable_trim MeasureTheory.ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim
+#align measure_theory.ae_eq_zero_of_forall_set_integral_eq_of_fin_strongly_measurable_trim MeasureTheory.ae_eq_zero_of_forall_setIntegral_eq_of_finStronglyMeasurable_trim
 -/
 
-#print MeasureTheory.Integrable.ae_eq_zero_of_forall_set_integral_eq_zero /-
-theorem Integrable.ae_eq_zero_of_forall_set_integral_eq_zero {f : α → E} (hf : Integrable f μ)
+#print MeasureTheory.Integrable.ae_eq_zero_of_forall_setIntegral_eq_zero /-
+theorem Integrable.ae_eq_zero_of_forall_setIntegral_eq_zero {f : α → E} (hf : Integrable f μ)
     (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 :=
   by
   have hf_Lp : mem_ℒp f 1 μ := mem_ℒp_one_iff_integrable.mpr hf
@@ -606,11 +606,11 @@ theorem Integrable.ae_eq_zero_of_forall_set_integral_eq_zero {f : α → E} (hf
   · intro s hs hμs
     rw [integral_congr_ae (ae_restrict_of_ae hf_f_Lp.symm)]
     exact hf_zero s hs hμs
-#align measure_theory.integrable.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.Integrable.ae_eq_zero_of_forall_set_integral_eq_zero
+#align measure_theory.integrable.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.Integrable.ae_eq_zero_of_forall_setIntegral_eq_zero
 -/
 
-#print MeasureTheory.Integrable.ae_eq_of_forall_set_integral_eq /-
-theorem Integrable.ae_eq_of_forall_set_integral_eq (f g : α → E) (hf : Integrable f μ)
+#print MeasureTheory.Integrable.ae_eq_of_forall_setIntegral_eq /-
+theorem Integrable.ae_eq_of_forall_setIntegral_eq (f g : α → E) (hf : Integrable f μ)
     (hg : Integrable g μ)
     (hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) :
     f =ᵐ[μ] g := by
@@ -621,7 +621,7 @@ theorem Integrable.ae_eq_of_forall_set_integral_eq (f g : α → E) (hf : Integr
     rw [integral_sub' hf.integrable_on hg.integrable_on]
     exact sub_eq_zero.mpr (hfg s hs hμs)
   exact integrable.ae_eq_zero_of_forall_set_integral_eq_zero (hf.sub hg) hfg'
-#align measure_theory.integrable.ae_eq_of_forall_set_integral_eq MeasureTheory.Integrable.ae_eq_of_forall_set_integral_eq
+#align measure_theory.integrable.ae_eq_of_forall_set_integral_eq MeasureTheory.Integrable.ae_eq_of_forall_setIntegral_eq
 -/
 
 end AeEqOfForallSetIntegralEq

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

@@ -233,7 +233,7 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
     have L2 : ∀ᶠ n : ℕ in (at_top : Filter ℕ), g x ≤ (n : ℝ≥0) :=
       haveI : tendsto (fun n : ℕ => ((n : ℝ≥0) : ℝ≥0∞)) at_top (𝓝 ∞) :=
         by
-        simp only [ENNReal.coe_nat]
+        simp only [ENNReal.coe_natCast]
         exact ENNReal.tendsto_nat_nhds_top
       eventually_ge_of_tendsto_gt (hx.trans_le le_top) this
     apply Set.mem_iUnion.2

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

@@ -55,7 +55,7 @@ namespace MeasureTheory
 
 section AeEqOfForall
 
-variable {α E 𝕜 : Type _} {m : MeasurableSpace α} {μ : Measure α} [IsROrC 𝕜]
+variable {α E 𝕜 : Type _} {m : MeasurableSpace α} {μ : Measure α} [RCLike 𝕜]
 
 #print MeasureTheory.ae_eq_zero_of_forall_inner /-
 theorem ae_eq_zero_of_forall_inner [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
@@ -108,7 +108,7 @@ theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedS
       _ ≤ 1 * ‖(x : E) - a‖ := (ContinuousLinearMap.le_of_opNorm_le _ (hs x).1 _)
       _ < ‖a‖ / 2 := by rw [one_mul]; rwa [dist_eq_norm'] at hx
       _ < ‖(x : E)‖ := I
-      _ = ‖s x x‖ := by rw [(hs x).2, IsROrC.norm_coe_norm]
+      _ = ‖s x x‖ := by rw [(hs x).2, RCLike.norm_coe_norm]
   have hfs : ∀ y : d, ∀ᵐ x ∂μ, ⟪f x, s y⟫ = (0 : 𝕜) := fun y => hf (s y)
   have hf' : ∀ᵐ x ∂μ, ∀ y : d, ⟪f x, s y⟫ = (0 : 𝕜) := by rwa [ae_all_iff]
   filter_upwards [hf', h't] with x hx h'x

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

@@ -99,14 +99,14 @@ theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedS
     have I : ‖a‖ / 2 < ‖(x : E)‖ :=
       by
       have : ‖a‖ ≤ ‖(x : E)‖ + ‖a - x‖ := norm_le_insert' _ _
-      have : ‖a - x‖ < ‖a‖ / 2 := by rwa [dist_eq_norm] at hx 
+      have : ‖a - x‖ < ‖a‖ / 2 := by rwa [dist_eq_norm] at hx
       linarith
     intro h
     apply lt_irrefl ‖s x x‖
     calc
       ‖s x x‖ = ‖s x (x - a)‖ := by simp only [h, sub_zero, ContinuousLinearMap.map_sub]
       _ ≤ 1 * ‖(x : E) - a‖ := (ContinuousLinearMap.le_of_opNorm_le _ (hs x).1 _)
-      _ < ‖a‖ / 2 := by rw [one_mul]; rwa [dist_eq_norm'] at hx 
+      _ < ‖a‖ / 2 := by rw [one_mul]; rwa [dist_eq_norm'] at hx
       _ < ‖(x : E)‖ := I
       _ = ‖s x x‖ := by rw [(hs x).2, IsROrC.norm_coe_norm]
   have hfs : ∀ y : d, ∀ᵐ x ∂μ, ⟪f x, s y⟫ = (0 : 𝕜) := fun y => hf (s y)
@@ -157,7 +157,7 @@ theorem ae_const_le_iff_forall_lt_measure_zero {β} [LinearOrder β] [Topologica
     obtain ⟨b, b_up, bc⟩ : ∃ b : β, b ∈ upperBounds (Set.Iio c) ∧ b < c := by
       simpa [IsLUB, IsLeast, this, lowerBounds] using H
     exact measure_mono_null (fun x hx => b_up hx) (hc b bc)
-  push_neg at H h 
+  push_neg at H h
   obtain ⟨u, u_mono, u_lt, u_lim, -⟩ :
     ∃ u : ℕ → β,
       StrictMono u ∧ (∀ n : ℕ, u n < c) ∧ tendsto u at_top (nhds c) ∧ ∀ n : ℕ, u n ∈ Set.Iio c :=
@@ -223,12 +223,12 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
   have μs : ∀ n, μ (s n) = 0 := fun n => A _ _ _ (u_pos n)
   have B : {x | f x ≤ g x}ᶜ ⊆ ⋃ n, s n := by
     intro x hx
-    simp at hx 
+    simp at hx
     have L1 : ∀ᶠ n in at_top, g x + u n ≤ f x :=
       by
       have : tendsto (fun n => g x + u n) at_top (𝓝 (g x + (0 : ℝ≥0))) :=
         tendsto_const_nhds.add (ENNReal.tendsto_coe.2 u_lim)
-      simp at this 
+      simp at this
       exact eventually_le_of_tendsto_lt hx this
     have L2 : ∀ᶠ n : ℕ in (at_top : Filter ℕ), g x ≤ (n : ℝ≥0) :=
       haveI : tendsto (fun n : ℕ => ((n : ℝ≥0) : ℝ≥0∞)) at_top (𝓝 ∞) :=
@@ -282,7 +282,7 @@ theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : Str
       μ s ≤ μ {x | (c : ℝ≥0∞) ≤ ‖f x‖₊} := by
         apply measure_mono
         intro x hx
-        simp only [Set.mem_setOf_eq] at hx 
+        simp only [Set.mem_setOf_eq] at hx
         simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,
           Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe] using hx
       _ ≤ (∫⁻ x, ‖f x‖₊ ∂μ) / c :=
@@ -296,7 +296,7 @@ theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : Str
         set_integral_mono_ae_restrict hf.integrable_on (integrable_on_const.mpr (Or.inr mus)) _
       rw [eventually_le, ae_restrict_iff hs]
       exact eventually_of_forall fun x hxs => hxs
-    rwa [set_integral_const, smul_eq_mul, mul_comm] at h_const_le 
+    rwa [set_integral_const, smul_eq_mul, mul_comm] at h_const_le
   by_contra
   refine' (lt_self_iff_false (∫ x in s, f x ∂μ)).mp (h_int_gt.trans_lt _)
   refine' (mul_neg_iff.mpr (Or.inr ⟨hb_neg, _⟩)).trans_le _
@@ -346,7 +346,7 @@ theorem ae_nonneg_restrict_of_forall_set_integral_nonneg_inter {f : α → ℝ}
   refine' ae_nonneg_of_forall_set_integral_nonneg hf fun s hs h's => _
   simp_rw [measure.restrict_restrict hs]
   apply hf_zero s hs
-  rwa [measure.restrict_apply hs] at h's 
+  rwa [measure.restrict_apply hs] at h's
 #align measure_theory.ae_nonneg_restrict_of_forall_set_integral_nonneg_inter MeasureTheory.ae_nonneg_restrict_of_forall_set_integral_nonneg_inter
 -/
 
@@ -378,10 +378,10 @@ theorem AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg {f : α
   refine'
     ae_nonneg_of_forall_set_integral_nonneg_of_sigma_finite (fun s hs hμts => _) fun s hs hμts => _
   · rw [integrable_on, measure.restrict_restrict hs]
-    rw [measure.restrict_apply hs] at hμts 
+    rw [measure.restrict_apply hs] at hμts
     exact hf_int_finite (s ∩ t) (hs.inter hf.measurable_set) hμts
   · rw [measure.restrict_restrict hs]
-    rw [measure.restrict_apply hs] at hμts 
+    rw [measure.restrict_apply hs] at hμts
     exact hf_zero (s ∩ t) (hs.inter hf.measurable_set) hμts
 #align measure_theory.ae_fin_strongly_measurable.ae_nonneg_of_forall_set_integral_nonneg MeasureTheory.AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg
 -/
@@ -414,7 +414,7 @@ theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real {f : α → ℝ}
         (fun s hs hμs => (hf_zero s hs hμs).symm.le) ht hμt⟩
   suffices h_neg : 0 ≤ᵐ[μ.restrict t] -f
   · refine' h_neg.mono fun x hx => _
-    rw [Pi.neg_apply] at hx 
+    rw [Pi.neg_apply] at hx
     simpa using hx
   refine'
     ae_nonneg_restrict_of_forall_set_integral_nonneg (fun s hs hμs => (hf_int_finite s hs hμs).neg)
@@ -512,11 +512,11 @@ theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero {f : 
   refine' ae_eq_zero_of_forall_set_integral_eq_of_sigma_finite _ _
   · intro s hs hμs
     rw [integrable_on, measure.restrict_restrict hs]
-    rw [measure.restrict_apply hs] at hμs 
+    rw [measure.restrict_apply hs] at hμs
     exact hf_int_finite _ (hs.inter hf.measurable_set) hμs
   · intro s hs hμs
     rw [measure.restrict_restrict hs]
-    rw [measure.restrict_apply hs] at hμs 
+    rw [measure.restrict_apply hs] at hμs
     exact hf_zero _ (hs.inter hf.measurable_set) hμs
 #align measure_theory.ae_fin_strongly_measurable.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero
 -/
@@ -568,7 +568,7 @@ theorem ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim (hm :
     (hf : FinStronglyMeasurable f (μ.trim hm)) : f =ᵐ[μ] 0 :=
   by
   obtain ⟨t, ht_meas, htf_zero, htμ⟩ := hf.exists_set_sigma_finite
-  haveI : sigma_finite ((μ.restrict t).trim hm) := by rwa [restrict_trim hm μ ht_meas] at htμ 
+  haveI : sigma_finite ((μ.restrict t).trim hm) := by rwa [restrict_trim hm μ ht_meas] at htμ
   have htf_zero : f =ᵐ[μ.restrict (tᶜ)] 0 :=
     by
     rw [eventually_eq, ae_restrict_iff' (MeasurableSet.compl (hm _ ht_meas))]
@@ -581,13 +581,13 @@ theorem ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim (hm :
   · intro s hs hμs
     rw [integrable_on, restrict_trim hm (μ.restrict t) hs, measure.restrict_restrict (hm s hs)]
     rw [← restrict_trim hm μ ht_meas, measure.restrict_apply hs,
-      trim_measurable_set_eq hm (hs.inter ht_meas)] at hμs 
+      trim_measurable_set_eq hm (hs.inter ht_meas)] at hμs
     refine' integrable.trim hm _ hf_meas_m
     exact hf_int_finite _ (hs.inter ht_meas) hμs
   · intro s hs hμs
     rw [restrict_trim hm (μ.restrict t) hs, measure.restrict_restrict (hm s hs)]
     rw [← restrict_trim hm μ ht_meas, measure.restrict_apply hs,
-      trim_measurable_set_eq hm (hs.inter ht_meas)] at hμs 
+      trim_measurable_set_eq hm (hs.inter ht_meas)] at hμs
     rw [← integral_trim hm hf_meas_m]
     exact hf_zero _ (hs.inter ht_meas) hμs
 #align measure_theory.ae_eq_zero_of_forall_set_integral_eq_of_fin_strongly_measurable_trim MeasureTheory.ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim

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

@@ -120,7 +120,8 @@ theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedS
 theorem ae_eq_zero_of_forall_dual [NormedAddCommGroup E] [NormedSpace 𝕜 E]
     [SecondCountableTopology E] {f : α → E} (hf : ∀ c : Dual 𝕜 E, (fun x => ⟪f x, c⟫) =ᵐ[μ] 0) :
     f =ᵐ[μ] 0 :=
-  ae_eq_zero_of_forall_dual_of_isSeparable 𝕜 (isSeparable_of_separableSpace (Set.univ : Set E)) hf
+  ae_eq_zero_of_forall_dual_of_isSeparable 𝕜
+    (TopologicalSpace.IsSeparable.of_separableSpace (Set.univ : Set E)) hf
     (eventually_of_forall fun x => Set.mem_univ _)
 #align measure_theory.ae_eq_zero_of_forall_dual MeasureTheory.ae_eq_zero_of_forall_dual
 -/

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

@@ -105,7 +105,7 @@ theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedS
     apply lt_irrefl ‖s x x‖
     calc
       ‖s x x‖ = ‖s x (x - a)‖ := by simp only [h, sub_zero, ContinuousLinearMap.map_sub]
-      _ ≤ 1 * ‖(x : E) - a‖ := (ContinuousLinearMap.le_of_op_norm_le _ (hs x).1 _)
+      _ ≤ 1 * ‖(x : E) - a‖ := (ContinuousLinearMap.le_of_opNorm_le _ (hs x).1 _)
       _ < ‖a‖ / 2 := by rw [one_mul]; rwa [dist_eq_norm'] at hx 
       _ < ‖(x : E)‖ := I
       _ = ‖s x x‖ := by rw [(hs x).2, IsROrC.norm_coe_norm]

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

@@ -627,8 +627,8 @@ end AeEqOfForallSetIntegralEq
 
 section Lintegral
 
-#print MeasureTheory.AeMeasurable.ae_eq_of_forall_set_lintegral_eq /-
-theorem AeMeasurable.ae_eq_of_forall_set_lintegral_eq {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
+#print MeasureTheory.AEMeasurable.ae_eq_of_forall_set_lintegral_eq /-
+theorem AEMeasurable.ae_eq_of_forall_set_lintegral_eq {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
     (hg : AEMeasurable g μ) (hfi : ∫⁻ x, f x ∂μ ≠ ∞) (hgi : ∫⁻ x, g x ∂μ ≠ ∞)
     (hfg : ∀ ⦃s⦄, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) : f =ᵐ[μ] g :=
   by
@@ -650,7 +650,7 @@ theorem AeMeasurable.ae_eq_of_forall_set_lintegral_eq {f g : α → ℝ≥0∞}
   exacts [ae_of_all _ fun x => ENNReal.toReal_nonneg,
     hg.ennreal_to_real.restrict.ae_strongly_measurable, ae_of_all _ fun x => ENNReal.toReal_nonneg,
     hf.ennreal_to_real.restrict.ae_strongly_measurable]
-#align measure_theory.ae_measurable.ae_eq_of_forall_set_lintegral_eq MeasureTheory.AeMeasurable.ae_eq_of_forall_set_lintegral_eq
+#align measure_theory.ae_measurable.ae_eq_of_forall_set_lintegral_eq MeasureTheory.AEMeasurable.ae_eq_of_forall_set_lintegral_eq
 -/
 
 end Lintegral

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

@@ -3,10 +3,10 @@ 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.Analysis.InnerProductSpace.Basic
-import Mathbin.Analysis.NormedSpace.Dual
-import Mathbin.MeasureTheory.Function.StronglyMeasurable.Lp
-import Mathbin.MeasureTheory.Integral.SetIntegral
+import Analysis.InnerProductSpace.Basic
+import Analysis.NormedSpace.Dual
+import MeasureTheory.Function.StronglyMeasurable.Lp
+import MeasureTheory.Integral.SetIntegral
 
 #align_import measure_theory.function.ae_eq_of_integral from "leanprover-community/mathlib"@"c20927220ef87bb4962ba08bf6da2ce3cf50a6dd"
 

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

@@ -2,17 +2,14 @@
 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 measure_theory.function.ae_eq_of_integral
-! leanprover-community/mathlib commit c20927220ef87bb4962ba08bf6da2ce3cf50a6dd
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.InnerProductSpace.Basic
 import Mathbin.Analysis.NormedSpace.Dual
 import Mathbin.MeasureTheory.Function.StronglyMeasurable.Lp
 import Mathbin.MeasureTheory.Integral.SetIntegral
 
+#align_import measure_theory.function.ae_eq_of_integral from "leanprover-community/mathlib"@"c20927220ef87bb4962ba08bf6da2ce3cf50a6dd"
+
 /-! # From equality of integrals to equality of functions
 
 > THIS FILE IS SYNCHRONIZED WITH MATHLIB4.

