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

@@ -69,10 +69,10 @@ theorem MeasureTheory.Memℒp.evariance_lt_top [IsFiniteMeasure μ] (hX : Memℒ
     evariance X μ < ∞ :=
   by
   have := ENNReal.pow_lt_top (hX.sub <| mem_ℒp_const <| μ[X]).2 2
-  rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, ← ENNReal.rpow_two] at this 
-  simp only [Pi.sub_apply, ENNReal.toReal_bit0, ENNReal.one_toReal, one_div] at this 
-  rw [← ENNReal.rpow_mul, inv_mul_cancel (two_ne_zero : (2 : ℝ) ≠ 0), ENNReal.rpow_one] at this 
-  simp_rw [ENNReal.rpow_two] at this 
+  rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, ← ENNReal.rpow_two] at this
+  simp only [Pi.sub_apply, ENNReal.toReal_bit0, ENNReal.one_toReal, one_div] at this
+  rw [← ENNReal.rpow_mul, inv_mul_cancel (two_ne_zero : (2 : ℝ) ≠ 0), ENNReal.rpow_one] at this
+  simp_rw [ENNReal.rpow_two] at this
   exact this
 #align measure_theory.mem_ℒp.evariance_lt_top MeasureTheory.Memℒp.evariance_lt_top
 -/
@@ -81,7 +81,7 @@ theorem MeasureTheory.Memℒp.evariance_lt_top [IsFiniteMeasure μ] (hX : Memℒ
 theorem evariance_eq_top [IsFiniteMeasure μ] (hXm : AEStronglyMeasurable X μ) (hX : ¬Memℒp X 2 μ) :
     evariance X μ = ∞ := by
   by_contra h
-  rw [← Ne.def, ← lt_top_iff_ne_top] at h 
+  rw [← Ne.def, ← lt_top_iff_ne_top] at h
   have : mem_ℒp (fun ω => X ω - μ[X]) 2 μ :=
     by
     refine' ⟨hXm.sub ae_strongly_measurable_const, _⟩
@@ -171,7 +171,7 @@ theorem evariance_eq_zero_iff (hX : AEMeasurable X μ) : evariance X μ = 0 ↔
   rw [evariance, lintegral_eq_zero_iff']
   constructor <;> intro hX <;> filter_upwards [hX] with ω hω
   · simp only [Pi.zero_apply, pow_eq_zero_iff, Nat.succ_pos', ENNReal.coe_eq_zero, nnnorm_eq_zero,
-      sub_eq_zero] at hω 
+      sub_eq_zero] at hω
     exact hω
   · rw [hω]
     simp
@@ -302,12 +302,12 @@ theorem evariance_def' [IsProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ
   · symm
     rw [evariance_eq_top hX hℒ, ENNReal.sub_eq_top_iff]
     refine' ⟨_, ENNReal.ofReal_ne_top⟩
-    rw [mem_ℒp, not_and] at hℒ 
+    rw [mem_ℒp, not_and] at hℒ
     specialize hℒ hX
     simp only [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, not_lt, top_le_iff,
       ENNReal.toReal_bit0, ENNReal.one_toReal, ENNReal.rpow_two, one_div, ENNReal.rpow_eq_top_iff,
       inv_lt_zero, inv_pos, zero_lt_bit0, zero_lt_one, and_true_iff, Classical.or_iff_not_imp_left,
-      not_and_or] at hℒ 
+      not_and_or] at hℒ
     exact hℒ fun _ => zero_le_two
 #align probability_theory.evariance_def' ProbabilityTheory.evariance_def'
 -/