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

@@ -60,6 +60,7 @@ section AeEqOfForall
 
 variable {α E 𝕜 : Type _} {m : MeasurableSpace α} {μ : Measure α} [IsROrC 𝕜]
 
+#print MeasureTheory.ae_eq_zero_of_forall_inner /-
 theorem ae_eq_zero_of_forall_inner [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
     [SecondCountableTopology E] {f : α → E} (hf : ∀ c : E, (fun x => (inner c (f x) : 𝕜)) =ᵐ[μ] 0) :
     f =ᵐ[μ] 0 := by
@@ -72,12 +73,13 @@ theorem ae_eq_zero_of_forall_inner [NormedAddCommGroup E] [InnerProductSpace 
     isClosed_eq (continuous_id.inner continuous_const) continuous_const
   exact @isClosed_property ℕ E _ s (fun c => inner c (f x) = (0 : 𝕜)) hs h_closed (fun n => hx n) _
 #align measure_theory.ae_eq_zero_of_forall_inner MeasureTheory.ae_eq_zero_of_forall_inner
+-/
 
--- mathport name: «expr⟪ , ⟫»
 local notation "⟪" x ", " y "⟫" => y x
 
 variable (𝕜)
 
+#print MeasureTheory.ae_eq_zero_of_forall_dual_of_isSeparable /-
 theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedSpace 𝕜 E]
     {t : Set E} (ht : TopologicalSpace.IsSeparable t) {f : α → E}
     (hf : ∀ c : Dual 𝕜 E, (fun x => ⟪f x, c⟫) =ᵐ[μ] 0) (h't : ∀ᵐ x ∂μ, f x ∈ t) : f =ᵐ[μ] 0 :=
@@ -115,13 +117,16 @@ theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedS
   filter_upwards [hf', h't] with x hx h'x
   exact A (f x) h'x hx
 #align measure_theory.ae_eq_zero_of_forall_dual_of_is_separable MeasureTheory.ae_eq_zero_of_forall_dual_of_isSeparable
+-/
 
+#print MeasureTheory.ae_eq_zero_of_forall_dual /-
 theorem ae_eq_zero_of_forall_dual [NormedAddCommGroup E] [NormedSpace 𝕜 E]
     [SecondCountableTopology E] {f : α → E} (hf : ∀ c : Dual 𝕜 E, (fun x => ⟪f x, c⟫) =ᵐ[μ] 0) :
     f =ᵐ[μ] 0 :=
   ae_eq_zero_of_forall_dual_of_isSeparable 𝕜 (isSeparable_of_separableSpace (Set.univ : Set E)) hf
     (eventually_of_forall fun x => Set.mem_univ _)
 #align measure_theory.ae_eq_zero_of_forall_dual MeasureTheory.ae_eq_zero_of_forall_dual
+-/
 
 variable {𝕜}
 
@@ -132,6 +137,7 @@ variable {α E : Type _} {m m0 : MeasurableSpace α} {μ : Measure α} {s t : Se
 
 section AeEqOfForallSetIntegralEq
 
+#print MeasureTheory.ae_const_le_iff_forall_lt_measure_zero /-
 theorem ae_const_le_iff_forall_lt_measure_zero {β} [LinearOrder β] [TopologicalSpace β]
     [OrderTopology β] [FirstCountableTopology β] (f : α → β) (c : β) :
     (∀ᵐ x ∂μ, c ≤ f x) ↔ ∀ b < c, μ {x | f x ≤ b} = 0 :=
@@ -169,11 +175,13 @@ theorem ae_const_le_iff_forall_lt_measure_zero {β} [LinearOrder β] [Topologica
   intro n
   exact hc _ (u_lt n)
 #align measure_theory.ae_const_le_iff_forall_lt_measure_zero MeasureTheory.ae_const_le_iff_forall_lt_measure_zero
+-/
 
 section ENNReal
 
 open scoped Topology
 
+#print MeasureTheory.ae_le_of_forall_set_lintegral_le_of_sigmaFinite /-
 theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g : α → ℝ≥0∞}
     (hf : Measurable f) (hg : Measurable g)
     (h : ∀ s, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ) : f ≤ᵐ[μ] g :=
@@ -238,7 +246,9 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
     _ ≤ ∑' n, μ (s n) := (measure_Union_le _)
     _ = 0 := by simp only [μs, tsum_zero]
 #align measure_theory.ae_le_of_forall_set_lintegral_le_of_sigma_finite MeasureTheory.ae_le_of_forall_set_lintegral_le_of_sigmaFinite
+-/
 
+#print MeasureTheory.ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite /-
 theorem ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite [SigmaFinite μ] {f g : α → ℝ≥0∞}
     (hf : Measurable f) (hg : Measurable g)
     (h : ∀ s, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) : f =ᵐ[μ] g :=
@@ -249,6 +259,7 @@ theorem ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite [SigmaFinite μ] {f g :
     ae_le_of_forall_set_lintegral_le_of_sigma_finite hg hf fun s hs h's => ge_of_eq (h s hs h's)
   filter_upwards [A, B] with x using le_antisymm
 #align measure_theory.ae_eq_of_forall_set_lintegral_eq_of_sigma_finite MeasureTheory.ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite
+-/
 
 end ENNReal
 
@@ -256,6 +267,7 @@ section Real
 
 variable {f : α → ℝ}
 
+#print MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable /-
 /-- Don't use this lemma. Use `ae_nonneg_of_forall_set_integral_nonneg`. -/
 theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : StronglyMeasurable f)
     (hf : Integrable f μ) (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) :
@@ -296,7 +308,9 @@ theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : Str
   · exact hμs_eq_zero
   · exact absurd hμs_eq_top mus.ne
 #align measure_theory.ae_nonneg_of_forall_set_integral_nonneg_of_strongly_measurable MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable
+-/
 
+#print MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg /-
 theorem ae_nonneg_of_forall_set_integral_nonneg (hf : Integrable f μ)
     (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f :=
   by
@@ -312,7 +326,9 @@ theorem ae_nonneg_of_forall_set_integral_nonneg (hf : Integrable f μ)
           hf'_zero).trans
       hf_ae.symm.le
 #align measure_theory.ae_nonneg_of_forall_set_integral_nonneg MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg
+-/
 
+#print MeasureTheory.ae_le_of_forall_set_integral_le /-
 theorem ae_le_of_forall_set_integral_le {f g : α → ℝ} (hf : Integrable f μ) (hg : Integrable g μ)
     (hf_le : ∀ s, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ ≤ ∫ x in s, g x ∂μ) : f ≤ᵐ[μ] g :=
   by
@@ -321,7 +337,9 @@ theorem ae_le_of_forall_set_integral_le {f g : α → ℝ} (hf : Integrable f μ
   rw [integral_sub' hg.integrable_on hf.integrable_on, sub_nonneg]
   exact hf_le s hs
 #align measure_theory.ae_le_of_forall_set_integral_le MeasureTheory.ae_le_of_forall_set_integral_le
+-/
 
+#print MeasureTheory.ae_nonneg_restrict_of_forall_set_integral_nonneg_inter /-
 theorem ae_nonneg_restrict_of_forall_set_integral_nonneg_inter {f : α → ℝ} {t : Set α}
     (hf : IntegrableOn f t μ)
     (hf_zero : ∀ s, MeasurableSet s → μ (s ∩ t) < ∞ → 0 ≤ ∫ x in s ∩ t, f x ∂μ) :
@@ -332,7 +350,9 @@ theorem ae_nonneg_restrict_of_forall_set_integral_nonneg_inter {f : α → ℝ}
   apply hf_zero s hs
   rwa [measure.restrict_apply hs] at h's 
 #align measure_theory.ae_nonneg_restrict_of_forall_set_integral_nonneg_inter MeasureTheory.ae_nonneg_restrict_of_forall_set_integral_nonneg_inter
+-/
 
+#print MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite /-
 theorem ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite [SigmaFinite μ] {f : α → ℝ}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f :=
@@ -345,7 +365,9 @@ theorem ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite [SigmaFinite μ]
     hf_zero _ (s_meas.inter t_meas)
       (lt_of_le_of_lt (measure_mono (Set.inter_subset_right _ _)) t_lt_top)
 #align measure_theory.ae_nonneg_of_forall_set_integral_nonneg_of_sigma_finite MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite
+-/
 
+#print MeasureTheory.AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg /-
 theorem AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg {f : α → ℝ}
     (hf : AEFinStronglyMeasurable f μ)
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
@@ -364,7 +386,9 @@ theorem AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg {f : α
     rw [measure.restrict_apply hs] at hμts 
     exact hf_zero (s ∩ t) (hs.inter hf.measurable_set) hμts
 #align measure_theory.ae_fin_strongly_measurable.ae_nonneg_of_forall_set_integral_nonneg MeasureTheory.AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg
+-/
 
+#print MeasureTheory.ae_nonneg_restrict_of_forall_set_integral_nonneg /-
 theorem ae_nonneg_restrict_of_forall_set_integral_nonneg {f : α → ℝ}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) {t : Set α}
@@ -376,7 +400,9 @@ theorem ae_nonneg_restrict_of_forall_set_integral_nonneg {f : α → ℝ}
   refine' hf_zero (s ∩ t) (hs.inter ht) _
   exact (measure_mono (Set.inter_subset_right s t)).trans_lt (lt_top_iff_ne_top.mpr hμt)
 #align measure_theory.ae_nonneg_restrict_of_forall_set_integral_nonneg MeasureTheory.ae_nonneg_restrict_of_forall_set_integral_nonneg
+-/
 
+#print MeasureTheory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real /-
 theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real {f : α → ℝ}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) {t : Set α}
@@ -399,9 +425,11 @@ theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real {f : α → ℝ}
   rw [integral_neg, neg_nonneg]
   exact (hf_zero s hs hμs).le
 #align measure_theory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real MeasureTheory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real
+-/
 
 end Real
 
+#print MeasureTheory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero /-
 theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero {f : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) {t : Set α}
@@ -417,7 +445,9 @@ theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero {f : α → E}
     rw [ContinuousLinearMap.integral_comp_comm c (hf_int_finite s hs hμs), hf_zero s hs hμs]
     exact ContinuousLinearMap.map_zero _
 #align measure_theory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero MeasureTheory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero
+-/
 
+#print MeasureTheory.ae_eq_restrict_of_forall_set_integral_eq /-
 theorem ae_eq_restrict_of_forall_set_integral_eq {f g : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
@@ -434,7 +464,9 @@ theorem ae_eq_restrict_of_forall_set_integral_eq {f g : α → E}
     (hf_int_finite s hs hμs).sub (hg_int_finite s hs hμs)
   exact ae_eq_zero_restrict_of_forall_set_integral_eq_zero hfg_int hfg' ht hμt
 #align measure_theory.ae_eq_restrict_of_forall_set_integral_eq MeasureTheory.ae_eq_restrict_of_forall_set_integral_eq
+-/
 
+#print MeasureTheory.ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite /-
 theorem ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 :=
@@ -449,7 +481,9 @@ theorem ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f
   rw [← measure.restrict_apply' h_meas_n]
   exact ae_eq_zero_restrict_of_forall_set_integral_eq_zero hf_int_finite hf_zero h_meas_n hμn.ne
 #align measure_theory.ae_eq_zero_of_forall_set_integral_eq_of_sigma_finite MeasureTheory.ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite
+-/
 
+#print MeasureTheory.ae_eq_of_forall_set_integral_eq_of_sigmaFinite /-
 theorem ae_eq_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f g : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
@@ -465,7 +499,9 @@ theorem ae_eq_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f g : 
     (hf_int_finite s hs hμs).sub (hg_int_finite s hs hμs)
   exact ae_eq_zero_of_forall_set_integral_eq_of_sigma_finite hfg_int hfg
 #align measure_theory.ae_eq_of_forall_set_integral_eq_of_sigma_finite MeasureTheory.ae_eq_of_forall_set_integral_eq_of_sigmaFinite
+-/
 
+#print MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero /-
 theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero {f : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
@@ -485,7 +521,9 @@ theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero {f : 
     rw [measure.restrict_apply hs] at hμs 
     exact hf_zero _ (hs.inter hf.measurable_set) hμs
 #align measure_theory.ae_fin_strongly_measurable.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero
+-/
 
+#print MeasureTheory.AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq /-
 theorem AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq {f g : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
@@ -502,14 +540,18 @@ theorem AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq {f g : α → E}
     (hf_int_finite s hs hμs).sub (hg_int_finite s hs hμs)
   exact (hf.sub hg).ae_eq_zero_of_forall_set_integral_eq_zero hfg_int hfg
 #align measure_theory.ae_fin_strongly_measurable.ae_eq_of_forall_set_integral_eq MeasureTheory.AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq
+-/
 
+#print MeasureTheory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero /-
 theorem Lp.ae_eq_zero_of_forall_set_integral_eq_zero (f : Lp E p μ) (hp_ne_zero : p ≠ 0)
     (hp_ne_top : p ≠ ∞) (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 :=
   AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero hf_int_finite hf_zero
     (Lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).AEFinStronglyMeasurable
 #align measure_theory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero
+-/
 
+#print MeasureTheory.Lp.ae_eq_of_forall_set_integral_eq /-
 theorem Lp.ae_eq_of_forall_set_integral_eq (f g : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
@@ -519,7 +561,9 @@ theorem Lp.ae_eq_of_forall_set_integral_eq (f g : Lp E p μ) (hp_ne_zero : p ≠
     (Lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).AEFinStronglyMeasurable
     (Lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).AEFinStronglyMeasurable
 #align measure_theory.Lp.ae_eq_of_forall_set_integral_eq MeasureTheory.Lp.ae_eq_of_forall_set_integral_eq
+-/
 
+#print MeasureTheory.ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim /-
 theorem ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim (hm : m ≤ m0) {f : α → E}
     (hf_int_finite : ∀ s, measurable_set[m] s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s : Set α, measurable_set[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
@@ -549,7 +593,9 @@ theorem ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim (hm :
     rw [← integral_trim hm hf_meas_m]
     exact hf_zero _ (hs.inter ht_meas) hμs
 #align measure_theory.ae_eq_zero_of_forall_set_integral_eq_of_fin_strongly_measurable_trim MeasureTheory.ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim
+-/
 
+#print MeasureTheory.Integrable.ae_eq_zero_of_forall_set_integral_eq_zero /-
 theorem Integrable.ae_eq_zero_of_forall_set_integral_eq_zero {f : α → E} (hf : Integrable f μ)
     (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 :=
   by
@@ -563,7 +609,9 @@ theorem Integrable.ae_eq_zero_of_forall_set_integral_eq_zero {f : α → E} (hf
     rw [integral_congr_ae (ae_restrict_of_ae hf_f_Lp.symm)]
     exact hf_zero s hs hμs
 #align measure_theory.integrable.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.Integrable.ae_eq_zero_of_forall_set_integral_eq_zero
+-/
 
+#print MeasureTheory.Integrable.ae_eq_of_forall_set_integral_eq /-
 theorem Integrable.ae_eq_of_forall_set_integral_eq (f g : α → E) (hf : Integrable f μ)
     (hg : Integrable g μ)
     (hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) :
@@ -576,11 +624,13 @@ theorem Integrable.ae_eq_of_forall_set_integral_eq (f g : α → E) (hf : Integr
     exact sub_eq_zero.mpr (hfg s hs hμs)
   exact integrable.ae_eq_zero_of_forall_set_integral_eq_zero (hf.sub hg) hfg'
 #align measure_theory.integrable.ae_eq_of_forall_set_integral_eq MeasureTheory.Integrable.ae_eq_of_forall_set_integral_eq
+-/
 
 end AeEqOfForallSetIntegralEq
 
 section Lintegral
 
+#print MeasureTheory.AeMeasurable.ae_eq_of_forall_set_lintegral_eq /-
 theorem AeMeasurable.ae_eq_of_forall_set_lintegral_eq {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
     (hg : AEMeasurable g μ) (hfi : ∫⁻ x, f x ∂μ ≠ ∞) (hgi : ∫⁻ x, g x ∂μ ≠ ∞)
     (hfg : ∀ ⦃s⦄, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) : f =ᵐ[μ] g :=
@@ -604,6 +654,7 @@ theorem AeMeasurable.ae_eq_of_forall_set_lintegral_eq {f g : α → ℝ≥0∞}
     hg.ennreal_to_real.restrict.ae_strongly_measurable, ae_of_all _ fun x => ENNReal.toReal_nonneg,
     hf.ennreal_to_real.restrict.ae_strongly_measurable]
 #align measure_theory.ae_measurable.ae_eq_of_forall_set_lintegral_eq MeasureTheory.AeMeasurable.ae_eq_of_forall_set_lintegral_eq
+-/
 
 end Lintegral
 

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

@@ -176,7 +176,7 @@ open scoped Topology
 
 theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g : α → ℝ≥0∞}
     (hf : Measurable f) (hg : Measurable g)
-    (h : ∀ s, MeasurableSet s → μ s < ∞ → (∫⁻ x in s, f x ∂μ) ≤ ∫⁻ x in s, g x ∂μ) : f ≤ᵐ[μ] g :=
+    (h : ∀ s, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ) : f ≤ᵐ[μ] g :=
   by
   have A :
     ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ε ≤ f x ∧ g x ≤ N} ∩ spanning_sets μ p) = 0 :=
@@ -190,18 +190,18 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
       exact (A.inter B).inter (measurable_spanning_sets μ p)
     have s_lt_top : μ s < ∞ :=
       (measure_mono (Set.inter_subset_right _ _)).trans_lt (measure_spanning_sets_lt_top μ p)
-    have A : (∫⁻ x in s, g x ∂μ) + ε * μ s ≤ (∫⁻ x in s, g x ∂μ) + 0 :=
+    have A : ∫⁻ x in s, g x ∂μ + ε * μ s ≤ ∫⁻ x in s, g x ∂μ + 0 :=
       calc
-        (∫⁻ x in s, g x ∂μ) + ε * μ s = (∫⁻ x in s, g x ∂μ) + ∫⁻ x in s, ε ∂μ := by
+        ∫⁻ x in s, g x ∂μ + ε * μ s = ∫⁻ x in s, g x ∂μ + ∫⁻ x in s, ε ∂μ := by
           simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, measure.restrict_apply]
         _ = ∫⁻ x in s, g x + ε ∂μ := (lintegral_add_right _ measurable_const).symm
         _ ≤ ∫⁻ x in s, f x ∂μ :=
           (set_lintegral_mono (hg.add measurable_const) hf fun x hx => hx.1.1)
-        _ ≤ (∫⁻ x in s, g x ∂μ) + 0 := by rw [add_zero]; exact h s s_meas s_lt_top
-    have B : (∫⁻ x in s, g x ∂μ) ≠ ∞ := by
+        _ ≤ ∫⁻ x in s, g x ∂μ + 0 := by rw [add_zero]; exact h s s_meas s_lt_top
+    have B : ∫⁻ x in s, g x ∂μ ≠ ∞ := by
       apply ne_of_lt
       calc
-        (∫⁻ x in s, g x ∂μ) ≤ ∫⁻ x in s, N ∂μ :=
+        ∫⁻ x in s, g x ∂μ ≤ ∫⁻ x in s, N ∂μ :=
           set_lintegral_mono hg measurable_const fun x hx => hx.1.2
         _ = N * μ s := by
           simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, measure.restrict_apply]
@@ -241,7 +241,7 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
 
 theorem ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite [SigmaFinite μ] {f g : α → ℝ≥0∞}
     (hf : Measurable f) (hg : Measurable g)
-    (h : ∀ s, MeasurableSet s → μ s < ∞ → (∫⁻ x in s, f x ∂μ) = ∫⁻ x in s, g x ∂μ) : f =ᵐ[μ] g :=
+    (h : ∀ s, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) : f =ᵐ[μ] g :=
   by
   have A : f ≤ᵐ[μ] g :=
     ae_le_of_forall_set_lintegral_le_of_sigma_finite hf hg fun s hs h's => le_of_eq (h s hs h's)
@@ -278,9 +278,9 @@ theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : Str
       _ ≤ (∫⁻ x, ‖f x‖₊ ∂μ) / c :=
         (meas_ge_le_lintegral_div hfm.ae_measurable.ennnorm c_pos ENNReal.coe_ne_top)
       _ < ∞ := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos
-  have h_int_gt : (∫ x in s, f x ∂μ) ≤ b * (μ s).toReal :=
+  have h_int_gt : ∫ x in s, f x ∂μ ≤ b * (μ s).toReal :=
     by
-    have h_const_le : (∫ x in s, f x ∂μ) ≤ ∫ x in s, b ∂μ :=
+    have h_const_le : ∫ x in s, f x ∂μ ≤ ∫ x in s, b ∂μ :=
       by
       refine'
         set_integral_mono_ae_restrict hf.integrable_on (integrable_on_const.mpr (Or.inr mus)) _
@@ -314,7 +314,7 @@ theorem ae_nonneg_of_forall_set_integral_nonneg (hf : Integrable f μ)
 #align measure_theory.ae_nonneg_of_forall_set_integral_nonneg MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg
 
 theorem ae_le_of_forall_set_integral_le {f g : α → ℝ} (hf : Integrable f μ) (hg : Integrable g μ)
-    (hf_le : ∀ s, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) ≤ ∫ x in s, g x ∂μ) : f ≤ᵐ[μ] g :=
+    (hf_le : ∀ s, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ ≤ ∫ x in s, g x ∂μ) : f ≤ᵐ[μ] g :=
   by
   rw [← eventually_sub_nonneg]
   refine' ae_nonneg_of_forall_set_integral_nonneg (hg.sub hf) fun s hs => _
@@ -379,7 +379,7 @@ theorem ae_nonneg_restrict_of_forall_set_integral_nonneg {f : α → ℝ}
 
 theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real {f : α → ℝ}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
-    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = 0) {t : Set α}
+    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) {t : Set α}
     (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] 0 :=
   by
   suffices h_and : f ≤ᵐ[μ.restrict t] 0 ∧ 0 ≤ᵐ[μ.restrict t] f
@@ -404,7 +404,7 @@ end Real
 
 theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero {f : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
-    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = 0) {t : Set α}
+    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) {t : Set α}
     (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] 0 :=
   by
   rcases(hf_int_finite t ht hμt.lt_top).AEStronglyMeasurable.isSeparable_ae_range with
@@ -421,11 +421,11 @@ theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero {f : α → E}
 theorem ae_eq_restrict_of_forall_set_integral_eq {f g : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
-    (hfg_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = ∫ x in s, g x ∂μ)
+    (hfg_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ)
     {t : Set α} (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] g :=
   by
   rw [← sub_ae_eq_zero]
-  have hfg' : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, (f - g) x ∂μ) = 0 :=
+  have hfg' : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, (f - g) x ∂μ = 0 :=
     by
     intro s hs hμs
     rw [integral_sub' (hf_int_finite s hs hμs) (hg_int_finite s hs hμs)]
@@ -437,7 +437,7 @@ theorem ae_eq_restrict_of_forall_set_integral_eq {f g : α → E}
 
 theorem ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
-    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = 0) : f =ᵐ[μ] 0 :=
+    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 :=
   by
   let S := spanning_sets μ
   rw [← @measure.restrict_univ _ _ μ, ← Union_spanning_sets μ, eventually_eq, ae_iff,
@@ -453,10 +453,10 @@ theorem ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f
 theorem ae_eq_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f g : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
-    (hfg_eq : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = ∫ x in s, g x ∂μ) :
+    (hfg_eq : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) :
     f =ᵐ[μ] g := by
   rw [← sub_ae_eq_zero]
-  have hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, (f - g) x ∂μ) = 0 :=
+  have hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, (f - g) x ∂μ = 0 :=
     by
     intro s hs hμs
     rw [integral_sub' (hf_int_finite s hs hμs) (hg_int_finite s hs hμs),
@@ -468,7 +468,7 @@ theorem ae_eq_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f g : 
 
 theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero {f : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
-    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = 0)
+    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
     (hf : AEFinStronglyMeasurable f μ) : f =ᵐ[μ] 0 :=
   by
   let t := hf.sigma_finite_set
@@ -489,11 +489,11 @@ theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero {f : 
 theorem AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq {f g : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
-    (hfg_eq : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = ∫ x in s, g x ∂μ)
+    (hfg_eq : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ)
     (hf : AEFinStronglyMeasurable f μ) (hg : AEFinStronglyMeasurable g μ) : f =ᵐ[μ] g :=
   by
   rw [← sub_ae_eq_zero]
-  have hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, (f - g) x ∂μ) = 0 :=
+  have hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, (f - g) x ∂μ = 0 :=
     by
     intro s hs hμs
     rw [integral_sub' (hf_int_finite s hs hμs) (hg_int_finite s hs hμs),
@@ -505,7 +505,7 @@ theorem AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq {f g : α → E}
 