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

@@ -378,7 +378,67 @@ variances. -/
 theorem IndepFun.variance_sum [IsProbabilityMeasure (ℙ : Measure Ω)] {ι : Type _} {X : ι → Ω → ℝ}
     {s : Finset ι} (hs : ∀ i ∈ s, Memℒp (X i) 2)
     (h : Set.Pairwise ↑s fun i j => IndepFun (X i) (X j)) :
-    Var[∑ i in s, X i] = ∑ i in s, Var[X i] := by classical
+    Var[∑ i in s, X i] = ∑ i in s, Var[X i] := by
+  classical
+  induction' s using Finset.induction_on with k s ks IH
+  · simp only [Finset.sum_empty, variance_zero]
+  rw [variance_def' (mem_ℒp_finset_sum' _ hs), sum_insert ks, sum_insert ks]
+  simp only [add_sq']
+  calc
+    𝔼[X k ^ 2 + (∑ i in s, X i) ^ 2 + 2 * X k * ∑ i in s, X i] - 𝔼[X k + ∑ i in s, X i] ^ 2 =
+        𝔼[X k ^ 2] + 𝔼[(∑ i in s, X i) ^ 2] + 𝔼[2 * X k * ∑ i in s, X i] -
+          (𝔼[X k] + 𝔼[∑ i in s, X i]) ^ 2 :=
+      by
+      rw [integral_add', integral_add', integral_add']
+      · exact mem_ℒp.integrable one_le_two (hs _ (mem_insert_self _ _))
+      · apply integrable_finset_sum' _ fun i hi => _
+        exact mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi))
+      · exact mem_ℒp.integrable_sq (hs _ (mem_insert_self _ _))
+      · apply mem_ℒp.integrable_sq
+        exact mem_ℒp_finset_sum' _ fun i hi => hs _ (mem_insert_of_mem hi)
+      · apply integrable.add
+        · exact mem_ℒp.integrable_sq (hs _ (mem_insert_self _ _))
+        · apply mem_ℒp.integrable_sq
+          exact mem_ℒp_finset_sum' _ fun i hi => hs _ (mem_insert_of_mem hi)
+      · rw [mul_assoc]
+        apply integrable.const_mul _ (2 : ℝ)
+        simp only [mul_sum, sum_apply, Pi.mul_apply]
+        apply integrable_finset_sum _ fun i hi => _
+        apply
+          indep_fun.integrable_mul _ (mem_ℒp.integrable one_le_two (hs _ (mem_insert_self _ _)))
+            (mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi)))
+        apply h (mem_insert_self _ _) (mem_insert_of_mem hi)
+        exact fun hki => ks (hki.symm ▸ hi)
+    _ =
+        Var[X k] + Var[∑ i in s, X i] +
+          (𝔼[2 * X k * ∑ i in s, X i] - 2 * 𝔼[X k] * 𝔼[∑ i in s, X i]) :=
+      by
+      rw [variance_def' (hs _ (mem_insert_self _ _)),
+        variance_def' (mem_ℒp_finset_sum' _ fun i hi => hs _ (mem_insert_of_mem hi))]
+      ring
+    _ = Var[X k] + Var[∑ i in s, X i] :=
+      by
+      simp only [mul_assoc, integral_mul_left, Pi.mul_apply, Pi.bit0_apply, Pi.one_apply, sum_apply,
+        add_right_eq_self, mul_sum]
+      rw [integral_finset_sum s fun i hi => _]; swap
+      · apply integrable.const_mul _ (2 : ℝ)
+        apply
+          indep_fun.integrable_mul _ (mem_ℒp.integrable one_le_two (hs _ (mem_insert_self _ _)))
+            (mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi)))
+        apply h (mem_insert_self _ _) (mem_insert_of_mem hi)
+        exact fun hki => ks (hki.symm ▸ hi)
+      rw [integral_finset_sum s fun i hi =>
+          mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi)),
+        mul_sum, mul_sum, ← sum_sub_distrib]
+      apply Finset.sum_eq_zero fun i hi => _
+      rw [integral_mul_left, indep_fun.integral_mul', sub_self]
+      · apply h (mem_insert_self _ _) (mem_insert_of_mem hi)
+        exact fun hki => ks (hki.symm ▸ hi)
+      · exact mem_ℒp.ae_strongly_measurable (hs _ (mem_insert_self _ _))
+      · exact mem_ℒp.ae_strongly_measurable (hs _ (mem_insert_of_mem hi))
+    _ = Var[X k] + ∑ i in s, Var[X i] := by
+      rw [IH (fun i hi => hs i (mem_insert_of_mem hi))
+          (h.mono (by simp only [coe_insert, Set.subset_insert]))]
 #align probability_theory.indep_fun.variance_sum ProbabilityTheory.IndepFun.variance_sum
 -/
 