 theorem Lp.ae_eq_zero_of_forall_set_integral_eq_zero (f : Lp E p μ) (hp_ne_zero : p ≠ 0)
     (hp_ne_top : p ≠ ∞) (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
-    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = 0) : f =ᵐ[μ] 0 :=
+    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 :=
   AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero hf_int_finite hf_zero
     (Lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).AEFinStronglyMeasurable
 #align measure_theory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero
@@ -513,7 +513,7 @@ theorem Lp.ae_eq_zero_of_forall_set_integral_eq_zero (f : Lp E p μ) (hp_ne_zero
 theorem Lp.ae_eq_of_forall_set_integral_eq (f g : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
-    (hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = ∫ x in s, g x ∂μ) :
+    (hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) :
     f =ᵐ[μ] g :=
   AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq hf_int_finite hg_int_finite hfg
     (Lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).AEFinStronglyMeasurable
@@ -522,7 +522,7 @@ theorem Lp.ae_eq_of_forall_set_integral_eq (f g : Lp E p μ) (hp_ne_zero : p ≠
 
 theorem ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim (hm : m ≤ m0) {f : α → E}
     (hf_int_finite : ∀ s, measurable_set[m] s → μ s < ∞ → IntegrableOn f s μ)
-    (hf_zero : ∀ s : Set α, measurable_set[m] s → μ s < ∞ → (∫ x in s, f x ∂μ) = 0)
+    (hf_zero : ∀ s : Set α, measurable_set[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
     (hf : FinStronglyMeasurable f (μ.trim hm)) : f =ᵐ[μ] 0 :=
   by
   obtain ⟨t, ht_meas, htf_zero, htμ⟩ := hf.exists_set_sigma_finite
@@ -551,7 +551,7 @@ theorem ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim (hm :
 #align measure_theory.ae_eq_zero_of_forall_set_integral_eq_of_fin_strongly_measurable_trim MeasureTheory.ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim
 
 theorem Integrable.ae_eq_zero_of_forall_set_integral_eq_zero {f : α → E} (hf : Integrable f μ)
-    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = 0) : f =ᵐ[μ] 0 :=
+    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 :=
   by
   have hf_Lp : mem_ℒp f 1 μ := mem_ℒp_one_iff_integrable.mpr hf
   let f_Lp := hf_Lp.to_Lp f
@@ -566,10 +566,10 @@ theorem Integrable.ae_eq_zero_of_forall_set_integral_eq_zero {f : α → E} (hf
 
 theorem Integrable.ae_eq_of_forall_set_integral_eq (f g : α → E) (hf : Integrable f μ)
     (hg : Integrable g μ)
-    (hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = ∫ x in s, g x ∂μ) :
+    (hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) :
     f =ᵐ[μ] g := by
   rw [← sub_ae_eq_zero]
-  have hfg' : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, (f - g) x ∂μ) = 0 :=
+  have hfg' : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, (f - g) x ∂μ = 0 :=
     by
     intro s hs hμs
     rw [integral_sub' hf.integrable_on hg.integrable_on]
@@ -582,9 +582,8 @@ end AeEqOfForallSetIntegralEq
 section Lintegral
 
 theorem AeMeasurable.ae_eq_of_forall_set_lintegral_eq {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
-    (hg : AEMeasurable g μ) (hfi : (∫⁻ x, f x ∂μ) ≠ ∞) (hgi : (∫⁻ x, g x ∂μ) ≠ ∞)
-    (hfg : ∀ ⦃s⦄, MeasurableSet s → μ s < ∞ → (∫⁻ x in s, f x ∂μ) = ∫⁻ x in s, g x ∂μ) :
-    f =ᵐ[μ] g :=
+    (hg : AEMeasurable g μ) (hfi : ∫⁻ x, f x ∂μ ≠ ∞) (hgi : ∫⁻ x, g x ∂μ ≠ ∞)
+    (hfg : ∀ ⦃s⦄, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) : f =ᵐ[μ] g :=
   by
   refine'
     ENNReal.eventuallyEq_of_toReal_eventuallyEq (ae_lt_top' hf hfi).ne_of_lt

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -110,7 +110,6 @@ theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedS
       _ < ‖a‖ / 2 := by rw [one_mul]; rwa [dist_eq_norm'] at hx 
       _ < ‖(x : E)‖ := I
       _ = ‖s x x‖ := by rw [(hs x).2, IsROrC.norm_coe_norm]
-      
   have hfs : ∀ y : d, ∀ᵐ x ∂μ, ⟪f x, s y⟫ = (0 : 𝕜) := fun y => hf (s y)
   have hf' : ∀ᵐ x ∂μ, ∀ y : d, ⟪f x, s y⟫ = (0 : 𝕜) := by rwa [ae_all_iff]
   filter_upwards [hf', h't] with x hx h'x
@@ -199,7 +198,6 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
         _ ≤ ∫⁻ x in s, f x ∂μ :=
           (set_lintegral_mono (hg.add measurable_const) hf fun x hx => hx.1.1)
         _ ≤ (∫⁻ x in s, g x ∂μ) + 0 := by rw [add_zero]; exact h s s_meas s_lt_top
-        
     have B : (∫⁻ x in s, g x ∂μ) ≠ ∞ := by
       apply ne_of_lt
       calc
@@ -210,7 +208,6 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
         _ < ∞ := by
           simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top,
             WithTop.mul_eq_top_iff, Ne.def, not_false_iff, false_and_iff, or_self_iff]
-        
     have : (ε : ℝ≥0∞) * μ s ≤ 0 := ENNReal.le_of_add_le_add_left B A
     simpa only [ENNReal.coe_eq_zero, nonpos_iff_eq_zero, mul_eq_zero, εpos.ne', false_or_iff]
   obtain ⟨u, u_mono, u_pos, u_lim⟩ :
@@ -240,7 +237,6 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
     μ ({x : α | (fun x : α => f x ≤ g x) x}ᶜ) ≤ μ (⋃ n, s n) := measure_mono B
     _ ≤ ∑' n, μ (s n) := (measure_Union_le _)
     _ = 0 := by simp only [μs, tsum_zero]
-    
 #align measure_theory.ae_le_of_forall_set_lintegral_le_of_sigma_finite MeasureTheory.ae_le_of_forall_set_lintegral_le_of_sigmaFinite
 
 theorem ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite [SigmaFinite μ] {f g : α → ℝ≥0∞}
@@ -282,7 +278,6 @@ theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : Str
       _ ≤ (∫⁻ x, ‖f x‖₊ ∂μ) / c :=
         (meas_ge_le_lintegral_div hfm.ae_measurable.ennnorm c_pos ENNReal.coe_ne_top)
       _ < ∞ := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos
-      
   have h_int_gt : (∫ x in s, f x ∂μ) ≤ b * (μ s).toReal :=
     by
     have h_const_le : (∫ x in s, f x ∂μ) ≤ ∫ x in s, b ∂μ :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004

@@ -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 measure_theory.function.ae_eq_of_integral
-! leanprover-community/mathlib commit 915591b2bb3ea303648db07284a161a7f2a9e3d4
+! leanprover-community/mathlib commit c20927220ef87bb4962ba08bf6da2ce3cf50a6dd
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.MeasureTheory.Integral.SetIntegral
 
 /-! # From equality of integrals to equality of functions
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file provides various statements of the general form "if two functions have the same integral
 on all sets, then they are equal almost everywhere".
 The different lemmas use various hypotheses on the class of functions, on the target space or on the

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

@@ -65,7 +65,7 @@ theorem ae_eq_zero_of_forall_inner [NormedAddCommGroup E] [InnerProductSpace 
   have hf' : ∀ᵐ x ∂μ, ∀ n : ℕ, inner (s n) (f x) = (0 : 𝕜) := ae_all_iff.mpr fun n => hf (s n)
   refine' hf'.mono fun x hx => _
   rw [Pi.zero_apply, ← @inner_self_eq_zero 𝕜]
-  have h_closed : IsClosed { c : E | inner c (f x) = (0 : 𝕜) } :=
+  have h_closed : IsClosed {c : E | inner c (f x) = (0 : 𝕜)} :=
     isClosed_eq (continuous_id.inner continuous_const) continuous_const
   exact @isClosed_property ℕ E _ s (fun c => inner c (f x) = (0 : 𝕜)) hs h_closed (fun n => hx n) _
 #align measure_theory.ae_eq_zero_of_forall_inner MeasureTheory.ae_eq_zero_of_forall_inner
@@ -110,7 +110,7 @@ theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedS
       
   have hfs : ∀ y : d, ∀ᵐ x ∂μ, ⟪f x, s y⟫ = (0 : 𝕜) := fun y => hf (s y)
   have hf' : ∀ᵐ x ∂μ, ∀ y : d, ⟪f x, s y⟫ = (0 : 𝕜) := by rwa [ae_all_iff]
-  filter_upwards [hf', h't]with x hx h'x
+  filter_upwards [hf', h't] with x hx h'x
   exact A (f x) h'x hx
 #align measure_theory.ae_eq_zero_of_forall_dual_of_is_separable MeasureTheory.ae_eq_zero_of_forall_dual_of_isSeparable
 
@@ -132,7 +132,7 @@ section AeEqOfForallSetIntegralEq
 
 theorem ae_const_le_iff_forall_lt_measure_zero {β} [LinearOrder β] [TopologicalSpace β]
     [OrderTopology β] [FirstCountableTopology β] (f : α → β) (c : β) :
-    (∀ᵐ x ∂μ, c ≤ f x) ↔ ∀ b < c, μ { x | f x ≤ b } = 0 :=
+    (∀ᵐ x ∂μ, c ≤ f x) ↔ ∀ b < c, μ {x | f x ≤ b} = 0 :=
   by
   rw [ae_iff]
   push_neg
@@ -141,7 +141,7 @@ theorem ae_const_le_iff_forall_lt_measure_zero {β} [LinearOrder β] [Topologica
     exact measure_mono_null (fun y hy => (lt_of_le_of_lt hy hb : _)) h
   intro hc
   by_cases h : ∀ b, c ≤ b
-  · have : { a : α | f a < c } = ∅ :=
+  · have : {a : α | f a < c} = ∅ :=
       by
       apply Set.eq_empty_iff_forall_not_mem.2 fun x hx => _
       exact (lt_irrefl _ (lt_of_lt_of_le hx (h (f x)))).elim
@@ -151,12 +151,12 @@ theorem ae_const_le_iff_forall_lt_measure_zero {β} [LinearOrder β] [Topologica
     obtain ⟨b, b_up, bc⟩ : ∃ b : β, b ∈ upperBounds (Set.Iio c) ∧ b < c := by
       simpa [IsLUB, IsLeast, this, lowerBounds] using H
     exact measure_mono_null (fun x hx => b_up hx) (hc b bc)
-  push_neg  at H h 
+  push_neg at H h 
   obtain ⟨u, u_mono, u_lt, u_lim, -⟩ :
     ∃ u : ℕ → β,
       StrictMono u ∧ (∀ n : ℕ, u n < c) ∧ tendsto u at_top (nhds c) ∧ ∀ n : ℕ, u n ∈ Set.Iio c :=
     H.exists_seq_strict_mono_tendsto_of_not_mem (lt_irrefl c) h
-  have h_Union : { x | f x < c } = ⋃ n : ℕ, { x | f x ≤ u n } :=
+  have h_Union : {x | f x < c} = ⋃ n : ℕ, {x | f x ≤ u n} :=
     by
     ext1 x
     simp_rw [Set.mem_iUnion, Set.mem_setOf_eq]
@@ -177,14 +177,14 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
     (h : ∀ s, MeasurableSet s → μ s < ∞ → (∫⁻ x in s, f x ∂μ) ≤ ∫⁻ x in s, g x ∂μ) : f ≤ᵐ[μ] g :=
   by
   have A :
-    ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({ x | g x + ε ≤ f x ∧ g x ≤ N } ∩ spanning_sets μ p) = 0 :=
+    ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ε ≤ f x ∧ g x ≤ N} ∩ spanning_sets μ p) = 0 :=
     by
     intro ε N p εpos
-    let s := { x | g x + ε ≤ f x ∧ g x ≤ N } ∩ spanning_sets μ p
+    let s := {x | g x + ε ≤ f x ∧ g x ≤ N} ∩ spanning_sets μ p
     have s_meas : MeasurableSet s :=
       by
-      have A : MeasurableSet { x | g x + ε ≤ f x } := measurableSet_le (hg.add measurable_const) hf
-      have B : MeasurableSet { x | g x ≤ N } := measurableSet_le hg measurable_const
+      have A : MeasurableSet {x | g x + ε ≤ f x} := measurableSet_le (hg.add measurable_const) hf
+      have B : MeasurableSet {x | g x ≤ N} := measurableSet_le hg measurable_const
       exact (A.inter B).inter (measurable_spanning_sets μ p)
     have s_lt_top : μ s < ∞ :=
       (measure_mono (Set.inter_subset_right _ _)).trans_lt (measure_spanning_sets_lt_top μ p)
@@ -213,9 +213,9 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
   obtain ⟨u, u_mono, u_pos, u_lim⟩ :
     ∃ u : ℕ → ℝ≥0, StrictAnti u ∧ (∀ n, 0 < u n) ∧ tendsto u at_top (nhds 0) :=
     exists_seq_strictAnti_tendsto (0 : ℝ≥0)
-  let s := fun n : ℕ => { x | g x + u n ≤ f x ∧ g x ≤ (n : ℝ≥0) } ∩ spanning_sets μ n
+  let s := fun n : ℕ => {x | g x + u n ≤ f x ∧ g x ≤ (n : ℝ≥0)} ∩ spanning_sets μ n
   have μs : ∀ n, μ (s n) = 0 := fun n => A _ _ _ (u_pos n)
-  have B : { x | f x ≤ g x }ᶜ ⊆ ⋃ n, s n := by
+  have B : {x | f x ≤ g x}ᶜ ⊆ ⋃ n, s n := by
     intro x hx
     simp at hx 
     have L1 : ∀ᶠ n in at_top, g x + u n ≤ f x :=
@@ -234,7 +234,7 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
     exact ((L1.and L2).And (eventually_mem_spanning_sets μ x)).exists
   refine' le_antisymm _ bot_le
   calc
-    μ ({ x : α | (fun x : α => f x ≤ g x) x }ᶜ) ≤ μ (⋃ n, s n) := measure_mono B
+    μ ({x : α | (fun x : α => f x ≤ g x) x}ᶜ) ≤ μ (⋃ n, s n) := measure_mono B
     _ ≤ ∑' n, μ (s n) := (measure_Union_le _)
     _ = 0 := by simp only [μs, tsum_zero]
     
@@ -248,7 +248,7 @@ theorem ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite [SigmaFinite μ] {f g :
     ae_le_of_forall_set_lintegral_le_of_sigma_finite hf hg fun s hs h's => le_of_eq (h s hs h's)
   have B : g ≤ᵐ[μ] f :=
     ae_le_of_forall_set_lintegral_le_of_sigma_finite hg hf fun s hs h's => ge_of_eq (h s hs h's)
-  filter_upwards [A, B]with x using le_antisymm
+  filter_upwards [A, B] with x using le_antisymm
 #align measure_theory.ae_eq_of_forall_set_lintegral_eq_of_sigma_finite MeasureTheory.ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite
 
 end ENNReal
@@ -264,14 +264,13 @@ theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : Str
   simp_rw [eventually_le, Pi.zero_apply]
   rw [ae_const_le_iff_forall_lt_measure_zero]
   intro b hb_neg
-  let s := { x | f x ≤ b }
+  let s := {x | f x ≤ b}
   have hs : MeasurableSet s := hfm.measurable_set_le strongly_measurable_const
   have mus : μ s < ∞ := by
     let c : ℝ≥0 := ⟨|b|, abs_nonneg _⟩
     have c_pos : (c : ℝ≥0∞) ≠ 0 := by simpa using hb_neg.ne
     calc
-      μ s ≤ μ { x | (c : ℝ≥0∞) ≤ ‖f x‖₊ } :=
-        by
+      μ s ≤ μ {x | (c : ℝ≥0∞) ≤ ‖f x‖₊} := by
         apply measure_mono
         intro x hx
         simp only [Set.mem_setOf_eq] at hx 

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

@@ -97,14 +97,14 @@ theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedS
     have I : ‖a‖ / 2 < ‖(x : E)‖ :=
       by
       have : ‖a‖ ≤ ‖(x : E)‖ + ‖a - x‖ := norm_le_insert' _ _
-      have : ‖a - x‖ < ‖a‖ / 2 := by rwa [dist_eq_norm] at hx
+      have : ‖a - x‖ < ‖a‖ / 2 := by rwa [dist_eq_norm] at hx 
       linarith
     intro h
     apply lt_irrefl ‖s x x‖
     calc
       ‖s x x‖ = ‖s x (x - a)‖ := by simp only [h, sub_zero, ContinuousLinearMap.map_sub]
       _ ≤ 1 * ‖(x : E) - a‖ := (ContinuousLinearMap.le_of_op_norm_le _ (hs x).1 _)
-      _ < ‖a‖ / 2 := by rw [one_mul]; rwa [dist_eq_norm'] at hx
+      _ < ‖a‖ / 2 := by rw [one_mul]; rwa [dist_eq_norm'] at hx 
       _ < ‖(x : E)‖ := I
       _ = ‖s x x‖ := by rw [(hs x).2, IsROrC.norm_coe_norm]
       
@@ -151,7 +151,7 @@ theorem ae_const_le_iff_forall_lt_measure_zero {β} [LinearOrder β] [Topologica
     obtain ⟨b, b_up, bc⟩ : ∃ b : β, b ∈ upperBounds (Set.Iio c) ∧ b < c := by
       simpa [IsLUB, IsLeast, this, lowerBounds] using H
     exact measure_mono_null (fun x hx => b_up hx) (hc b bc)
-  push_neg  at H h
+  push_neg  at H h 
   obtain ⟨u, u_mono, u_lt, u_lim, -⟩ :
     ∃ u : ℕ → β,
       StrictMono u ∧ (∀ n : ℕ, u n < c) ∧ tendsto u at_top (nhds c) ∧ ∀ n : ℕ, u n ∈ Set.Iio c :=
@@ -217,12 +217,12 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
   have μs : ∀ n, μ (s n) = 0 := fun n => A _ _ _ (u_pos n)
   have B : { x | f x ≤ g x }ᶜ ⊆ ⋃ n, s n := by
     intro x hx
-    simp at hx
+    simp at hx 
     have L1 : ∀ᶠ n in at_top, g x + u n ≤ f x :=
       by
       have : tendsto (fun n => g x + u n) at_top (𝓝 (g x + (0 : ℝ≥0))) :=
         tendsto_const_nhds.add (ENNReal.tendsto_coe.2 u_lim)
-      simp at this
+      simp at this 
       exact eventually_le_of_tendsto_lt hx this
     have L2 : ∀ᶠ n : ℕ in (at_top : Filter ℕ), g x ≤ (n : ℝ≥0) :=
       haveI : tendsto (fun n : ℕ => ((n : ℝ≥0) : ℝ≥0∞)) at_top (𝓝 ∞) :=
@@ -274,7 +274,7 @@ theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : Str
         by
         apply measure_mono
         intro x hx
-        simp only [Set.mem_setOf_eq] at hx
+        simp only [Set.mem_setOf_eq] at hx 
         simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,
           Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe] using hx
       _ ≤ (∫⁻ x, ‖f x‖₊ ∂μ) / c :=
@@ -289,7 +289,7 @@ theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : Str
         set_integral_mono_ae_restrict hf.integrable_on (integrable_on_const.mpr (Or.inr mus)) _
       rw [eventually_le, ae_restrict_iff hs]
       exact eventually_of_forall fun x hxs => hxs
-    rwa [set_integral_const, smul_eq_mul, mul_comm] at h_const_le
+    rwa [set_integral_const, smul_eq_mul, mul_comm] at h_const_le 
   by_contra
   refine' (lt_self_iff_false (∫ x in s, f x ∂μ)).mp (h_int_gt.trans_lt _)
   refine' (mul_neg_iff.mpr (Or.inr ⟨hb_neg, _⟩)).trans_le _
@@ -333,7 +333,7 @@ theorem ae_nonneg_restrict_of_forall_set_integral_nonneg_inter {f : α → ℝ}
   refine' ae_nonneg_of_forall_set_integral_nonneg hf fun s hs h's => _
   simp_rw [measure.restrict_restrict hs]
   apply hf_zero s hs
-  rwa [measure.restrict_apply hs] at h's
+  rwa [measure.restrict_apply hs] at h's 
 #align measure_theory.ae_nonneg_restrict_of_forall_set_integral_nonneg_inter MeasureTheory.ae_nonneg_restrict_of_forall_set_integral_nonneg_inter
 
 theorem ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite [SigmaFinite μ] {f : α → ℝ}
@@ -361,10 +361,10 @@ theorem AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg {f : α
   refine'
     ae_nonneg_of_forall_set_integral_nonneg_of_sigma_finite (fun s hs hμts => _) fun s hs hμts => _
   · rw [integrable_on, measure.restrict_restrict hs]
-    rw [measure.restrict_apply hs] at hμts
+    rw [measure.restrict_apply hs] at hμts 
     exact hf_int_finite (s ∩ t) (hs.inter hf.measurable_set) hμts
   · rw [measure.restrict_restrict hs]
-    rw [measure.restrict_apply hs] at hμts
+    rw [measure.restrict_apply hs] at hμts 
     exact hf_zero (s ∩ t) (hs.inter hf.measurable_set) hμts
 #align measure_theory.ae_fin_strongly_measurable.ae_nonneg_of_forall_set_integral_nonneg MeasureTheory.AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg
 