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

@@ -378,67 +378,7 @@ variances. -/
 theorem IndepFun.variance_sum [IsProbabilityMeasure (ℙ : Measure Ω)] {ι : Type _} {X : ι → Ω → ℝ}
     {s : Finset ι} (hs : ∀ i ∈ s, Memℒp (X i) 2)
     (h : Set.Pairwise ↑s fun i j => IndepFun (X i) (X j)) :
-    Var[∑ i in s, X i] = ∑ i in s, Var[X i] := by
-  classical
-  induction' s using Finset.induction_on with k s ks IH
-  · simp only [Finset.sum_empty, variance_zero]
-  rw [variance_def' (mem_ℒp_finset_sum' _ hs), sum_insert ks, sum_insert ks]
-  simp only [add_sq']
-  calc
-    𝔼[X k ^ 2 + (∑ i in s, X i) ^ 2 + 2 * X k * ∑ i in s, X i] - 𝔼[X k + ∑ i in s, X i] ^ 2 =
-        𝔼[X k ^ 2] + 𝔼[(∑ i in s, X i) ^ 2] + 𝔼[2 * X k * ∑ i in s, X i] -
-          (𝔼[X k] + 𝔼[∑ i in s, X i]) ^ 2 :=
-      by
-      rw [integral_add', integral_add', integral_add']
-      · exact mem_ℒp.integrable one_le_two (hs _ (mem_insert_self _ _))
-      · apply integrable_finset_sum' _ fun i hi => _
-        exact mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi))
-      · exact mem_ℒp.integrable_sq (hs _ (mem_insert_self _ _))
-      · apply mem_ℒp.integrable_sq
-        exact mem_ℒp_finset_sum' _ fun i hi => hs _ (mem_insert_of_mem hi)
-      · apply integrable.add
-        · exact mem_ℒp.integrable_sq (hs _ (mem_insert_self _ _))
-        · apply mem_ℒp.integrable_sq
-          exact mem_ℒp_finset_sum' _ fun i hi => hs _ (mem_insert_of_mem hi)
-      · rw [mul_assoc]
-        apply integrable.const_mul _ (2 : ℝ)
-        simp only [mul_sum, sum_apply, Pi.mul_apply]
-        apply integrable_finset_sum _ fun i hi => _
-        apply
-          indep_fun.integrable_mul _ (mem_ℒp.integrable one_le_two (hs _ (mem_insert_self _ _)))
-            (mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi)))
-        apply h (mem_insert_self _ _) (mem_insert_of_mem hi)
-        exact fun hki => ks (hki.symm ▸ hi)
-    _ =
-        Var[X k] + Var[∑ i in s, X i] +
-          (𝔼[2 * X k * ∑ i in s, X i] - 2 * 𝔼[X k] * 𝔼[∑ i in s, X i]) :=
-      by
-      rw [variance_def' (hs _ (mem_insert_self _ _)),
-        variance_def' (mem_ℒp_finset_sum' _ fun i hi => hs _ (mem_insert_of_mem hi))]
-      ring
-    _ = Var[X k] + Var[∑ i in s, X i] :=
-      by
-      simp only [mul_assoc, integral_mul_left, Pi.mul_apply, Pi.bit0_apply, Pi.one_apply, sum_apply,
-        add_right_eq_self, mul_sum]
-      rw [integral_finset_sum s fun i hi => _]; swap
-      · apply integrable.const_mul _ (2 : ℝ)
-        apply
-          indep_fun.integrable_mul _ (mem_ℒp.integrable one_le_two (hs _ (mem_insert_self _ _)))
-            (mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi)))
-        apply h (mem_insert_self _ _) (mem_insert_of_mem hi)
-        exact fun hki => ks (hki.symm ▸ hi)
-      rw [integral_finset_sum s fun i hi =>
-          mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi)),
-        mul_sum, mul_sum, ← sum_sub_distrib]
-      apply Finset.sum_eq_zero fun i hi => _
-      rw [integral_mul_left, indep_fun.integral_mul', sub_self]
-      · apply h (mem_insert_self _ _) (mem_insert_of_mem hi)
-        exact fun hki => ks (hki.symm ▸ hi)
-      · exact mem_ℒp.ae_strongly_measurable (hs _ (mem_insert_self _ _))
-      · exact mem_ℒp.ae_strongly_measurable (hs _ (mem_insert_of_mem hi))
-    _ = Var[X k] + ∑ i in s, Var[X i] := by
-      rw [IH (fun i hi => hs i (mem_insert_of_mem hi))
-          (h.mono (by simp only [coe_insert, Set.subset_insert]))]
+    Var[∑ i in s, X i] = ∑ i in s, Var[X i] := by classical
 #align probability_theory.indep_fun.variance_sum ProbabilityTheory.IndepFun.variance_sum
 -/
 

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

@@ -306,7 +306,7 @@ theorem evariance_def' [IsProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ
     specialize hℒ hX
     simp only [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, not_lt, top_le_iff,
       ENNReal.toReal_bit0, ENNReal.one_toReal, ENNReal.rpow_two, one_div, ENNReal.rpow_eq_top_iff,
-      inv_lt_zero, inv_pos, zero_lt_bit0, zero_lt_one, and_true_iff, or_iff_not_imp_left,
+      inv_lt_zero, inv_pos, zero_lt_bit0, zero_lt_one, and_true_iff, Classical.or_iff_not_imp_left,
       not_and_or] at hℒ 
     exact hℒ fun _ => zero_le_two
 #align probability_theory.evariance_def' ProbabilityTheory.evariance_def'

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

@@ -3,9 +3,9 @@ Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Kexing Ying
 -/
-import Mathbin.Probability.Notation
-import Mathbin.Probability.Integration
-import Mathbin.MeasureTheory.Function.L2Space
+import Probability.Notation
+import Probability.Integration
+import MeasureTheory.Function.L2Space
 
 #align_import probability.variance from "leanprover-community/mathlib"@"1b089e3bdc3ce6b39cd472543474a0a137128c6c"
 

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

@@ -2,16 +2,13 @@
 Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Kexing Ying
-
-! This file was ported from Lean 3 source module probability.variance
-! leanprover-community/mathlib commit 1b089e3bdc3ce6b39cd472543474a0a137128c6c
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Probability.Notation
 import Mathbin.Probability.Integration
 import Mathbin.MeasureTheory.Function.L2Space
 
+#align_import probability.variance from "leanprover-community/mathlib"@"1b089e3bdc3ce6b39cd472543474a0a137128c6c"
+
 /-!
 # Variance of random variables
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/d30d31261cdb4d2f5e612eabc3c4bf45556350d5

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Kexing Ying
 
 ! This file was ported from Lean 3 source module probability.variance
-! leanprover-community/mathlib commit f0c8bf9245297a541f468be517f1bde6195105e9
+! leanprover-community/mathlib commit 1b089e3bdc3ce6b39cd472543474a0a137128c6c
 ! 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.Function.L2Space
 /-!
 # Variance of random variables
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 We define the variance of a real-valued random variable as `Var[X] = 𝔼[(X - 𝔼[X])^2]` (in the
 `probability_theory` locale).
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/f2ad3645af9effcdb587637dc28a6074edc813f9

@@ -46,20 +46,25 @@ open scoped BigOperators MeasureTheory ProbabilityTheory ENNReal NNReal
 
 namespace ProbabilityTheory
 
+#print ProbabilityTheory.evariance /-
 /-- The `ℝ≥0∞`-valued variance of a real-valued random variable defined as the Lebesgue integral of
 `(X - 𝔼[X])^2`. -/
 def evariance {Ω : Type _} {m : MeasurableSpace Ω} (X : Ω → ℝ) (μ : Measure Ω) : ℝ≥0∞ :=
   ∫⁻ ω, ‖X ω - μ[X]‖₊ ^ 2 ∂μ
 #align probability_theory.evariance ProbabilityTheory.evariance
+-/
 
+#print ProbabilityTheory.variance /-
 /-- The `ℝ`-valued variance of a real-valued random variable defined by applying `ennreal.to_real`
 to `evariance`. -/
 def variance {Ω : Type _} {m : MeasurableSpace Ω} (X : Ω → ℝ) (μ : Measure Ω) : ℝ :=
   (evariance X μ).toReal
 #align probability_theory.variance ProbabilityTheory.variance
+-/
 
 variable {Ω : Type _} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : Measure Ω}
 
+#print MeasureTheory.Memℒp.evariance_lt_top /-
 theorem MeasureTheory.Memℒp.evariance_lt_top [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
     evariance X μ < ∞ :=
   by
@@ -70,7 +75,9 @@ theorem MeasureTheory.Memℒp.evariance_lt_top [IsFiniteMeasure μ] (hX : Memℒ
   simp_rw [ENNReal.rpow_two] at this 
   exact this
 #align measure_theory.mem_ℒp.evariance_lt_top MeasureTheory.Memℒp.evariance_lt_top
+-/
 
+#print ProbabilityTheory.evariance_eq_top /-
 theorem evariance_eq_top [IsFiniteMeasure μ] (hXm : AEStronglyMeasurable X μ) (hX : ¬Memℒp X 2 μ) :
     evariance X μ = ∞ := by
   by_contra h
@@ -86,7 +93,9 @@ theorem evariance_eq_top [IsFiniteMeasure μ] (hXm : AEStronglyMeasurable X μ)
   ext ω
   rw [Pi.add_apply, sub_add_cancel]
 #align probability_theory.evariance_eq_top ProbabilityTheory.evariance_eq_top
+-/
 