@@ -393,7 +393,7 @@ theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real {f : α → ℝ}
         (fun s hs hμs => (hf_zero s hs hμs).symm.le) ht hμt⟩
   suffices h_neg : 0 ≤ᵐ[μ.restrict t] -f
   · refine' h_neg.mono fun x hx => _
-    rw [Pi.neg_apply] at hx
+    rw [Pi.neg_apply] at hx 
     simpa using hx
   refine'
     ae_nonneg_restrict_of_forall_set_integral_nonneg (fun s hs hμs => (hf_int_finite s hs hμs).neg)
@@ -481,11 +481,11 @@ theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero {f : 
   refine' ae_eq_zero_of_forall_set_integral_eq_of_sigma_finite _ _
   · intro s hs hμs
     rw [integrable_on, measure.restrict_restrict hs]
-    rw [measure.restrict_apply hs] at hμs
+    rw [measure.restrict_apply hs] at hμs 
     exact hf_int_finite _ (hs.inter hf.measurable_set) hμs
   · intro s hs hμs
     rw [measure.restrict_restrict hs]
-    rw [measure.restrict_apply hs] at hμs
+    rw [measure.restrict_apply hs] at hμs 
     exact hf_zero _ (hs.inter hf.measurable_set) hμs
 #align measure_theory.ae_fin_strongly_measurable.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero
 
@@ -529,7 +529,7 @@ theorem ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim (hm :
     (hf : FinStronglyMeasurable f (μ.trim hm)) : f =ᵐ[μ] 0 :=
   by
   obtain ⟨t, ht_meas, htf_zero, htμ⟩ := hf.exists_set_sigma_finite
-  haveI : sigma_finite ((μ.restrict t).trim hm) := by rwa [restrict_trim hm μ ht_meas] at htμ
+  haveI : sigma_finite ((μ.restrict t).trim hm) := by rwa [restrict_trim hm μ ht_meas] at htμ 
   have htf_zero : f =ᵐ[μ.restrict (tᶜ)] 0 :=
     by
     rw [eventually_eq, ae_restrict_iff' (MeasurableSet.compl (hm _ ht_meas))]
@@ -542,13 +542,13 @@ theorem ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim (hm :
   · intro s hs hμs
     rw [integrable_on, restrict_trim hm (μ.restrict t) hs, measure.restrict_restrict (hm s hs)]
     rw [← restrict_trim hm μ ht_meas, measure.restrict_apply hs,
-      trim_measurable_set_eq hm (hs.inter ht_meas)] at hμs
+      trim_measurable_set_eq hm (hs.inter ht_meas)] at hμs 
     refine' integrable.trim hm _ hf_meas_m
     exact hf_int_finite _ (hs.inter ht_meas) hμs
   · intro s hs hμs
     rw [restrict_trim hm (μ.restrict t) hs, measure.restrict_restrict (hm s hs)]
     rw [← restrict_trim hm μ ht_meas, measure.restrict_apply hs,
-      trim_measurable_set_eq hm (hs.inter ht_meas)] at hμs
+      trim_measurable_set_eq hm (hs.inter ht_meas)] at hμs 
     rw [← integral_trim hm hf_meas_m]
     exact hf_zero _ (hs.inter ht_meas) hμs
 #align measure_theory.ae_eq_zero_of_forall_set_integral_eq_of_fin_strongly_measurable_trim MeasureTheory.ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim
@@ -604,7 +604,7 @@ theorem AeMeasurable.ae_eq_of_forall_set_lintegral_eq {f g : α → ℝ≥0∞}
     · refine' ae_lt_top' hf.restrict (ne_of_lt (lt_of_le_of_lt _ hfi.lt_top))
       exact @set_lintegral_univ α _ μ f ▸ lintegral_mono_set (Set.subset_univ _)
   -- putting the proofs where they are used is extremely slow
-  exacts[ae_of_all _ fun x => ENNReal.toReal_nonneg,
+  exacts [ae_of_all _ fun x => ENNReal.toReal_nonneg,
     hg.ennreal_to_real.restrict.ae_strongly_measurable, ae_of_all _ fun x => ENNReal.toReal_nonneg,
     hf.ennreal_to_real.restrict.ae_strongly_measurable]
 #align measure_theory.ae_measurable.ae_eq_of_forall_set_lintegral_eq MeasureTheory.AeMeasurable.ae_eq_of_forall_set_lintegral_eq

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

@@ -506,22 +506,22 @@ theorem AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq {f g : α → E}
   exact (hf.sub hg).ae_eq_zero_of_forall_set_integral_eq_zero hfg_int hfg
 #align measure_theory.ae_fin_strongly_measurable.ae_eq_of_forall_set_integral_eq MeasureTheory.AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq
 
-theorem lp.ae_eq_zero_of_forall_set_integral_eq_zero (f : lp E p μ) (hp_ne_zero : p ≠ 0)
+theorem Lp.ae_eq_zero_of_forall_set_integral_eq_zero (f : Lp E p μ) (hp_ne_zero : p ≠ 0)
     (hp_ne_top : p ≠ ∞) (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = 0) : f =ᵐ[μ] 0 :=
   AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero hf_int_finite hf_zero
-    (lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).AEFinStronglyMeasurable
-#align measure_theory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.lp.ae_eq_zero_of_forall_set_integral_eq_zero
+    (Lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).AEFinStronglyMeasurable
+#align measure_theory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero
 
-theorem lp.ae_eq_of_forall_set_integral_eq (f g : lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
+theorem Lp.ae_eq_of_forall_set_integral_eq (f g : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
     (hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = ∫ x in s, g x ∂μ) :
     f =ᵐ[μ] g :=
   AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq hf_int_finite hg_int_finite hfg
-    (lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).AEFinStronglyMeasurable
-    (lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).AEFinStronglyMeasurable
-#align measure_theory.Lp.ae_eq_of_forall_set_integral_eq MeasureTheory.lp.ae_eq_of_forall_set_integral_eq
+    (Lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).AEFinStronglyMeasurable
+    (Lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).AEFinStronglyMeasurable
+#align measure_theory.Lp.ae_eq_of_forall_set_integral_eq MeasureTheory.Lp.ae_eq_of_forall_set_integral_eq
 
 theorem ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim (hm : m ≤ m0) {f : α → E}
     (hf_int_finite : ∀ s, measurable_set[m] s → μ s < ∞ → IntegrableOn f s μ)

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

@@ -49,7 +49,7 @@ Generally useful lemmas which are not related to integrals:
 
 open MeasureTheory TopologicalSpace NormedSpace Filter
 
-open ENNReal NNReal MeasureTheory
+open scoped ENNReal NNReal MeasureTheory
 
 namespace MeasureTheory
 
@@ -170,7 +170,7 @@ theorem ae_const_le_iff_forall_lt_measure_zero {β} [LinearOrder β] [Topologica
 
 section ENNReal
 
-open Topology
+open scoped Topology
 
 theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g : α → ℝ≥0∞}
     (hf : Measurable f) (hg : Measurable g)

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

@@ -104,9 +104,7 @@ theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedS
     calc
       ‖s x x‖ = ‖s x (x - a)‖ := by simp only [h, sub_zero, ContinuousLinearMap.map_sub]
       _ ≤ 1 * ‖(x : E) - a‖ := (ContinuousLinearMap.le_of_op_norm_le _ (hs x).1 _)
-      _ < ‖a‖ / 2 := by
-        rw [one_mul]
-        rwa [dist_eq_norm'] at hx
+      _ < ‖a‖ / 2 := by rw [one_mul]; rwa [dist_eq_norm'] at hx
       _ < ‖(x : E)‖ := I
       _ = ‖s x x‖ := by rw [(hs x).2, IsROrC.norm_coe_norm]
       
@@ -163,10 +161,8 @@ theorem ae_const_le_iff_forall_lt_measure_zero {β} [LinearOrder β] [Topologica
     ext1 x
     simp_rw [Set.mem_iUnion, Set.mem_setOf_eq]
     constructor <;> intro h
-    · obtain ⟨n, hn⟩ := ((tendsto_order.1 u_lim).1 _ h).exists
-      exact ⟨n, hn.le⟩
-    · obtain ⟨n, hn⟩ := h
-      exact hn.trans_lt (u_lt _)
+    · obtain ⟨n, hn⟩ := ((tendsto_order.1 u_lim).1 _ h).exists; exact ⟨n, hn.le⟩
+    · obtain ⟨n, hn⟩ := h; exact hn.trans_lt (u_lt _)
   rw [h_Union, measure_Union_null_iff]
   intro n
   exact hc _ (u_lt n)
@@ -199,9 +195,7 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
         _ = ∫⁻ x in s, g x + ε ∂μ := (lintegral_add_right _ measurable_const).symm
         _ ≤ ∫⁻ x in s, f x ∂μ :=
           (set_lintegral_mono (hg.add measurable_const) hf fun x hx => hx.1.1)
-        _ ≤ (∫⁻ x in s, g x ∂μ) + 0 := by
-          rw [add_zero]
-          exact h s s_meas s_lt_top
+        _ ≤ (∫⁻ x in s, g x ∂μ) + 0 := by rw [add_zero]; exact h s s_meas s_lt_top
         
     have B : (∫⁻ x in s, g x ∂μ) ≠ ∞ := by
       apply ne_of_lt
@@ -299,9 +293,7 @@ theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : Str
   by_contra
   refine' (lt_self_iff_false (∫ x in s, f x ∂μ)).mp (h_int_gt.trans_lt _)
   refine' (mul_neg_iff.mpr (Or.inr ⟨hb_neg, _⟩)).trans_le _
-  swap
-  · simp_rw [measure.restrict_restrict hs]
-    exact hf_zero s hs mus
+  swap; · simp_rw [measure.restrict_restrict hs]; exact hf_zero s hs mus
   refine' ENNReal.toReal_nonneg.lt_of_ne fun h_eq => h _
   cases' (ENNReal.toReal_eq_zero_iff _).mp h_eq.symm with hμs_eq_zero hμs_eq_top
   · exact hμs_eq_zero

mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828

@@ -357,8 +357,8 @@ theorem ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite [SigmaFinite μ]
       (lt_of_le_of_lt (measure_mono (Set.inter_subset_right _ _)) t_lt_top)
 #align measure_theory.ae_nonneg_of_forall_set_integral_nonneg_of_sigma_finite MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite
 
-theorem AeFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg {f : α → ℝ}
-    (hf : AeFinStronglyMeasurable f μ)
+theorem AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg {f : α → ℝ}
+    (hf : AEFinStronglyMeasurable f μ)
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f :=
   by
@@ -374,7 +374,7 @@ theorem AeFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg {f : α
   · rw [measure.restrict_restrict hs]
     rw [measure.restrict_apply hs] at hμts
     exact hf_zero (s ∩ t) (hs.inter hf.measurable_set) hμts
-#align measure_theory.ae_fin_strongly_measurable.ae_nonneg_of_forall_set_integral_nonneg MeasureTheory.AeFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg
+#align measure_theory.ae_fin_strongly_measurable.ae_nonneg_of_forall_set_integral_nonneg MeasureTheory.AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg
 
 theorem ae_nonneg_restrict_of_forall_set_integral_nonneg {f : α → ℝ}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
@@ -418,7 +418,7 @@ theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero {f : α → E}
     (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = 0) {t : Set α}
     (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] 0 :=
   by
-  rcases(hf_int_finite t ht hμt.lt_top).AeStronglyMeasurable.isSeparable_ae_range with
+  rcases(hf_int_finite t ht hμt.lt_top).AEStronglyMeasurable.isSeparable_ae_range with
     ⟨u, u_sep, hu⟩
   refine' ae_eq_zero_of_forall_dual_of_is_separable ℝ u_sep (fun c => _) hu
   refine' ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real _ _ ht hμt
@@ -477,10 +477,10 @@ theorem ae_eq_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f g : 
   exact ae_eq_zero_of_forall_set_integral_eq_of_sigma_finite hfg_int hfg
 #align measure_theory.ae_eq_of_forall_set_integral_eq_of_sigma_finite MeasureTheory.ae_eq_of_forall_set_integral_eq_of_sigmaFinite
 
-theorem AeFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero {f : α → E}
+theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero {f : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = 0)
-    (hf : AeFinStronglyMeasurable f μ) : f =ᵐ[μ] 0 :=
+    (hf : AEFinStronglyMeasurable f μ) : f =ᵐ[μ] 0 :=
   by
   let t := hf.sigma_finite_set
   suffices : f =ᵐ[μ.restrict t] 0
@@ -495,13 +495,13 @@ theorem AeFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero {f : 
     rw [measure.restrict_restrict hs]
     rw [measure.restrict_apply hs] at hμs
     exact hf_zero _ (hs.inter hf.measurable_set) hμs
-#align measure_theory.ae_fin_strongly_measurable.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.AeFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero
+#align measure_theory.ae_fin_strongly_measurable.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero
 
-theorem AeFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq {f g : α → E}
+theorem AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq {f g : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
     (hfg_eq : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = ∫ x in s, g x ∂μ)
-    (hf : AeFinStronglyMeasurable f μ) (hg : AeFinStronglyMeasurable g μ) : f =ᵐ[μ] g :=
+    (hf : AEFinStronglyMeasurable f μ) (hg : AEFinStronglyMeasurable g μ) : f =ᵐ[μ] g :=
   by
   rw [← sub_ae_eq_zero]
   have hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, (f - g) x ∂μ) = 0 :=
@@ -512,13 +512,13 @@ theorem AeFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq {f g : α → E}
   have hfg_int : ∀ s, MeasurableSet s → μ s < ∞ → integrable_on (f - g) s μ := fun s hs hμs =>
     (hf_int_finite s hs hμs).sub (hg_int_finite s hs hμs)
   exact (hf.sub hg).ae_eq_zero_of_forall_set_integral_eq_zero hfg_int hfg
-#align measure_theory.ae_fin_strongly_measurable.ae_eq_of_forall_set_integral_eq MeasureTheory.AeFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq
+#align measure_theory.ae_fin_strongly_measurable.ae_eq_of_forall_set_integral_eq MeasureTheory.AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq
 
 theorem lp.ae_eq_zero_of_forall_set_integral_eq_zero (f : lp E p μ) (hp_ne_zero : p ≠ 0)
     (hp_ne_top : p ≠ ∞) (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = 0) : f =ᵐ[μ] 0 :=
-  AeFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero hf_int_finite hf_zero
-    (lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).AeFinStronglyMeasurable
+  AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero hf_int_finite hf_zero
+    (lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).AEFinStronglyMeasurable
 #align measure_theory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.lp.ae_eq_zero_of_forall_set_integral_eq_zero
 
 theorem lp.ae_eq_of_forall_set_integral_eq (f g : lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
@@ -526,9 +526,9 @@ theorem lp.ae_eq_of_forall_set_integral_eq (f g : lp E p μ) (hp_ne_zero : p ≠
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
     (hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = ∫ x in s, g x ∂μ) :
     f =ᵐ[μ] g :=
-  AeFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq hf_int_finite hg_int_finite hfg
-    (lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).AeFinStronglyMeasurable
-    (lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).AeFinStronglyMeasurable
+  AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq hf_int_finite hg_int_finite hfg
+    (lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).AEFinStronglyMeasurable
+    (lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).AEFinStronglyMeasurable
 #align measure_theory.Lp.ae_eq_of_forall_set_integral_eq MeasureTheory.lp.ae_eq_of_forall_set_integral_eq
 
 theorem ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim (hm : m ≤ m0) {f : α → E}

mathlib commit https://github.com/leanprover-community/mathlib/commit/f51de8769c34652d82d1c8e5f8f18f8374782bed

@@ -4,10 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 
 ! This file was ported from Lean 3 source module measure_theory.function.ae_eq_of_integral
-! leanprover-community/mathlib commit 46b633fd842bef9469441c0209906f6dddd2b4f5
+! leanprover-community/mathlib commit 915591b2bb3ea303648db07284a161a7f2a9e3d4
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
+import Mathbin.Analysis.InnerProductSpace.Basic
 import Mathbin.Analysis.NormedSpace.Dual
 import Mathbin.MeasureTheory.Function.StronglyMeasurable.Lp
 import Mathbin.MeasureTheory.Integral.SetIntegral

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

@@ -160,7 +160,7 @@ theorem ae_const_le_iff_forall_lt_measure_zero {β} [LinearOrder β] [Topologica
   have h_Union : { x | f x < c } = ⋃ n : ℕ, { x | f x ≤ u n } :=
     by
     ext1 x
-    simp_rw [Set.mem_unionᵢ, Set.mem_setOf_eq]
+    simp_rw [Set.mem_iUnion, Set.mem_setOf_eq]
     constructor <;> intro h
     · obtain ⟨n, hn⟩ := ((tendsto_order.1 u_lim).1 _ h).exists
       exact ⟨n, hn.le⟩
@@ -235,7 +235,7 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
         simp only [ENNReal.coe_nat]
         exact ENNReal.tendsto_nat_nhds_top
       eventually_ge_of_tendsto_gt (hx.trans_le le_top) this
-    apply Set.mem_unionᵢ.2
+    apply Set.mem_iUnion.2
     exact ((L1.and L2).And (eventually_mem_spanning_sets μ x)).exists
   refine' le_antisymm _ bot_le
   calc
@@ -451,8 +451,8 @@ theorem ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f
   by
   let S := spanning_sets μ
   rw [← @measure.restrict_univ _ _ μ, ← Union_spanning_sets μ, eventually_eq, ae_iff,
-    measure.restrict_apply' (MeasurableSet.unionᵢ (measurable_spanning_sets μ))]
-  rw [Set.inter_unionᵢ, measure_Union_null_iff]
+    measure.restrict_apply' (MeasurableSet.iUnion (measurable_spanning_sets μ))]
+  rw [Set.inter_iUnion, measure_Union_null_iff]
   intro n
   have h_meas_n : MeasurableSet (S n) := measurable_spanning_sets μ n
   have hμn : μ (S n) < ∞ := measure_spanning_sets_lt_top μ n

mathlib commit https://github.com/leanprover-community/mathlib/commit/92c69b77c5a7dc0f7eeddb552508633305157caa

@@ -422,7 +422,7 @@ theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero {f : α → E}
   refine' ae_eq_zero_of_forall_dual_of_is_separable ℝ u_sep (fun c => _) hu
   refine' ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real _ _ ht hμt
   · intro s hs hμs
-    exact ContinuousLinearMap.integrableComp c (hf_int_finite s hs hμs)
+    exact ContinuousLinearMap.integrable_comp c (hf_int_finite s hs hμs)
   · intro s hs hμs
     rw [ContinuousLinearMap.integral_comp_comm c (hf_int_finite s hs hμs), hf_zero s hs hμs]
     exact ContinuousLinearMap.map_zero _
@@ -591,8 +591,8 @@ end AeEqOfForallSetIntegralEq
 
 section Lintegral
 
-theorem AeMeasurable.ae_eq_of_forall_set_lintegral_eq {f g : α → ℝ≥0∞} (hf : AeMeasurable f μ)
-    (hg : AeMeasurable g μ) (hfi : (∫⁻ x, f x ∂μ) ≠ ∞) (hgi : (∫⁻ x, g x ∂μ) ≠ ∞)
+theorem AeMeasurable.ae_eq_of_forall_set_lintegral_eq {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
+    (hg : AEMeasurable g μ) (hfi : (∫⁻ x, f x ∂μ) ≠ ∞) (hgi : (∫⁻ x, g x ∂μ) ≠ ∞)
     (hfg : ∀ ⦃s⦄, MeasurableSet s → μ s < ∞ → (∫⁻ x in s, f x ∂μ) = ∫⁻ x in s, g x ∂μ) :
     f =ᵐ[μ] g :=
   by

mathlib commit https://github.com/leanprover-community/mathlib/commit/55d771df074d0dd020139ee1cd4b95521422df9f

@@ -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 measure_theory.function.ae_eq_of_integral
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit 46b633fd842bef9469441c0209906f6dddd2b4f5
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -56,14 +56,14 @@ section AeEqOfForall
 
 variable {α E 𝕜 : Type _} {m : MeasurableSpace α} {μ : Measure α} [IsROrC 𝕜]
 
-theorem ae_eq_zero_of_forall_inner [InnerProductSpace 𝕜 E] [SecondCountableTopology E] {f : α → E}
-    (hf : ∀ c : E, (fun x => (inner c (f x) : 𝕜)) =ᵐ[μ] 0) : f =ᵐ[μ] 0 :=
-  by
+theorem ae_eq_zero_of_forall_inner [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
+    [SecondCountableTopology E] {f : α → E} (hf : ∀ c : E, (fun x => (inner c (f x) : 𝕜)) =ᵐ[μ] 0) :
+    f =ᵐ[μ] 0 := by
   let s := dense_seq E
   have hs : DenseRange s := dense_range_dense_seq E
   have hf' : ∀ᵐ x ∂μ, ∀ n : ℕ, inner (s n) (f x) = (0 : 𝕜) := ae_all_iff.mpr fun n => hf (s n)
   refine' hf'.mono fun x hx => _
-  rw [Pi.zero_apply, ← inner_self_eq_zero]
+  rw [Pi.zero_apply, ← @inner_self_eq_zero 𝕜]
   have h_closed : IsClosed { c : E | inner c (f x) = (0 : 𝕜) } :=
     isClosed_eq (continuous_id.inner continuous_const) continuous_const
   exact @isClosed_property ℕ E _ s (fun c => inner c (f x) = (0 : 𝕜)) hs h_closed (fun n => hx n) _

mathlib commit https://github.com/leanprover-community/mathlib/commit/4c586d291f189eecb9d00581aeb3dd998ac34442

@@ -102,7 +102,7 @@ theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedS
     apply lt_irrefl ‖s x x‖
     calc
       ‖s x x‖ = ‖s x (x - a)‖ := by simp only [h, sub_zero, ContinuousLinearMap.map_sub]
-      _ ≤ 1 * ‖(x : E) - a‖ := ContinuousLinearMap.le_of_op_norm_le _ (hs x).1 _
+      _ ≤ 1 * ‖(x : E) - a‖ := (ContinuousLinearMap.le_of_op_norm_le _ (hs x).1 _)
       _ < ‖a‖ / 2 := by
         rw [one_mul]
         rwa [dist_eq_norm'] at hx
@@ -196,7 +196,8 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
         (∫⁻ x in s, g x ∂μ) + ε * μ s = (∫⁻ x in s, g x ∂μ) + ∫⁻ x in s, ε ∂μ := by
           simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, measure.restrict_apply]
         _ = ∫⁻ x in s, g x + ε ∂μ := (lintegral_add_right _ measurable_const).symm
-        _ ≤ ∫⁻ x in s, f x ∂μ := set_lintegral_mono (hg.add measurable_const) hf fun x hx => hx.1.1
+        _ ≤ ∫⁻ x in s, f x ∂μ :=
+          (set_lintegral_mono (hg.add measurable_const) hf fun x hx => hx.1.1)
         _ ≤ (∫⁻ x in s, g x ∂μ) + 0 := by
           rw [add_zero]
           exact h s s_meas s_lt_top
@@ -239,7 +240,7 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
   refine' le_antisymm _ bot_le
   calc
     μ ({ x : α | (fun x : α => f x ≤ g x) x }ᶜ) ≤ μ (⋃ n, s n) := measure_mono B
-    _ ≤ ∑' n, μ (s n) := measure_Union_le _
+    _ ≤ ∑' n, μ (s n) := (measure_Union_le _)
     _ = 0 := by simp only [μs, tsum_zero]
     
 #align measure_theory.ae_le_of_forall_set_lintegral_le_of_sigma_finite MeasureTheory.ae_le_of_forall_set_lintegral_le_of_sigmaFinite
@@ -282,7 +283,7 @@ theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : Str
         simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,
           Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe] using hx
       _ ≤ (∫⁻ x, ‖f x‖₊ ∂μ) / c :=
-        meas_ge_le_lintegral_div hfm.ae_measurable.ennnorm c_pos ENNReal.coe_ne_top
+        (meas_ge_le_lintegral_div hfm.ae_measurable.ennnorm c_pos ENNReal.coe_ne_top)
       _ < ∞ := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos
       
   have h_int_gt : (∫ x in s, f x ∂μ) ≤ b * (μ s).toReal :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/eb0cb4511aaef0da2462207b67358a0e1fe1e2ee

@@ -48,7 +48,7 @@ Generally useful lemmas which are not related to integrals:
 
 open MeasureTheory TopologicalSpace NormedSpace Filter
 
-open Ennreal NNReal MeasureTheory
+open ENNReal NNReal MeasureTheory
 
 namespace MeasureTheory
 
@@ -171,7 +171,7 @@ theorem ae_const_le_iff_forall_lt_measure_zero {β} [LinearOrder β] [Topologica
   exact hc _ (u_lt n)
 #align measure_theory.ae_const_le_iff_forall_lt_measure_zero MeasureTheory.ae_const_le_iff_forall_lt_measure_zero
 
-section Ennreal
+section ENNReal
 
 open Topology
 
@@ -209,11 +209,11 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
         _ = N * μ s := by
           simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, measure.restrict_apply]
         _ < ∞ := by
-          simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, Ennreal.coe_ne_top,
+          simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top,
             WithTop.mul_eq_top_iff, Ne.def, not_false_iff, false_and_iff, or_self_iff]
         
-    have : (ε : ℝ≥0∞) * μ s ≤ 0 := Ennreal.le_of_add_le_add_left B A
-    simpa only [Ennreal.coe_eq_zero, nonpos_iff_eq_zero, mul_eq_zero, εpos.ne', false_or_iff]
+    have : (ε : ℝ≥0∞) * μ s ≤ 0 := ENNReal.le_of_add_le_add_left B A
+    simpa only [ENNReal.coe_eq_zero, nonpos_iff_eq_zero, mul_eq_zero, εpos.ne', false_or_iff]
   obtain ⟨u, u_mono, u_pos, u_lim⟩ :
     ∃ u : ℕ → ℝ≥0, StrictAnti u ∧ (∀ n, 0 < u n) ∧ tendsto u at_top (nhds 0) :=
     exists_seq_strictAnti_tendsto (0 : ℝ≥0)
@@ -225,14 +225,14 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
     have L1 : ∀ᶠ n in at_top, g x + u n ≤ f x :=
       by
       have : tendsto (fun n => g x + u n) at_top (𝓝 (g x + (0 : ℝ≥0))) :=
-        tendsto_const_nhds.add (Ennreal.tendsto_coe.2 u_lim)
+        tendsto_const_nhds.add (ENNReal.tendsto_coe.2 u_lim)
       simp at this
       exact eventually_le_of_tendsto_lt hx this
     have L2 : ∀ᶠ n : ℕ in (at_top : Filter ℕ), g x ≤ (n : ℝ≥0) :=
       haveI : tendsto (fun n : ℕ => ((n : ℝ≥0) : ℝ≥0∞)) at_top (𝓝 ∞) :=
         by
-        simp only [Ennreal.coe_nat]
-        exact Ennreal.tendsto_nat_nhds_top
+        simp only [ENNReal.coe_nat]
+        exact ENNReal.tendsto_nat_nhds_top
       eventually_ge_of_tendsto_gt (hx.trans_le le_top) this
     apply Set.mem_unionᵢ.2
     exact ((L1.and L2).And (eventually_mem_spanning_sets μ x)).exists
@@ -255,7 +255,7 @@ theorem ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite [SigmaFinite μ] {f g :
   filter_upwards [A, B]with x using le_antisymm
 #align measure_theory.ae_eq_of_forall_set_lintegral_eq_of_sigma_finite MeasureTheory.ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite
 
-end Ennreal
+end ENNReal
 
 section Real
 
@@ -280,10 +280,10 @@ theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : Str
         intro x hx
         simp only [Set.mem_setOf_eq] at hx
         simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,
-          Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, Ennreal.coe_le_coe] using hx
+          Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe] using hx
       _ ≤ (∫⁻ x, ‖f x‖₊ ∂μ) / c :=
-        meas_ge_le_lintegral_div hfm.ae_measurable.ennnorm c_pos Ennreal.coe_ne_top
-      _ < ∞ := Ennreal.div_lt_top (ne_of_lt hf.2) c_pos
+        meas_ge_le_lintegral_div hfm.ae_measurable.ennnorm c_pos ENNReal.coe_ne_top
+      _ < ∞ := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos
       
   have h_int_gt : (∫ x in s, f x ∂μ) ≤ b * (μ s).toReal :=
     by
@@ -300,8 +300,8 @@ theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : Str
   swap
   · simp_rw [measure.restrict_restrict hs]
     exact hf_zero s hs mus
-  refine' Ennreal.toReal_nonneg.lt_of_ne fun h_eq => h _
-  cases' (Ennreal.toReal_eq_zero_iff _).mp h_eq.symm with hμs_eq_zero hμs_eq_top
+  refine' ENNReal.toReal_nonneg.lt_of_ne fun h_eq => h _
+  cases' (ENNReal.toReal_eq_zero_iff _).mp h_eq.symm with hμs_eq_zero hμs_eq_top
   · exact hμs_eq_zero
   · exact absurd hμs_eq_top mus.ne
 #align measure_theory.ae_nonneg_of_forall_set_integral_nonneg_of_strongly_measurable MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable
@@ -566,7 +566,7 @@ theorem Integrable.ae_eq_zero_of_forall_set_integral_eq_zero {f : α → E} (hf
   let f_Lp := hf_Lp.to_Lp f
   have hf_f_Lp : f =ᵐ[μ] f_Lp := (mem_ℒp.coe_fn_to_Lp hf_Lp).symm
   refine' hf_f_Lp.trans _
-  refine' Lp.ae_eq_zero_of_forall_set_integral_eq_zero f_Lp one_ne_zero Ennreal.coe_ne_top _ _
+  refine' Lp.ae_eq_zero_of_forall_set_integral_eq_zero f_Lp one_ne_zero ENNReal.coe_ne_top _ _
   · exact fun s hs hμs => integrable.integrable_on (L1.integrable_coe_fn _)
   · intro s hs hμs
     rw [integral_congr_ae (ae_restrict_of_ae hf_f_Lp.symm)]
@@ -596,7 +596,7 @@ theorem AeMeasurable.ae_eq_of_forall_set_lintegral_eq {f g : α → ℝ≥0∞}
     f =ᵐ[μ] g :=
   by
   refine'
-    Ennreal.eventuallyEq_of_toReal_eventuallyEq (ae_lt_top' hf hfi).ne_of_lt
+    ENNReal.eventuallyEq_of_toReal_eventuallyEq (ae_lt_top' hf hfi).ne_of_lt
       (ae_lt_top' hg hgi).ne_of_lt
       (integrable.ae_eq_of_forall_set_integral_eq _ _
         (integrable_to_real_of_lintegral_ne_top hf hfi)
@@ -610,8 +610,8 @@ theorem AeMeasurable.ae_eq_of_forall_set_lintegral_eq {f g : α → ℝ≥0∞}
     · refine' ae_lt_top' hf.restrict (ne_of_lt (lt_of_le_of_lt _ hfi.lt_top))
       exact @set_lintegral_univ α _ μ f ▸ lintegral_mono_set (Set.subset_univ _)
   -- putting the proofs where they are used is extremely slow
-  exacts[ae_of_all _ fun x => Ennreal.toReal_nonneg,
-    hg.ennreal_to_real.restrict.ae_strongly_measurable, ae_of_all _ fun x => Ennreal.toReal_nonneg,
+  exacts[ae_of_all _ fun x => ENNReal.toReal_nonneg,
+    hg.ennreal_to_real.restrict.ae_strongly_measurable, ae_of_all _ fun x => ENNReal.toReal_nonneg,
     hf.ennreal_to_real.restrict.ae_strongly_measurable]
 #align measure_theory.ae_measurable.ae_eq_of_forall_set_lintegral_eq MeasureTheory.AeMeasurable.ae_eq_of_forall_set_lintegral_eq
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

Done with a global search and replace, and then (to fix the #align lines), replace (#align \S*)setIntegral with $1set_integral.

@@ -23,14 +23,14 @@ All results listed below apply to two functions `f, g`, together with two main h
 * `f` and `g` are integrable on all measurable sets with finite measure,
 * for all measurable sets `s` with finite measure, `∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ`.
 The conclusion is then `f =ᵐ[μ] g`. The main lemmas are:
-* `ae_eq_of_forall_set_integral_eq_of_sigmaFinite`: case of a sigma-finite measure.
-* `AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq`: for functions which are
+* `ae_eq_of_forall_setIntegral_eq_of_sigmaFinite`: case of a sigma-finite measure.
+* `AEFinStronglyMeasurable.ae_eq_of_forall_setIntegral_eq`: for functions which are
   `AEFinStronglyMeasurable`.
-* `Lp.ae_eq_of_forall_set_integral_eq`: for elements of `Lp`, for `0 < p < ∞`.
-* `Integrable.ae_eq_of_forall_set_integral_eq`: for integrable functions.
+* `Lp.ae_eq_of_forall_setIntegral_eq`: for elements of `Lp`, for `0 < p < ∞`.
+* `Integrable.ae_eq_of_forall_setIntegral_eq`: for integrable functions.
 
 For each of these results, we also provide a lemma about the equality of one function and 0. For
-example, `Lp.ae_eq_zero_of_forall_set_integral_eq_zero`.
+example, `Lp.ae_eq_zero_of_forall_setIntegral_eq_zero`.
 
 We also register the corresponding lemma for integrals of `ℝ≥0∞`-valued functions, in
 `ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite`.
@@ -256,8 +256,8 @@ section Real
 
 variable {f : α → ℝ}
 
-/-- Don't use this lemma. Use `ae_nonneg_of_forall_set_integral_nonneg`. -/
-theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : StronglyMeasurable f)
+/-- Don't use this lemma. Use `ae_nonneg_of_forall_setIntegral_nonneg`. -/
+theorem ae_nonneg_of_forall_setIntegral_nonneg_of_stronglyMeasurable (hfm : StronglyMeasurable f)
     (hf : Integrable f μ) (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) :
     0 ≤ᵐ[μ] f := by
   simp_rw [EventuallyLE, Pi.zero_apply]
@@ -281,10 +281,10 @@ theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : Str
   have h_int_gt : (∫ x in s, f x ∂μ) ≤ b * (μ s).toReal := by
     have h_const_le : (∫ x in s, f x ∂μ) ≤ ∫ _ in s, b ∂μ := by
       refine'
-        set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _
+        setIntegral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _
       rw [EventuallyLE, ae_restrict_iff hs]
       exact eventually_of_forall fun x hxs => hxs
-    rwa [set_integral_const, smul_eq_mul, mul_comm] at h_const_le
+    rwa [setIntegral_const, smul_eq_mul, mul_comm] at h_const_le
   by_contra h
   refine' (lt_self_iff_false (∫ x in s, f x ∂μ)).mp (h_int_gt.trans_lt _)
   refine' (mul_neg_iff.mpr (Or.inr ⟨hb_neg, _⟩)).trans_le _
@@ -294,54 +294,74 @@ theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : Str
   cases' (ENNReal.toReal_eq_zero_iff _).mp h_eq.symm with hμs_eq_zero hμs_eq_top
   · exact hμs_eq_zero
   · exact absurd hμs_eq_top mus.ne
-#align measure_theory.ae_nonneg_of_forall_set_integral_nonneg_of_strongly_measurable MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable
+#align measure_theory.ae_nonneg_of_forall_set_integral_nonneg_of_strongly_measurable MeasureTheory.ae_nonneg_of_forall_setIntegral_nonneg_of_stronglyMeasurable
 
-theorem ae_nonneg_of_forall_set_integral_nonneg (hf : Integrable f μ)
+@[deprecated]
+alias ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable :=
+  ae_nonneg_of_forall_setIntegral_nonneg_of_stronglyMeasurable -- deprecated on 2024-04-17
+
+theorem ae_nonneg_of_forall_setIntegral_nonneg (hf : Integrable f μ)
     (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f := by
   rcases hf.1 with ⟨f', hf'_meas, hf_ae⟩
   have hf'_integrable : Integrable f' μ := Integrable.congr hf hf_ae
   have hf'_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f' x ∂μ := by
     intro s hs h's
-    rw [set_integral_congr_ae hs (hf_ae.mono fun x hx _ => hx.symm)]
+    rw [setIntegral_congr_ae hs (hf_ae.mono fun x hx _ => hx.symm)]
     exact hf_zero s hs h's
   exact
-    (ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable hf'_meas hf'_integrable
+    (ae_nonneg_of_forall_setIntegral_nonneg_of_stronglyMeasurable hf'_meas hf'_integrable
           hf'_zero).trans
       hf_ae.symm.le
-#align measure_theory.ae_nonneg_of_forall_set_integral_nonneg MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg
+#align measure_theory.ae_nonneg_of_forall_set_integral_nonneg MeasureTheory.ae_nonneg_of_forall_setIntegral_nonneg
+
+@[deprecated]
+alias ae_nonneg_of_forall_set_integral_nonneg :=
+  ae_nonneg_of_forall_setIntegral_nonneg -- deprecated on 2024-04-17
 
-theorem ae_le_of_forall_set_integral_le {f g : α → ℝ} (hf : Integrable f μ) (hg : Integrable g μ)
+theorem ae_le_of_forall_setIntegral_le {f g : α → ℝ} (hf : Integrable f μ) (hg : Integrable g μ)
     (hf_le : ∀ s, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) ≤ ∫ x in s, g x ∂μ) :
     f ≤ᵐ[μ] g := by
   rw [← eventually_sub_nonneg]
-  refine' ae_nonneg_of_forall_set_integral_nonneg (hg.sub hf) fun s hs => _
+  refine' ae_nonneg_of_forall_setIntegral_nonneg (hg.sub hf) fun s hs => _
   rw [integral_sub' hg.integrableOn hf.integrableOn, sub_nonneg]
   exact hf_le s hs
-#align measure_theory.ae_le_of_forall_set_integral_le MeasureTheory.ae_le_of_forall_set_integral_le
+#align measure_theory.ae_le_of_forall_set_integral_le MeasureTheory.ae_le_of_forall_setIntegral_le
+
+@[deprecated]
+alias ae_le_of_forall_set_integral_le :=
+  ae_le_of_forall_setIntegral_le -- deprecated on 2024-04-17
 
-theorem ae_nonneg_restrict_of_forall_set_integral_nonneg_inter {f : α → ℝ} {t : Set α}
+theorem ae_nonneg_restrict_of_forall_setIntegral_nonneg_inter {f : α → ℝ} {t : Set α}
     (hf : IntegrableOn f t μ)
     (hf_zero : ∀ s, MeasurableSet s → μ (s ∩ t) < ∞ → 0 ≤ ∫ x in s ∩ t, f x ∂μ) :
     0 ≤ᵐ[μ.restrict t] f := by
-  refine' ae_nonneg_of_forall_set_integral_nonneg hf fun s hs h's => _
+  refine' ae_nonneg_of_forall_setIntegral_nonneg hf fun s hs h's => _
   simp_rw [Measure.restrict_restrict hs]
   apply hf_zero s hs
   rwa [Measure.restrict_apply hs] at h's
-#align measure_theory.ae_nonneg_restrict_of_forall_set_integral_nonneg_inter MeasureTheory.ae_nonneg_restrict_of_forall_set_integral_nonneg_inter
+#align measure_theory.ae_nonneg_restrict_of_forall_set_integral_nonneg_inter MeasureTheory.ae_nonneg_restrict_of_forall_setIntegral_nonneg_inter
 
-theorem ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite [SigmaFinite μ] {f : α → ℝ}
+@[deprecated]
+alias ae_nonneg_restrict_of_forall_set_integral_nonneg_inter :=
+  ae_nonneg_restrict_of_forall_setIntegral_nonneg_inter -- deprecated on 2024-04-17
+
+theorem ae_nonneg_of_forall_setIntegral_nonneg_of_sigmaFinite [SigmaFinite μ] {f : α → ℝ}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f := by
   apply ae_of_forall_measure_lt_top_ae_restrict
   intro t t_meas t_lt_top
-  apply ae_nonneg_restrict_of_forall_set_integral_nonneg_inter (hf_int_finite t t_meas t_lt_top)
+  apply ae_nonneg_restrict_of_forall_setIntegral_nonneg_inter (hf_int_finite t t_meas t_lt_top)
   intro s s_meas _
   exact
     hf_zero _ (s_meas.inter t_meas)
       (lt_of_le_of_lt (measure_mono (Set.inter_subset_right _ _)) t_lt_top)
-#align measure_theory.ae_nonneg_of_forall_set_integral_nonneg_of_sigma_finite MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite
+#align measure_theory.ae_nonneg_of_forall_set_integral_nonneg_of_sigma_finite MeasureTheory.ae_nonneg_of_forall_setIntegral_nonneg_of_sigmaFinite
+
+@[deprecated]
+alias ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite :=
+  ae_nonneg_of_forall_setIntegral_nonneg_of_sigmaFinite -- deprecated on 2024-04-17
 
-theorem AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg {f : α → ℝ}
+theorem AEFinStronglyMeasurable.ae_nonneg_of_forall_setIntegral_nonneg {f : α → ℝ}
     (hf : AEFinStronglyMeasurable f μ)
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f := by
@@ -350,27 +370,35 @@ theorem AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg {f : α
     ae_of_ae_restrict_of_ae_restrict_compl _ this hf.ae_eq_zero_compl.symm.le
   haveI : SigmaFinite (μ.restrict t) := hf.sigmaFinite_restrict
   refine'
-    ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite (fun s hs hμts => _) fun s hs hμts => _
+    ae_nonneg_of_forall_setIntegral_nonneg_of_sigmaFinite (fun s hs hμts => _) fun s hs hμts => _
   · rw [IntegrableOn, Measure.restrict_restrict hs]
     rw [Measure.restrict_apply hs] at hμts
     exact hf_int_finite (s ∩ t) (hs.inter hf.measurableSet) hμts
   · rw [Measure.restrict_restrict hs]
     rw [Measure.restrict_apply hs] at hμts
     exact hf_zero (s ∩ t) (hs.inter hf.measurableSet) hμts
-#align measure_theory.ae_fin_strongly_measurable.ae_nonneg_of_forall_set_integral_nonneg MeasureTheory.AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg
+#align measure_theory.ae_fin_strongly_measurable.ae_nonneg_of_forall_set_integral_nonneg MeasureTheory.AEFinStronglyMeasurable.ae_nonneg_of_forall_setIntegral_nonneg
 
-theorem ae_nonneg_restrict_of_forall_set_integral_nonneg {f : α → ℝ}
+@[deprecated]
+alias AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg :=
+  AEFinStronglyMeasurable.ae_nonneg_of_forall_setIntegral_nonneg -- deprecated on 2024-04-17
+
+theorem ae_nonneg_restrict_of_forall_setIntegral_nonneg {f : α → ℝ}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) {t : Set α}
     (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : 0 ≤ᵐ[μ.restrict t] f := by
   refine'
-    ae_nonneg_restrict_of_forall_set_integral_nonneg_inter
+    ae_nonneg_restrict_of_forall_setIntegral_nonneg_inter
       (hf_int_finite t ht (lt_top_iff_ne_top.mpr hμt)) fun s hs _ => _
   refine' hf_zero (s ∩ t) (hs.inter ht) _
   exact (measure_mono (Set.inter_subset_right s t)).trans_lt (lt_top_iff_ne_top.mpr hμt)
-#align measure_theory.ae_nonneg_restrict_of_forall_set_integral_nonneg MeasureTheory.ae_nonneg_restrict_of_forall_set_integral_nonneg
+#align measure_theory.ae_nonneg_restrict_of_forall_set_integral_nonneg MeasureTheory.ae_nonneg_restrict_of_forall_setIntegral_nonneg
+
+@[deprecated]
+alias ae_nonneg_restrict_of_forall_set_integral_nonneg :=
+  ae_nonneg_restrict_of_forall_setIntegral_nonneg -- deprecated on 2024-04-17
 
-theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real {f : α → ℝ}
+theorem ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real {f : α → ℝ}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) {t : Set α}
     (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] 0 := by
@@ -378,38 +406,46 @@ theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real {f : α → ℝ}
     h_and.1.mp (h_and.2.mono fun x hx1 hx2 => le_antisymm hx2 hx1)
   refine'
     ⟨_,
-      ae_nonneg_restrict_of_forall_set_integral_nonneg hf_int_finite
+      ae_nonneg_restrict_of_forall_setIntegral_nonneg hf_int_finite
         (fun s hs hμs => (hf_zero s hs hμs).symm.le) ht hμt⟩
   suffices h_neg : 0 ≤ᵐ[μ.restrict t] -f by
     refine' h_neg.mono fun x hx => _
     rw [Pi.neg_apply] at hx
     simpa using hx
   refine'
-    ae_nonneg_restrict_of_forall_set_integral_nonneg (fun s hs hμs => (hf_int_finite s hs hμs).neg)
+    ae_nonneg_restrict_of_forall_setIntegral_nonneg (fun s hs hμs => (hf_int_finite s hs hμs).neg)
       (fun s hs hμs => _) ht hμt
   simp_rw [Pi.neg_apply]
   rw [integral_neg, neg_nonneg]
   exact (hf_zero s hs hμs).le
-#align measure_theory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real MeasureTheory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real
+#align measure_theory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real MeasureTheory.ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real
+
+@[deprecated]
+alias ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real :=
+  ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real -- deprecated on 2024-04-17
 
 end Real
 
-theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero {f : α → E}
+theorem ae_eq_zero_restrict_of_forall_setIntegral_eq_zero {f : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) {t : Set α}
     (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] 0 := by
   rcases (hf_int_finite t ht hμt.lt_top).aestronglyMeasurable.isSeparable_ae_range with
     ⟨u, u_sep, hu⟩
   refine' ae_eq_zero_of_forall_dual_of_isSeparable ℝ u_sep (fun c => _) hu
-  refine' ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real _ _ ht hμt
+  refine' ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real _ _ ht hμt
   · intro s hs hμs
     exact ContinuousLinearMap.integrable_comp c (hf_int_finite s hs hμs)
   · intro s hs hμs
     rw [ContinuousLinearMap.integral_comp_comm c (hf_int_finite s hs hμs), hf_zero s hs hμs]
     exact ContinuousLinearMap.map_zero _
-#align measure_theory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero MeasureTheory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero
+#align measure_theory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero MeasureTheory.ae_eq_zero_restrict_of_forall_setIntegral_eq_zero
 
-theorem ae_eq_restrict_of_forall_set_integral_eq {f g : α → E}
+@[deprecated]
+alias ae_eq_zero_restrict_of_forall_set_integral_eq_zero :=
+  ae_eq_zero_restrict_of_forall_setIntegral_eq_zero -- deprecated on 2024-04-17
+
+theorem ae_eq_restrict_of_forall_setIntegral_eq {f g : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
     (hfg_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ)
@@ -421,10 +457,14 @@ theorem ae_eq_restrict_of_forall_set_integral_eq {f g : α → E}
     exact sub_eq_zero.mpr (hfg_zero s hs hμs)
   have hfg_int : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn (f - g) s μ := fun s hs hμs =>
     (hf_int_finite s hs hμs).sub (hg_int_finite s hs hμs)
-  exact ae_eq_zero_restrict_of_forall_set_integral_eq_zero hfg_int hfg' ht hμt
-#align measure_theory.ae_eq_restrict_of_forall_set_integral_eq MeasureTheory.ae_eq_restrict_of_forall_set_integral_eq
+  exact ae_eq_zero_restrict_of_forall_setIntegral_eq_zero hfg_int hfg' ht hμt
+#align measure_theory.ae_eq_restrict_of_forall_set_integral_eq MeasureTheory.ae_eq_restrict_of_forall_setIntegral_eq
+
+@[deprecated]
+alias ae_eq_restrict_of_forall_set_integral_eq :=
+  ae_eq_restrict_of_forall_setIntegral_eq -- deprecated on 2024-04-17
 
-theorem ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f : α → E}
+theorem ae_eq_zero_of_forall_setIntegral_eq_of_sigmaFinite [SigmaFinite μ] {f : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 := by
   let S := spanningSets μ
@@ -435,10 +475,14 @@ theorem ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f
   have h_meas_n : MeasurableSet (S n) := measurable_spanningSets μ n
   have hμn : μ (S n) < ∞ := measure_spanningSets_lt_top μ n
   rw [← Measure.restrict_apply' h_meas_n]
-  exact ae_eq_zero_restrict_of_forall_set_integral_eq_zero hf_int_finite hf_zero h_meas_n hμn.ne
-#align measure_theory.ae_eq_zero_of_forall_set_integral_eq_of_sigma_finite MeasureTheory.ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite
+  exact ae_eq_zero_restrict_of_forall_setIntegral_eq_zero hf_int_finite hf_zero h_meas_n hμn.ne
+#align measure_theory.ae_eq_zero_of_forall_set_integral_eq_of_sigma_finite MeasureTheory.ae_eq_zero_of_forall_setIntegral_eq_of_sigmaFinite
 
-theorem ae_eq_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f g : α → E}
+@[deprecated]
+alias ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite :=
+  ae_eq_zero_of_forall_setIntegral_eq_of_sigmaFinite -- deprecated on 2024-04-17
+
+theorem ae_eq_of_forall_setIntegral_eq_of_sigmaFinite [SigmaFinite μ] {f g : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
     (hfg_eq : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) :
@@ -450,10 +494,14 @@ theorem ae_eq_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f g : 
       sub_eq_zero.mpr (hfg_eq s hs hμs)]
   have hfg_int : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn (f - g) s μ := fun s hs hμs =>
     (hf_int_finite s hs hμs).sub (hg_int_finite s hs hμs)
-  exact ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite hfg_int hfg
-#align measure_theory.ae_eq_of_forall_set_integral_eq_of_sigma_finite MeasureTheory.ae_eq_of_forall_set_integral_eq_of_sigmaFinite
+  exact ae_eq_zero_of_forall_setIntegral_eq_of_sigmaFinite hfg_int hfg
+#align measure_theory.ae_eq_of_forall_set_integral_eq_of_sigma_finite MeasureTheory.ae_eq_of_forall_setIntegral_eq_of_sigmaFinite
+
+@[deprecated]
+alias ae_eq_of_forall_set_integral_eq_of_sigmaFinite :=
+  ae_eq_of_forall_setIntegral_eq_of_sigmaFinite -- deprecated on 2024-04-17
 
-theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero {f : α → E}
+theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_setIntegral_eq_zero {f : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
     (hf : AEFinStronglyMeasurable f μ) : f =ᵐ[μ] 0 := by
@@ -461,7 +509,7 @@ theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero {f : 
   suffices f =ᵐ[μ.restrict t] 0 from
     ae_of_ae_restrict_of_ae_restrict_compl _ this hf.ae_eq_zero_compl
   haveI : SigmaFinite (μ.restrict t) := hf.sigmaFinite_restrict
-  refine' ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite _ _
+  refine' ae_eq_zero_of_forall_setIntegral_eq_of_sigmaFinite _ _
   · intro s hs hμs
     rw [IntegrableOn, Measure.restrict_restrict hs]
     rw [Measure.restrict_apply hs] at hμs
@@ -470,9 +518,13 @@ theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero {f : 
     rw [Measure.restrict_restrict hs]
     rw [Measure.restrict_apply hs] at hμs
     exact hf_zero _ (hs.inter hf.measurableSet) hμs
-#align measure_theory.ae_fin_strongly_measurable.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero
+#align measure_theory.ae_fin_strongly_measurable.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_of_forall_setIntegral_eq_zero
+
+@[deprecated]
+alias AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero :=
+  AEFinStronglyMeasurable.ae_eq_zero_of_forall_setIntegral_eq_zero -- deprecated on 2024-04-17
 
-theorem AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq {f g : α → E}
+theorem AEFinStronglyMeasurable.ae_eq_of_forall_setIntegral_eq {f g : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
     (hfg_eq : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ)
@@ -484,29 +536,41 @@ theorem AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq {f g : α → E}
       sub_eq_zero.mpr (hfg_eq s hs hμs)]
   have hfg_int : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn (f - g) s μ := fun s hs hμs =>
     (hf_int_finite s hs hμs).sub (hg_int_finite s hs hμs)
-  exact (hf.sub hg).ae_eq_zero_of_forall_set_integral_eq_zero hfg_int hfg
-#align measure_theory.ae_fin_strongly_measurable.ae_eq_of_forall_set_integral_eq MeasureTheory.AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq
+  exact (hf.sub hg).ae_eq_zero_of_forall_setIntegral_eq_zero hfg_int hfg
+#align measure_theory.ae_fin_strongly_measurable.ae_eq_of_forall_set_integral_eq MeasureTheory.AEFinStronglyMeasurable.ae_eq_of_forall_setIntegral_eq
 
-theorem Lp.ae_eq_zero_of_forall_set_integral_eq_zero (f : Lp E p μ) (hp_ne_zero : p ≠ 0)
+@[deprecated]
+alias AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq :=
+  AEFinStronglyMeasurable.ae_eq_of_forall_setIntegral_eq -- deprecated on 2024-04-17
+
+theorem Lp.ae_eq_zero_of_forall_setIntegral_eq_zero (f : Lp E p μ) (hp_ne_zero : p ≠ 0)
     (hp_ne_top : p ≠ ∞) (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 :=
-  AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero hf_int_finite hf_zero
+  AEFinStronglyMeasurable.ae_eq_zero_of_forall_setIntegral_eq_zero hf_int_finite hf_zero
     (Lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).aefinStronglyMeasurable
 set_option linter.uppercaseLean3 false in
-#align measure_theory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero
+#align measure_theory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.Lp.ae_eq_zero_of_forall_setIntegral_eq_zero
+
+@[deprecated]
+alias Lp.ae_eq_zero_of_forall_set_integral_eq_zero :=
+  Lp.ae_eq_zero_of_forall_setIntegral_eq_zero -- deprecated on 2024-04-17
 
-theorem Lp.ae_eq_of_forall_set_integral_eq (f g : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
+theorem Lp.ae_eq_of_forall_setIntegral_eq (f g : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
     (hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) :
     f =ᵐ[μ] g :=
-  AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq hf_int_finite hg_int_finite hfg
+  AEFinStronglyMeasurable.ae_eq_of_forall_setIntegral_eq hf_int_finite hg_int_finite hfg
     (Lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).aefinStronglyMeasurable
     (Lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).aefinStronglyMeasurable
 set_option linter.uppercaseLean3 false in
-#align measure_theory.Lp.ae_eq_of_forall_set_integral_eq MeasureTheory.Lp.ae_eq_of_forall_set_integral_eq
+#align measure_theory.Lp.ae_eq_of_forall_set_integral_eq MeasureTheory.Lp.ae_eq_of_forall_setIntegral_eq
 
-theorem ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim (hm : m ≤ m0) {f : α → E}
+@[deprecated]
+alias Lp.ae_eq_of_forall_set_integral_eq :=
+  Lp.ae_eq_of_forall_setIntegral_eq -- deprecated on 2024-04-17
+
+theorem ae_eq_zero_of_forall_setIntegral_eq_of_finStronglyMeasurable_trim (hm : m ≤ m0) {f : α → E}
     (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
     (hf : FinStronglyMeasurable f (μ.trim hm)) : f =ᵐ[μ] 0 := by
@@ -519,7 +583,7 @@ theorem ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim (hm :
   suffices f =ᵐ[μ.restrict t] 0 from
     ae_of_ae_restrict_of_ae_restrict_compl _ this htf_zero
   refine' measure_eq_zero_of_trim_eq_zero hm _
-  refine' ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite _ _
+  refine' ae_eq_zero_of_forall_setIntegral_eq_of_sigmaFinite _ _
   · intro s hs hμs
     unfold IntegrableOn
     rw [restrict_trim hm (μ.restrict t) hs, Measure.restrict_restrict (hm s hs)]
@@ -533,22 +597,30 @@ theorem ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim (hm :
       trim_measurableSet_eq hm (hs.inter ht_meas)] at hμs
     rw [← integral_trim hm hf_meas_m]
     exact hf_zero _ (hs.inter ht_meas) hμs
-#align measure_theory.ae_eq_zero_of_forall_set_integral_eq_of_fin_strongly_measurable_trim MeasureTheory.ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim
+#align measure_theory.ae_eq_zero_of_forall_set_integral_eq_of_fin_strongly_measurable_trim MeasureTheory.ae_eq_zero_of_forall_setIntegral_eq_of_finStronglyMeasurable_trim
+
+@[deprecated]
+alias ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim :=
+  ae_eq_zero_of_forall_setIntegral_eq_of_finStronglyMeasurable_trim -- deprecated on 2024-04-17
 
-theorem Integrable.ae_eq_zero_of_forall_set_integral_eq_zero {f : α → E} (hf : Integrable f μ)
+theorem Integrable.ae_eq_zero_of_forall_setIntegral_eq_zero {f : α → E} (hf : Integrable f μ)
     (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 := by
   have hf_Lp : Memℒp f 1 μ := memℒp_one_iff_integrable.mpr hf
   let f_Lp := hf_Lp.toLp f
   have hf_f_Lp : f =ᵐ[μ] f_Lp := (Memℒp.coeFn_toLp hf_Lp).symm
   refine' hf_f_Lp.trans _
-  refine' Lp.ae_eq_zero_of_forall_set_integral_eq_zero f_Lp one_ne_zero ENNReal.coe_ne_top _ _
+  refine' Lp.ae_eq_zero_of_forall_setIntegral_eq_zero f_Lp one_ne_zero ENNReal.coe_ne_top _ _
   · exact fun s _ _ => Integrable.integrableOn (L1.integrable_coeFn _)
   · intro s hs hμs
     rw [integral_congr_ae (ae_restrict_of_ae hf_f_Lp.symm)]
     exact hf_zero s hs hμs
-#align measure_theory.integrable.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.Integrable.ae_eq_zero_of_forall_set_integral_eq_zero
+#align measure_theory.integrable.ae_eq_zero_of_forall_set_integral_eq_zero MeasureTheory.Integrable.ae_eq_zero_of_forall_setIntegral_eq_zero
+
+@[deprecated]
+alias Integrable.ae_eq_zero_of_forall_set_integral_eq_zero :=
+  Integrable.ae_eq_zero_of_forall_setIntegral_eq_zero -- deprecated on 2024-04-17
 
-theorem Integrable.ae_eq_of_forall_set_integral_eq (f g : α → E) (hf : Integrable f μ)
+theorem Integrable.ae_eq_of_forall_setIntegral_eq (f g : α → E) (hf : Integrable f μ)
     (hg : Integrable g μ)
     (hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) :
     f =ᵐ[μ] g := by
@@ -557,18 +629,22 @@ theorem Integrable.ae_eq_of_forall_set_integral_eq (f g : α → E) (hf : Integr
     intro s hs hμs
     rw [integral_sub' hf.integrableOn hg.integrableOn]
     exact sub_eq_zero.mpr (hfg s hs hμs)
-  exact Integrable.ae_eq_zero_of_forall_set_integral_eq_zero (hf.sub hg) hfg'
-#align measure_theory.integrable.ae_eq_of_forall_set_integral_eq MeasureTheory.Integrable.ae_eq_of_forall_set_integral_eq
+  exact Integrable.ae_eq_zero_of_forall_setIntegral_eq_zero (hf.sub hg) hfg'
+#align measure_theory.integrable.ae_eq_of_forall_set_integral_eq MeasureTheory.Integrable.ae_eq_of_forall_setIntegral_eq
+
+@[deprecated]
+alias Integrable.ae_eq_of_forall_set_integral_eq :=
+  Integrable.ae_eq_of_forall_setIntegral_eq -- deprecated on 2024-04-17
 
 variable {β : Type*} [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β]
 
 /-- If an integrable function has zero integral on all closed sets, then it is zero
 almost everwhere. -/
-lemma ae_eq_zero_of_forall_set_integral_isClosed_eq_zero {μ : Measure β} {f : β → E}
+lemma ae_eq_zero_of_forall_setIntegral_isClosed_eq_zero {μ : Measure β} {f : β → E}
     (hf : Integrable f μ) (h'f : ∀ (s : Set β), IsClosed s → ∫ x in s, f x ∂μ = 0) :
     f =ᵐ[μ] 0 := by
   suffices ∀ s, MeasurableSet s → ∫ x in s, f x ∂μ = 0 from
-    hf.ae_eq_zero_of_forall_set_integral_eq_zero (fun s hs _ ↦ this s hs)
+    hf.ae_eq_zero_of_forall_setIntegral_eq_zero (fun s hs _ ↦ this s hs)
   have A : ∀ (t : Set β), MeasurableSet t → ∫ (x : β) in t, f x ∂μ = 0
       → ∫ (x : β) in tᶜ, f x ∂μ = 0 := by
     intro t t_meas ht
@@ -581,13 +657,17 @@ lemma ae_eq_zero_of_forall_set_integral_isClosed_eq_zero {μ : Measure β} {f :
   · rw [integral_iUnion g_meas g_disj hf.integrableOn]
     simp [hg]
 
+@[deprecated]
+alias ae_eq_zero_of_forall_set_integral_isClosed_eq_zero :=
+  ae_eq_zero_of_forall_setIntegral_isClosed_eq_zero -- deprecated on 2024-04-17
+
 /-- If an integrable function has zero integral on all compact sets in a sigma-compact space, then
 it is zero almost everwhere. -/
-lemma ae_eq_zero_of_forall_set_integral_isCompact_eq_zero
+lemma ae_eq_zero_of_forall_setIntegral_isCompact_eq_zero
     [SigmaCompactSpace β] [T2Space β] {μ : Measure β} {f : β → E} (hf : Integrable f μ)
     (h'f : ∀ (s : Set β), IsCompact s → ∫ x in s, f x ∂μ = 0) :
     f =ᵐ[μ] 0 := by
-  apply ae_eq_zero_of_forall_set_integral_isClosed_eq_zero hf (fun s hs ↦ ?_)
+  apply ae_eq_zero_of_forall_setIntegral_isClosed_eq_zero hf (fun s hs ↦ ?_)
   let t : ℕ → Set β := fun n ↦ compactCovering β n ∩ s
   suffices H : Tendsto (fun n ↦ ∫ x in t n, f x ∂μ) atTop (𝓝 (∫ x in s, f x ∂μ)) by
     have A : ∀ n, ∫ x in t n, f x ∂μ = 0 :=
@@ -596,7 +676,7 @@ lemma ae_eq_zero_of_forall_set_integral_isCompact_eq_zero
     exact H.symm
   have B : s = ⋃ n, t n := by rw [← Set.iUnion_inter, iUnion_compactCovering, Set.univ_inter]
   rw [B]
-  apply tendsto_set_integral_of_monotone
+  apply tendsto_setIntegral_of_monotone
   · intros n
     exact ((isCompact_compactCovering β n).inter_right hs).isClosed.measurableSet
   · intros m n hmn
@@ -605,13 +685,13 @@ lemma ae_eq_zero_of_forall_set_integral_isCompact_eq_zero
 
 /-- If a locally integrable function has zero integral on all compact sets in a sigma-compact space,
 then it is zero almost everwhere. -/
-lemma ae_eq_zero_of_forall_set_integral_isCompact_eq_zero'
+lemma ae_eq_zero_of_forall_setIntegral_isCompact_eq_zero'
     [SigmaCompactSpace β] [T2Space β] {μ : Measure β} {f : β → E} (hf : LocallyIntegrable f μ)
     (h'f : ∀ (s : Set β), IsCompact s → ∫ x in s, f x ∂μ = 0) :
     f =ᵐ[μ] 0 := by
   rw [← Measure.restrict_univ (μ := μ), ← iUnion_compactCovering]
   apply (ae_restrict_iUnion_iff _ _).2 (fun n ↦ ?_)
-  apply ae_eq_zero_of_forall_set_integral_isCompact_eq_zero
+  apply ae_eq_zero_of_forall_setIntegral_isCompact_eq_zero
   · exact hf.integrableOn_isCompact (isCompact_compactCovering β n)
   · intro s hs
     rw [Measure.restrict_restrict hs.measurableSet]
@@ -628,7 +708,7 @@ theorem AEMeasurable.ae_eq_of_forall_set_lintegral_eq {f g : α → ℝ≥0∞}
   refine'
     ENNReal.eventuallyEq_of_toReal_eventuallyEq (ae_lt_top' hf hfi).ne_of_lt
       (ae_lt_top' hg hgi).ne_of_lt
-      (Integrable.ae_eq_of_forall_set_integral_eq _ _
+      (Integrable.ae_eq_of_forall_setIntegral_eq _ _
         (integrable_toReal_of_lintegral_ne_top hf hfi)
         (integrable_toReal_of_lintegral_ne_top hg hgi) fun s hs hs' => _)
   rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae]

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

@@ -96,7 +96,7 @@ theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedS
     apply lt_irrefl ‖s x x‖
     calc
       ‖s x x‖ = ‖s x (x - a)‖ := by simp only [h, sub_zero, ContinuousLinearMap.map_sub]
-      _ ≤ 1 * ‖(x : E) - a‖ := (ContinuousLinearMap.le_of_opNorm_le _ (hs x).1 _)
+      _ ≤ 1 * ‖(x : E) - a‖ := ContinuousLinearMap.le_of_opNorm_le _ (hs x).1 _
       _ < ‖a‖ / 2 := by rw [one_mul]; rwa [dist_eq_norm'] at hx
       _ < ‖(x : E)‖ := I
       _ = ‖s x x‖ := by rw [(hs x).2, RCLike.norm_coe_norm]
@@ -217,7 +217,7 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
   refine' le_antisymm _ bot_le
   calc
     μ {x : α | (fun x : α => f x ≤ g x) x}ᶜ ≤ μ (⋃ n, s n) := measure_mono B
-    _ ≤ ∑' n, μ (s n) := (measure_iUnion_le _)
+    _ ≤ ∑' n, μ (s n) := measure_iUnion_le _
     _ = 0 := by simp only [μs, tsum_zero]
 #align measure_theory.ae_le_of_forall_set_lintegral_le_of_sigma_finite MeasureTheory.ae_le_of_forall_set_lintegral_le_of_sigmaFinite
 

This is less exhaustive than its sibling #11486 because edge cases are harder to classify. No fundamental difficulty, just me being a bit fast and lazy.

Reduce the diff of #11203

@@ -209,7 +209,7 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
       exact eventually_le_of_tendsto_lt hx this
     have L2 : ∀ᶠ n : ℕ in (atTop : Filter ℕ), g x ≤ (n : ℝ≥0) :=
       haveI : Tendsto (fun n : ℕ => ((n : ℝ≥0) : ℝ≥0∞)) atTop (𝓝 ∞) := by
-        simp only [ENNReal.coe_nat]
+        simp only [ENNReal.coe_natCast]
         exact ENNReal.tendsto_nat_nhds_top
       eventually_ge_of_tendsto_gt (hx.trans_le le_top) this
     apply Set.mem_iUnion.2

Purely automatic replacement. If this is in any way controversial; I'm happy to just close this PR.

@@ -563,7 +563,7 @@ theorem Integrable.ae_eq_of_forall_set_integral_eq (f g : α → E) (hf : Integr
 variable {β : Type*} [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β]
 
 /-- If an integrable function has zero integral on all closed sets, then it is zero
-almost everwhere.-/
+almost everwhere. -/
 lemma ae_eq_zero_of_forall_set_integral_isClosed_eq_zero {μ : Measure β} {f : β → E}
     (hf : Integrable f μ) (h'f : ∀ (s : Set β), IsClosed s → ∫ x in s, f x ∂μ = 0) :
     f =ᵐ[μ] 0 := by

Quoting [@digama0](https://github.com/digama0):

These were actually never meant to go in the file, they are basically debugging information and only useful on significantly broken mathport files. You can safely remove all of them.

@@ -67,7 +67,6 @@ theorem ae_eq_zero_of_forall_inner [NormedAddCommGroup E] [InnerProductSpace 
   exact @isClosed_property ℕ E _ s (fun c => inner c (f x) = (0 : 𝕜)) hs h_closed (fun n => hx n) _
 #align measure_theory.ae_eq_zero_of_forall_inner MeasureTheory.ae_eq_zero_of_forall_inner
 
--- mathport name: «expr⟪ , ⟫»
 local notation "⟪" x ", " y "⟫" => y x
 
 variable (𝕜)

@@ -83,7 +83,7 @@ theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedS
   have A : ∀ a : E, a ∈ t → (∀ x, ⟪a, s x⟫ = (0 : 𝕜)) → a = 0 := by
     intro a hat ha
     contrapose! ha
-    have a_pos : 0 < ‖a‖ := by simp only [ha, norm_pos_iff, Ne.def, not_false_iff]
+    have a_pos : 0 < ‖a‖ := by simp only [ha, norm_pos_iff, Ne, not_false_iff]
     have a_mem : a ∈ closure d := hd hat
     obtain ⟨x, hx⟩ : ∃ x : d, dist a x < ‖a‖ / 2 := by
       rcases Metric.mem_closure_iff.1 a_mem (‖a‖ / 2) (half_pos a_pos) with ⟨x, h'x, hx⟩
@@ -192,7 +192,7 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
           simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]
         _ < ∞ := by
           simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top,
-            ENNReal.mul_eq_top, Ne.def, not_false_iff, false_and_iff, or_self_iff]
+            ENNReal.mul_eq_top, Ne, not_false_iff, false_and_iff, or_self_iff]
     have : (ε : ℝ≥0∞) * μ s ≤ 0 := ENNReal.le_of_add_le_add_left B A
     simpa only [ENNReal.coe_eq_zero, nonpos_iff_eq_zero, mul_eq_zero, εpos.ne', false_or_iff]
   obtain ⟨u, _, u_pos, u_lim⟩ :

IsROrC contains data, which goes against the expectation that classes prefixed with Is are prop-valued. People have been complaining about this on and off, so this PR renames IsROrC to RCLike.

@@ -52,7 +52,7 @@ namespace MeasureTheory
 
 section AeEqOfForall
 
-variable {α E 𝕜 : Type*} {m : MeasurableSpace α} {μ : Measure α} [IsROrC 𝕜]
+variable {α E 𝕜 : Type*} {m : MeasurableSpace α} {μ : Measure α} [RCLike 𝕜]
 
 theorem ae_eq_zero_of_forall_inner [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
     [SecondCountableTopology E] {f : α → E} (hf : ∀ c : E, (fun x => (inner c (f x) : 𝕜)) =ᵐ[μ] 0) :
@@ -100,7 +100,7 @@ theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedS
       _ ≤ 1 * ‖(x : E) - a‖ := (ContinuousLinearMap.le_of_opNorm_le _ (hs x).1 _)
       _ < ‖a‖ / 2 := by rw [one_mul]; rwa [dist_eq_norm'] at hx
       _ < ‖(x : E)‖ := I
-      _ = ‖s x x‖ := by rw [(hs x).2, IsROrC.norm_coe_norm]
+      _ = ‖s x x‖ := by rw [(hs x).2, RCLike.norm_coe_norm]
   have hfs : ∀ y : d, ∀ᵐ x ∂μ, ⟪f x, s y⟫ = (0 : 𝕜) := fun y => hf (s y)
   have hf' : ∀ᵐ x ∂μ, ∀ y : d, ⟪f x, s y⟫ = (0 : 𝕜) := by rwa [ae_all_iff]
   filter_upwards [hf', h't] with x hx h'x

Co-authored-by: Patrick Massot <patrickmassot@free.fr> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -268,13 +268,13 @@ theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : Str
   have hs : MeasurableSet s := hfm.measurableSet_le stronglyMeasurable_const
   have mus : μ s < ∞ := by
     let c : ℝ≥0 := ⟨|b|, abs_nonneg _⟩
-    have c_pos : (c : ℝ≥0∞) ≠ 0 := by simpa [← NNReal.coe_eq_zero] using hb_neg.ne
+    have c_pos : (c : ℝ≥0∞) ≠ 0 := by simpa [c, ← NNReal.coe_eq_zero] using hb_neg.ne
     calc
       μ s ≤ μ {x | (c : ℝ≥0∞) ≤ ‖f x‖₊} := by
         apply measure_mono
         intro x hx
-        simp only [Set.mem_setOf_eq] at hx
-        simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,
+        simp only [s, Set.mem_setOf_eq] at hx
+        simpa only [c, nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,
           Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx
       _ ≤ (∫⁻ x, ‖f x‖₊ ∂μ) / c :=
         (meas_ge_le_lintegral_div hfm.aemeasurable.ennnorm c_pos ENNReal.coe_ne_top)

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>

@@ -375,14 +375,14 @@ theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real {f : α → ℝ}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) {t : Set α}
     (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] 0 := by
-  suffices h_and : f ≤ᵐ[μ.restrict t] 0 ∧ 0 ≤ᵐ[μ.restrict t] f
-  exact h_and.1.mp (h_and.2.mono fun x hx1 hx2 => le_antisymm hx2 hx1)
+  suffices h_and : f ≤ᵐ[μ.restrict t] 0 ∧ 0 ≤ᵐ[μ.restrict t] f from
+    h_and.1.mp (h_and.2.mono fun x hx1 hx2 => le_antisymm hx2 hx1)
   refine'
     ⟨_,
       ae_nonneg_restrict_of_forall_set_integral_nonneg hf_int_finite
         (fun s hs hμs => (hf_zero s hs hμs).symm.le) ht hμt⟩
-  suffices h_neg : 0 ≤ᵐ[μ.restrict t] -f
-  · refine' h_neg.mono fun x hx => _
+  suffices h_neg : 0 ≤ᵐ[μ.restrict t] -f by
+    refine' h_neg.mono fun x hx => _
     rw [Pi.neg_apply] at hx
     simpa using hx
   refine'
@@ -517,8 +517,8 @@ theorem ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim (hm :
     rw [EventuallyEq, ae_restrict_iff' (MeasurableSet.compl (hm _ ht_meas))]
     exact eventually_of_forall htf_zero
   have hf_meas_m : StronglyMeasurable[m] f := hf.stronglyMeasurable
-  suffices : f =ᵐ[μ.restrict t] 0
-  exact ae_of_ae_restrict_of_ae_restrict_compl _ this htf_zero
+  suffices f =ᵐ[μ.restrict t] 0 from
+    ae_of_ae_restrict_of_ae_restrict_compl _ this htf_zero
   refine' measure_eq_zero_of_trim_eq_zero hm _
   refine' ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite _ _
   · intro s hs hμs
@@ -568,8 +568,8 @@ almost everwhere.-/
 lemma ae_eq_zero_of_forall_set_integral_isClosed_eq_zero {μ : Measure β} {f : β → E}
     (hf : Integrable f μ) (h'f : ∀ (s : Set β), IsClosed s → ∫ x in s, f x ∂μ = 0) :
     f =ᵐ[μ] 0 := by
-  suffices : ∀ s, MeasurableSet s → ∫ x in s, f x ∂μ = 0
-  · exact hf.ae_eq_zero_of_forall_set_integral_eq_zero (fun s hs _ ↦ this s hs)
+  suffices ∀ s, MeasurableSet s → ∫ x in s, f x ∂μ = 0 from
+    hf.ae_eq_zero_of_forall_set_integral_eq_zero (fun s hs _ ↦ this s hs)
   have A : ∀ (t : Set β), MeasurableSet t → ∫ (x : β) in t, f x ∂μ = 0
       → ∫ (x : β) in tᶜ, f x ∂μ = 0 := by
     intro t t_meas ht
@@ -590,8 +590,8 @@ lemma ae_eq_zero_of_forall_set_integral_isCompact_eq_zero
     f =ᵐ[μ] 0 := by
   apply ae_eq_zero_of_forall_set_integral_isClosed_eq_zero hf (fun s hs ↦ ?_)
   let t : ℕ → Set β := fun n ↦ compactCovering β n ∩ s
-  suffices H : Tendsto (fun n ↦ ∫ x in t n, f x ∂μ) atTop (𝓝 (∫ x in s, f x ∂μ))
-  · have A : ∀ n, ∫ x in t n, f x ∂μ = 0 :=
+  suffices H : Tendsto (fun n ↦ ∫ x in t n, f x ∂μ) atTop (𝓝 (∫ x in s, f x ∂μ)) by
+    have A : ∀ n, ∫ x in t n, f x ∂μ = 0 :=
       fun n ↦ h'f _ (IsCompact.inter_right (isCompact_compactCovering β n) hs)
     simp_rw [A, tendsto_const_nhds_iff] at H
     exact H.symm

Also fix GeneralizedContinuedFraction.of_convergence: it worked for the Preorder.topology only.

@@ -145,7 +145,7 @@ theorem ae_const_le_iff_forall_lt_measure_zero {β} [LinearOrder β] [Topologica
   push_neg at H h
   obtain ⟨u, _, u_lt, u_lim, -⟩ :
     ∃ u : ℕ → β,
-      StrictMono u ∧ (∀ n : ℕ, u n < c) ∧ Tendsto u atTop (nhds c) ∧ ∀ n : ℕ, u n ∈ Set.Iio c :=
+      StrictMono u ∧ (∀ n : ℕ, u n < c) ∧ Tendsto u atTop (𝓝 c) ∧ ∀ n : ℕ, u n ∈ Set.Iio c :=
     H.exists_seq_strictMono_tendsto_of_not_mem (lt_irrefl c) h
   have h_Union : {x | f x < c} = ⋃ n : ℕ, {x | f x ≤ u n} := by
     ext1 x
@@ -196,7 +196,7 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
     have : (ε : ℝ≥0∞) * μ s ≤ 0 := ENNReal.le_of_add_le_add_left B A
     simpa only [ENNReal.coe_eq_zero, nonpos_iff_eq_zero, mul_eq_zero, εpos.ne', false_or_iff]
   obtain ⟨u, _, u_pos, u_lim⟩ :
-    ∃ u : ℕ → ℝ≥0, StrictAnti u ∧ (∀ n, 0 < u n) ∧ Tendsto u atTop (nhds 0) :=
+    ∃ u : ℕ → ℝ≥0, StrictAnti u ∧ (∀ n, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
     exists_seq_strictAnti_tendsto (0 : ℝ≥0)
   let s := fun n : ℕ => {x | g x + u n ≤ f x ∧ g x ≤ (n : ℝ≥0)} ∩ spanningSets μ n
   have μs : ∀ n, μ (s n) = 0 := fun n => A _ _ _ (u_pos n)

• upgrade isSeparable_iUnion to an Iff lemma, restore the original version as IsSeparable.iUnion;
• add isSeparable_union and isSeparable_closure;
• upgrade isSeparable_pi from [Finite ι] to [Countable ι], add IsSeparable.univ_pi version;
• add Dense.isSeparable_iff and isSeparable_range;
• rename isSeparable_of_separableSpace_subtype to IsSeparable.of_subtype;
• rename isSeparable_of_separableSpace to IsSeparable.of_separableSpace.

@@ -110,7 +110,7 @@ theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedS
 theorem ae_eq_zero_of_forall_dual [NormedAddCommGroup E] [NormedSpace 𝕜 E]
     [SecondCountableTopology E] {f : α → E} (hf : ∀ c : Dual 𝕜 E, (fun x => ⟪f x, c⟫) =ᵐ[μ] 0) :
     f =ᵐ[μ] 0 :=
-  ae_eq_zero_of_forall_dual_of_isSeparable 𝕜 (isSeparable_of_separableSpace (Set.univ : Set E)) hf
+  ae_eq_zero_of_forall_dual_of_isSeparable 𝕜 (.of_separableSpace Set.univ) hf
     (eventually_of_forall fun _ => Set.mem_univ _)
 #align measure_theory.ae_eq_zero_of_forall_dual MeasureTheory.ae_eq_zero_of_forall_dual
 

Co-authored-by: adomani <adomani@gmail.com>

@@ -97,7 +97,7 @@ theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedS
     apply lt_irrefl ‖s x x‖
     calc
       ‖s x x‖ = ‖s x (x - a)‖ := by simp only [h, sub_zero, ContinuousLinearMap.map_sub]
-      _ ≤ 1 * ‖(x : E) - a‖ := (ContinuousLinearMap.le_of_op_norm_le _ (hs x).1 _)
+      _ ≤ 1 * ‖(x : E) - a‖ := (ContinuousLinearMap.le_of_opNorm_le _ (hs x).1 _)
       _ < ‖a‖ / 2 := by rw [one_mul]; rwa [dist_eq_norm'] at hx
       _ < ‖(x : E)‖ := I
       _ = ‖s x x‖ := by rw [(hs x).2, IsROrC.norm_coe_norm]

Various results about rnDeriv, notably rnDeriv_add, rnDeriv_smul_left and rnDeriv_smul_right. These results describe the Radon-Nikodym derivatives of sums and scaling of measures.

These lemmas were already there for signed measures, but not for Measure. The proofs for signed measures use that addition is cancelative (μ + ν₁ = μ + ν₂ ↔ ν₁ = ν₂). This is not true in general for measures, but is true when μ is mutually singular with the two other measures or when μ is sigma-finite, which is enough for these proofs.

Co-authored-by: RemyDegenne <remydegenne@gmail.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -623,7 +623,7 @@ end AeEqOfForallSetIntegralEq
 section Lintegral
 
 theorem AEMeasurable.ae_eq_of_forall_set_lintegral_eq {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
-    (hg : AEMeasurable g μ) (hfi : (∫⁻ x, f x ∂μ) ≠ ∞) (hgi : (∫⁻ x, g x ∂μ) ≠ ∞)
+    (hg : AEMeasurable g μ) (hfi : ∫⁻ x, f x ∂μ ≠ ∞) (hgi : ∫⁻ x, g x ∂μ ≠ ∞)
     (hfg : ∀ ⦃s⦄, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) :
     f =ᵐ[μ] g := by
   refine'

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.

@@ -648,4 +648,25 @@ theorem AEMeasurable.ae_eq_of_forall_set_lintegral_eq {f g : α → ℝ≥0∞}
 
 end Lintegral
 
+section WithDensity
+
+variable {m : MeasurableSpace α} {μ : Measure α}
+
+theorem withDensity_eq_iff_of_sigmaFinite [SigmaFinite μ] {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
+    (hg : AEMeasurable g μ) : μ.withDensity f = μ.withDensity g ↔ f =ᵐ[μ] g :=
+  ⟨fun hfg ↦ by
+    refine ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite₀ hf hg fun s hs _ ↦ ?_
+    rw [← withDensity_apply f hs, ← withDensity_apply g hs, ← hfg], withDensity_congr_ae⟩
+
+theorem withDensity_eq_iff {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
+    (hg : AEMeasurable g μ) (hfi : ∫⁻ x, f x ∂μ ≠ ∞) :
+    μ.withDensity f = μ.withDensity g ↔ f =ᵐ[μ] g :=
+  ⟨fun hfg ↦ by
+    refine AEMeasurable.ae_eq_of_forall_set_lintegral_eq hf hg hfi ?_ fun s hs _ ↦ ?_
+    · rwa [← set_lintegral_univ, ← withDensity_apply g MeasurableSet.univ, ← hfg,
+        withDensity_apply f MeasurableSet.univ, set_lintegral_univ]
+    · rw [← withDensity_apply f hs, ← withDensity_apply g hs, ← hfg], withDensity_congr_ae⟩
+
+end WithDensity
+
 end MeasureTheory

@@ -222,14 +222,33 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
     _ = 0 := by simp only [μs, tsum_zero]
 #align measure_theory.ae_le_of_forall_set_lintegral_le_of_sigma_finite MeasureTheory.ae_le_of_forall_set_lintegral_le_of_sigmaFinite
 
-theorem ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite [SigmaFinite μ] {f g : α → ℝ≥0∞}
-    (hf : Measurable f) (hg : Measurable g)
+theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite₀ [SigmaFinite μ]
+    {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
+    (h : ∀ s, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ) :
+    f ≤ᵐ[μ] g := by
+  have h' : ∀ s, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, hf.mk f x ∂μ ≤ ∫⁻ x in s, hg.mk g x ∂μ := by
+    refine fun s hs hμs ↦ (set_lintegral_congr_fun hs ?_).trans_le
+      ((h s hs hμs).trans_eq (set_lintegral_congr_fun hs ?_))
+    · filter_upwards [hf.ae_eq_mk] with a ha using fun _ ↦ ha.symm
+    · filter_upwards [hg.ae_eq_mk] with a ha using fun _ ↦ ha
+  filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk,
+    ae_le_of_forall_set_lintegral_le_of_sigmaFinite hf.measurable_mk hg.measurable_mk h']
+    with a haf hag ha
+  rwa [haf, hag]
+
+theorem ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite₀ [SigmaFinite μ]
+    {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
     (h : ∀ s, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) : f =ᵐ[μ] g := by
   have A : f ≤ᵐ[μ] g :=
-    ae_le_of_forall_set_lintegral_le_of_sigmaFinite hf hg fun s hs h's => le_of_eq (h s hs h's)
+    ae_le_of_forall_set_lintegral_le_of_sigmaFinite₀ hf hg fun s hs h's => le_of_eq (h s hs h's)
   have B : g ≤ᵐ[μ] f :=
-    ae_le_of_forall_set_lintegral_le_of_sigmaFinite hg hf fun s hs h's => ge_of_eq (h s hs h's)
+    ae_le_of_forall_set_lintegral_le_of_sigmaFinite₀ hg hf fun s hs h's => ge_of_eq (h s hs h's)
   filter_upwards [A, B] with x using le_antisymm
+
+theorem ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite [SigmaFinite μ] {f g : α → ℝ≥0∞}
+    (hf : Measurable f) (hg : Measurable g)
+    (h : ∀ s, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) : f =ᵐ[μ] g :=
+  ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite₀ hf.aemeasurable hg.aemeasurable h
 #align measure_theory.ae_eq_of_forall_set_lintegral_eq_of_sigma_finite MeasureTheory.ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite
 
 end ENNReal

Replace rcases( with rcases (. Same thing for convert( and congrm(. No other change.

@@ -380,7 +380,7 @@ theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero {f : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) {t : Set α}
     (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] 0 := by
-  rcases(hf_int_finite t ht hμt.lt_top).aestronglyMeasurable.isSeparable_ae_range with
+  rcases (hf_int_finite t ht hμt.lt_top).aestronglyMeasurable.isSeparable_ae_range with
     ⟨u, u_sep, hu⟩
   refine' ae_eq_zero_of_forall_dual_of_isSeparable ℝ u_sep (fun c => _) hu
   refine' ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real _ _ ht hμt

@@ -603,7 +603,7 @@ end AeEqOfForallSetIntegralEq
 
 section Lintegral
 
-theorem AeMeasurable.ae_eq_of_forall_set_lintegral_eq {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
+theorem AEMeasurable.ae_eq_of_forall_set_lintegral_eq {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
     (hg : AEMeasurable g μ) (hfi : (∫⁻ x, f x ∂μ) ≠ ∞) (hgi : (∫⁻ x, g x ∂μ) ≠ ∞)
     (hfg : ∀ ⦃s⦄, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) :
     f =ᵐ[μ] g := by
@@ -625,7 +625,7 @@ theorem AeMeasurable.ae_eq_of_forall_set_lintegral_eq {f g : α → ℝ≥0∞}
   exacts [ae_of_all _ fun x => ENNReal.toReal_nonneg,
     hg.ennreal_toReal.restrict.aestronglyMeasurable, ae_of_all _ fun x => ENNReal.toReal_nonneg,
     hf.ennreal_toReal.restrict.aestronglyMeasurable]
-#align measure_theory.ae_measurable.ae_eq_of_forall_set_lintegral_eq MeasureTheory.AeMeasurable.ae_eq_of_forall_set_lintegral_eq
+#align measure_theory.ae_measurable.ae_eq_of_forall_set_lintegral_eq MeasureTheory.AEMeasurable.ae_eq_of_forall_set_lintegral_eq
 
 end Lintegral
 

Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -270,7 +270,8 @@ theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : Str
   by_contra h
   refine' (lt_self_iff_false (∫ x in s, f x ∂μ)).mp (h_int_gt.trans_lt _)
   refine' (mul_neg_iff.mpr (Or.inr ⟨hb_neg, _⟩)).trans_le _
-  swap; · simp_rw [Measure.restrict_restrict hs]; exact hf_zero s hs mus
+  swap
+  · exact hf_zero s hs mus
   refine' ENNReal.toReal_nonneg.lt_of_ne fun h_eq => h _
   cases' (ENNReal.toReal_eq_zero_iff _).mp h_eq.symm with hμs_eq_zero hμs_eq_top
   · exact hμs_eq_zero

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

This has nice performance benefits.

@@ -52,7 +52,7 @@ namespace MeasureTheory
 
 section AeEqOfForall
 
-variable {α E 𝕜 : Type _} {m : MeasurableSpace α} {μ : Measure α} [IsROrC 𝕜]
+variable {α E 𝕜 : Type*} {m : MeasurableSpace α} {μ : Measure α} [IsROrC 𝕜]
 
 theorem ae_eq_zero_of_forall_inner [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
     [SecondCountableTopology E] {f : α → E} (hf : ∀ c : E, (fun x => (inner c (f x) : 𝕜)) =ᵐ[μ] 0) :
@@ -118,7 +118,7 @@ variable {𝕜}
 
 end AeEqOfForall
 
-variable {α E : Type _} {m m0 : MeasurableSpace α} {μ : Measure α} {s t : Set α}
+variable {α E : Type*} {m m0 : MeasurableSpace α} {μ : Measure α} {s t : Set α}
   [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {p : ℝ≥0∞}
 
 section AeEqOfForallSetIntegralEq
@@ -541,7 +541,7 @@ theorem Integrable.ae_eq_of_forall_set_integral_eq (f g : α → E) (hf : Integr
   exact Integrable.ae_eq_zero_of_forall_set_integral_eq_zero (hf.sub hg) hfg'
 #align measure_theory.integrable.ae_eq_of_forall_set_integral_eq MeasureTheory.Integrable.ae_eq_of_forall_set_integral_eq
 
-variable {β : Type _} [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β]
+variable {β : Type*} [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β]
 
 /-- If an integrable function has zero integral on all closed sets, then it is zero
 almost everwhere.-/

We show that, if a locally integrable function has zero integral on all compact sets, then it vanishes almost everywhere.

@@ -46,7 +46,7 @@ Generally useful lemmas which are not related to integrals:
 
 open MeasureTheory TopologicalSpace NormedSpace Filter
 
-open scoped ENNReal NNReal MeasureTheory
+open scoped ENNReal NNReal MeasureTheory Topology
 
 namespace MeasureTheory
 
@@ -541,6 +541,63 @@ theorem Integrable.ae_eq_of_forall_set_integral_eq (f g : α → E) (hf : Integr
   exact Integrable.ae_eq_zero_of_forall_set_integral_eq_zero (hf.sub hg) hfg'
 #align measure_theory.integrable.ae_eq_of_forall_set_integral_eq MeasureTheory.Integrable.ae_eq_of_forall_set_integral_eq
 
+variable {β : Type _} [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β]
+
+/-- If an integrable function has zero integral on all closed sets, then it is zero
+almost everwhere.-/
+lemma ae_eq_zero_of_forall_set_integral_isClosed_eq_zero {μ : Measure β} {f : β → E}
+    (hf : Integrable f μ) (h'f : ∀ (s : Set β), IsClosed s → ∫ x in s, f x ∂μ = 0) :
+    f =ᵐ[μ] 0 := by
+  suffices : ∀ s, MeasurableSet s → ∫ x in s, f x ∂μ = 0
+  · exact hf.ae_eq_zero_of_forall_set_integral_eq_zero (fun s hs _ ↦ this s hs)
+  have A : ∀ (t : Set β), MeasurableSet t → ∫ (x : β) in t, f x ∂μ = 0
+      → ∫ (x : β) in tᶜ, f x ∂μ = 0 := by
+    intro t t_meas ht
+    have I : ∫ x, f x ∂μ = 0 := by rw [← integral_univ]; exact h'f _ isClosed_univ
+    simpa [ht, I] using integral_add_compl t_meas hf
+  intro s hs
+  refine MeasurableSet.induction_on_open (fun U hU ↦ ?_) A (fun g g_disj g_meas hg ↦ ?_) hs
+  · rw [← compl_compl U]
+    exact A _ hU.measurableSet.compl (h'f _ hU.isClosed_compl)
+  · rw [integral_iUnion g_meas g_disj hf.integrableOn]
+    simp [hg]
+
+/-- If an integrable function has zero integral on all compact sets in a sigma-compact space, then
+it is zero almost everwhere. -/
+lemma ae_eq_zero_of_forall_set_integral_isCompact_eq_zero
+    [SigmaCompactSpace β] [T2Space β] {μ : Measure β} {f : β → E} (hf : Integrable f μ)
+    (h'f : ∀ (s : Set β), IsCompact s → ∫ x in s, f x ∂μ = 0) :
+    f =ᵐ[μ] 0 := by
+  apply ae_eq_zero_of_forall_set_integral_isClosed_eq_zero hf (fun s hs ↦ ?_)
+  let t : ℕ → Set β := fun n ↦ compactCovering β n ∩ s
+  suffices H : Tendsto (fun n ↦ ∫ x in t n, f x ∂μ) atTop (𝓝 (∫ x in s, f x ∂μ))
+  · have A : ∀ n, ∫ x in t n, f x ∂μ = 0 :=
+      fun n ↦ h'f _ (IsCompact.inter_right (isCompact_compactCovering β n) hs)
+    simp_rw [A, tendsto_const_nhds_iff] at H
+    exact H.symm
+  have B : s = ⋃ n, t n := by rw [← Set.iUnion_inter, iUnion_compactCovering, Set.univ_inter]
+  rw [B]
+  apply tendsto_set_integral_of_monotone
+  · intros n
+    exact ((isCompact_compactCovering β n).inter_right hs).isClosed.measurableSet
+  · intros m n hmn
+    exact Set.inter_subset_inter_left _ (compactCovering_subset β hmn)
+  · exact hf.integrableOn
+
+/-- If a locally integrable function has zero integral on all compact sets in a sigma-compact space,
+then it is zero almost everwhere. -/
+lemma ae_eq_zero_of_forall_set_integral_isCompact_eq_zero'
+    [SigmaCompactSpace β] [T2Space β] {μ : Measure β} {f : β → E} (hf : LocallyIntegrable f μ)
+    (h'f : ∀ (s : Set β), IsCompact s → ∫ x in s, f x ∂μ = 0) :
+    f =ᵐ[μ] 0 := by
+  rw [← Measure.restrict_univ (μ := μ), ← iUnion_compactCovering]
+  apply (ae_restrict_iUnion_iff _ _).2 (fun n ↦ ?_)
+  apply ae_eq_zero_of_forall_set_integral_isCompact_eq_zero
+  · exact hf.integrableOn_isCompact (isCompact_compactCovering β n)
+  · intro s hs
+    rw [Measure.restrict_restrict hs.measurableSet]
+    exact h'f _ (hs.inter (isCompact_compactCovering β n))
+
 end AeEqOfForallSetIntegralEq
 
 section Lintegral

• depends on: #5966

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

@@ -2,17 +2,14 @@
 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 measure_theory.function.ae_eq_of_integral
-! leanprover-community/mathlib commit 915591b2bb3ea303648db07284a161a7f2a9e3d4
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.InnerProductSpace.Basic
 import Mathlib.Analysis.NormedSpace.Dual
 import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
 import Mathlib.MeasureTheory.Integral.SetIntegral
 
+#align_import measure_theory.function.ae_eq_of_integral from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
+
 /-! # From equality of integrals to equality of functions
 
 This file provides various statements of the general form "if two functions have the same integral

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

@@ -220,7 +220,7 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
     exact ((L1.and L2).and (eventually_mem_spanningSets μ x)).exists
   refine' le_antisymm _ bot_le
   calc
-    μ ({x : α | (fun x : α => f x ≤ g x) x}ᶜ) ≤ μ (⋃ n, s n) := measure_mono B
+    μ {x : α | (fun x : α => f x ≤ g x) x}ᶜ ≤ μ (⋃ n, s n) := measure_mono B
     _ ≤ ∑' n, μ (s n) := (measure_iUnion_le _)
     _ = 0 := by simp only [μs, tsum_zero]
 #align measure_theory.ae_le_of_forall_set_lintegral_le_of_sigma_finite MeasureTheory.ae_le_of_forall_set_lintegral_le_of_sigmaFinite
@@ -496,7 +496,7 @@ theorem ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim (hm :
     (hf : FinStronglyMeasurable f (μ.trim hm)) : f =ᵐ[μ] 0 := by
   obtain ⟨t, ht_meas, htf_zero, htμ⟩ := hf.exists_set_sigmaFinite
   haveI : SigmaFinite ((μ.restrict t).trim hm) := by rwa [restrict_trim hm μ ht_meas] at htμ
-  have htf_zero : f =ᵐ[μ.restrict (tᶜ)] 0 := by
+  have htf_zero : f =ᵐ[μ.restrict tᶜ] 0 := by
     rw [EventuallyEq, ae_restrict_iff' (MeasurableSet.compl (hm _ ht_meas))]
     exact eventually_of_forall htf_zero
   have hf_meas_m : StronglyMeasurable[m] f := hf.stronglyMeasurable

@@ -227,7 +227,7 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
 
 theorem ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite [SigmaFinite μ] {f g : α → ℝ≥0∞}
     (hf : Measurable f) (hg : Measurable g)
-    (h : ∀ s, MeasurableSet s → μ s < ∞ → (∫⁻ x in s, f x ∂μ) = ∫⁻ x in s, g x ∂μ) : f =ᵐ[μ] g := by
+    (h : ∀ s, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) : f =ᵐ[μ] g := by
   have A : f ≤ᵐ[μ] g :=
     ae_le_of_forall_set_lintegral_le_of_sigmaFinite hf hg fun s hs h's => le_of_eq (h s hs h's)
   have B : g ≤ᵐ[μ] f :=
@@ -356,7 +356,7 @@ theorem ae_nonneg_restrict_of_forall_set_integral_nonneg {f : α → ℝ}
 
 theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real {f : α → ℝ}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
-    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = 0) {t : Set α}
+    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) {t : Set α}
     (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] 0 := by
   suffices h_and : f ≤ᵐ[μ.restrict t] 0 ∧ 0 ≤ᵐ[μ.restrict t] f
   exact h_and.1.mp (h_and.2.mono fun x hx1 hx2 => le_antisymm hx2 hx1)
@@ -380,7 +380,7 @@ end Real
 
 theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero {f : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
-    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = 0) {t : Set α}
+    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) {t : Set α}
     (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] 0 := by
   rcases(hf_int_finite t ht hμt.lt_top).aestronglyMeasurable.isSeparable_ae_range with
     ⟨u, u_sep, hu⟩
@@ -396,7 +396,7 @@ theorem ae_eq_zero_restrict_of_forall_set_integral_eq_zero {f : α → E}
 theorem ae_eq_restrict_of_forall_set_integral_eq {f g : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
-    (hfg_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = ∫ x in s, g x ∂μ)
+    (hfg_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ)
     {t : Set α} (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] g := by
   rw [← sub_ae_eq_zero]
   have hfg' : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, (f - g) x ∂μ) = 0 := by
@@ -410,7 +410,7 @@ theorem ae_eq_restrict_of_forall_set_integral_eq {f g : α → E}
 
 theorem ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
-    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = 0) : f =ᵐ[μ] 0 := by
+    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 := by
   let S := spanningSets μ
   rw [← @Measure.restrict_univ _ _ μ, ← iUnion_spanningSets μ, EventuallyEq, ae_iff,
     Measure.restrict_apply' (MeasurableSet.iUnion (measurable_spanningSets μ))]
@@ -425,7 +425,7 @@ theorem ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f
 theorem ae_eq_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f g : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
-    (hfg_eq : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = ∫ x in s, g x ∂μ) :
+    (hfg_eq : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) :
     f =ᵐ[μ] g := by
   rw [← sub_ae_eq_zero]
   have hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, (f - g) x ∂μ) = 0 := by
@@ -439,7 +439,7 @@ theorem ae_eq_of_forall_set_integral_eq_of_sigmaFinite [SigmaFinite μ] {f g : 
 
 theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero {f : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
-    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = 0)
+    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
     (hf : AEFinStronglyMeasurable f μ) : f =ᵐ[μ] 0 := by
   let t := hf.sigmaFiniteSet
   suffices f =ᵐ[μ.restrict t] 0 from
@@ -459,7 +459,7 @@ theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero {f : 
 theorem AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq {f g : α → E}
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
-    (hfg_eq : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = ∫ x in s, g x ∂μ)
+    (hfg_eq : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ)
     (hf : AEFinStronglyMeasurable f μ) (hg : AEFinStronglyMeasurable g μ) : f =ᵐ[μ] g := by
   rw [← sub_ae_eq_zero]
   have hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, (f - g) x ∂μ) = 0 := by
@@ -473,7 +473,7 @@ theorem AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq {f g : α → E}
 
 theorem Lp.ae_eq_zero_of_forall_set_integral_eq_zero (f : Lp E p μ) (hp_ne_zero : p ≠ 0)
     (hp_ne_top : p ≠ ∞) (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
-    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = 0) : f =ᵐ[μ] 0 :=
+    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 :=
   AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero hf_int_finite hf_zero
     (Lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).aefinStronglyMeasurable
 set_option linter.uppercaseLean3 false in
@@ -482,7 +482,7 @@ set_option linter.uppercaseLean3 false in
 theorem Lp.ae_eq_of_forall_set_integral_eq (f g : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
-    (hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = ∫ x in s, g x ∂μ) :
+    (hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) :
     f =ᵐ[μ] g :=
   AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq hf_int_finite hg_int_finite hfg
     (Lp.finStronglyMeasurable _ hp_ne_zero hp_ne_top).aefinStronglyMeasurable
@@ -492,7 +492,7 @@ set_option linter.uppercaseLean3 false in
 
 theorem ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim (hm : m ≤ m0) {f : α → E}
     (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
-    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → (∫ x in s, f x ∂μ) = 0)
+    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
     (hf : FinStronglyMeasurable f (μ.trim hm)) : f =ᵐ[μ] 0 := by
   obtain ⟨t, ht_meas, htf_zero, htμ⟩ := hf.exists_set_sigmaFinite
   haveI : SigmaFinite ((μ.restrict t).trim hm) := by rwa [restrict_trim hm μ ht_meas] at htμ
@@ -520,7 +520,7 @@ theorem ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim (hm :
 #align measure_theory.ae_eq_zero_of_forall_set_integral_eq_of_fin_strongly_measurable_trim MeasureTheory.ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim
 
 theorem Integrable.ae_eq_zero_of_forall_set_integral_eq_zero {f : α → E} (hf : Integrable f μ)
-    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = 0) : f =ᵐ[μ] 0 := by
+    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 := by
   have hf_Lp : Memℒp f 1 μ := memℒp_one_iff_integrable.mpr hf
   let f_Lp := hf_Lp.toLp f
   have hf_f_Lp : f =ᵐ[μ] f_Lp := (Memℒp.coeFn_toLp hf_Lp).symm
@@ -534,7 +534,7 @@ theorem Integrable.ae_eq_zero_of_forall_set_integral_eq_zero {f : α → E} (hf
 
 theorem Integrable.ae_eq_of_forall_set_integral_eq (f g : α → E) (hf : Integrable f μ)
     (hg : Integrable g μ)
-    (hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = ∫ x in s, g x ∂μ) :
+    (hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) :
     f =ᵐ[μ] g := by
   rw [← sub_ae_eq_zero]
   have hfg' : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, (f - g) x ∂μ) = 0 := by
@@ -550,7 +550,7 @@ section Lintegral
 
 theorem AeMeasurable.ae_eq_of_forall_set_lintegral_eq {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
     (hg : AEMeasurable g μ) (hfi : (∫⁻ x, f x ∂μ) ≠ ∞) (hgi : (∫⁻ x, g x ∂μ) ≠ ∞)
-    (hfg : ∀ ⦃s⦄, MeasurableSet s → μ s < ∞ → (∫⁻ x in s, f x ∂μ) = ∫⁻ x in s, g x ∂μ) :
+    (hfg : ∀ ⦃s⦄, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) :
     f =ᵐ[μ] g := by
   refine'
     ENNReal.eventuallyEq_of_toReal_eventuallyEq (ae_lt_top' hf hfi).ne_of_lt

@@ -205,11 +205,11 @@ theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g :
   have μs : ∀ n, μ (s n) = 0 := fun n => A _ _ _ (u_pos n)
   have B : {x | f x ≤ g x}ᶜ ⊆ ⋃ n, s n := by
     intro x hx
-    simp at hx
+    simp only [Set.mem_compl_iff, Set.mem_setOf, not_le] at hx
     have L1 : ∀ᶠ n in atTop, g x + u n ≤ f x := by
       have : Tendsto (fun n => g x + u n) atTop (𝓝 (g x + (0 : ℝ≥0))) :=
         tendsto_const_nhds.add (ENNReal.tendsto_coe.2 u_lim)
-      simp at this
+      simp only [ENNReal.coe_zero, add_zero] at this
       exact eventually_le_of_tendsto_lt hx this
     have L2 : ∀ᶠ n : ℕ in (atTop : Filter ℕ), g x ≤ (n : ℝ≥0) :=
       haveI : Tendsto (fun n : ℕ => ((n : ℝ≥0) : ℝ≥0∞)) atTop (𝓝 ∞) := by
@@ -330,8 +330,8 @@ theorem AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg {f : α
     (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
     (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f := by
   let t := hf.sigmaFiniteSet
-  suffices : 0 ≤ᵐ[μ.restrict t] f
-  exact ae_of_ae_restrict_of_ae_restrict_compl _ this hf.ae_eq_zero_compl.symm.le
+  suffices 0 ≤ᵐ[μ.restrict t] f from
+    ae_of_ae_restrict_of_ae_restrict_compl _ this hf.ae_eq_zero_compl.symm.le
   haveI : SigmaFinite (μ.restrict t) := hf.sigmaFinite_restrict
   refine'
     ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite (fun s hs hμts => _) fun s hs hμts => _
@@ -442,8 +442,8 @@ theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero {f : 
     (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) = 0)
     (hf : AEFinStronglyMeasurable f μ) : f =ᵐ[μ] 0 := by
   let t := hf.sigmaFiniteSet
-  suffices : f =ᵐ[μ.restrict t] 0
-  exact ae_of_ae_restrict_of_ae_restrict_compl _ this hf.ae_eq_zero_compl
+  suffices f =ᵐ[μ.restrict t] 0 from
+    ae_of_ae_restrict_of_ae_restrict_compl _ this hf.ae_eq_zero_compl
   haveI : SigmaFinite (μ.restrict t) := hf.sigmaFinite_restrict
   refine' ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite _ _
   · intro s hs hμs