+#print ProbabilityTheory.evariance_lt_top_iff_memℒp /-
 theorem evariance_lt_top_iff_memℒp [IsFiniteMeasure μ] (hX : AEStronglyMeasurable X μ) :
     evariance X μ < ∞ ↔ Memℒp X 2 μ :=
   by
@@ -95,12 +104,16 @@ theorem evariance_lt_top_iff_memℒp [IsFiniteMeasure μ] (hX : AEStronglyMeasur
   rw [not_lt, top_le_iff]
   exact evariance_eq_top hX
 #align probability_theory.evariance_lt_top_iff_mem_ℒp ProbabilityTheory.evariance_lt_top_iff_memℒp
+-/
 
+#print MeasureTheory.Memℒp.ofReal_variance_eq /-
 theorem MeasureTheory.Memℒp.ofReal_variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
     ENNReal.ofReal (variance X μ) = evariance X μ := by rw [variance, ENNReal.ofReal_toReal];
   exact hX.evariance_lt_top.ne
 #align measure_theory.mem_ℒp.of_real_variance_eq MeasureTheory.Memℒp.ofReal_variance_eq
+-/
 
+#print ProbabilityTheory.evariance_eq_lintegral_ofReal /-
 theorem evariance_eq_lintegral_ofReal (X : Ω → ℝ) (μ : Measure Ω) :
     evariance X μ = ∫⁻ ω, ENNReal.ofReal ((X ω - μ[X]) ^ 2) ∂μ :=
   by
@@ -111,7 +124,9 @@ theorem evariance_eq_lintegral_ofReal (X : Ω → ℝ) (μ : Measure Ω) :
   congr
   exact (Real.toNNReal_eq_nnnorm_of_nonneg <| sq_nonneg _).symm
 #align probability_theory.evariance_eq_lintegral_of_real ProbabilityTheory.evariance_eq_lintegral_ofReal
+-/
 
+#print MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero /-
 theorem MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero (hX : Memℒp X 2 μ) (hXint : μ[X] = 0) :
     variance X μ = μ[X ^ 2] :=
   by
@@ -126,7 +141,9 @@ theorem MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero (hX : Memℒp X 2
       Real.rpow_two, pow_bit0_abs]
   · exact ae_of_all _ fun ω => pow_two_nonneg _
 #align measure_theory.mem_ℒp.variance_eq_of_integral_eq_zero MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero
+-/
 
+#print MeasureTheory.Memℒp.variance_eq /-
 theorem MeasureTheory.Memℒp.variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
     variance X μ = μ[(X - fun ω => μ[X]) ^ 2] :=
   by
@@ -140,11 +157,15 @@ theorem MeasureTheory.Memℒp.variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2
       Real.rpow_two, pow_bit0_abs]
   · exact ae_of_all _ fun ω => pow_two_nonneg _
 #align measure_theory.mem_ℒp.variance_eq MeasureTheory.Memℒp.variance_eq
+-/
 
+#print ProbabilityTheory.evariance_zero /-
 @[simp]
 theorem evariance_zero : evariance 0 μ = 0 := by simp [evariance]
 #align probability_theory.evariance_zero ProbabilityTheory.evariance_zero
+-/
 
+#print ProbabilityTheory.evariance_eq_zero_iff /-
 theorem evariance_eq_zero_iff (hX : AEMeasurable X μ) : evariance X μ = 0 ↔ X =ᵐ[μ] fun ω => μ[X] :=
   by
   rw [evariance, lintegral_eq_zero_iff']
@@ -156,7 +177,9 @@ theorem evariance_eq_zero_iff (hX : AEMeasurable X μ) : evariance X μ = 0 ↔
     simp
   · measurability
 #align probability_theory.evariance_eq_zero_iff ProbabilityTheory.evariance_eq_zero_iff
+-/
 
+#print ProbabilityTheory.evariance_mul /-
 theorem evariance_mul (c : ℝ) (X : Ω → ℝ) (μ : Measure Ω) :
     evariance (fun ω => c * X ω) μ = ENNReal.ofReal (c ^ 2) * evariance X μ :=
   by
@@ -172,30 +195,40 @@ theorem evariance_mul (c : ℝ) (X : Ω → ℝ) (μ : Measure Ω) :
   rw [mul_comm]
   simp_rw [← smul_eq_mul, ← integral_smul_const, smul_eq_mul, mul_comm]
 #align probability_theory.evariance_mul ProbabilityTheory.evariance_mul
+-/
 
 scoped notation "eVar[" X "]" => ProbabilityTheory.evariance X MeasureTheory.MeasureSpace.volume
 
+#print ProbabilityTheory.variance_zero /-
 @[simp]
 theorem variance_zero (μ : Measure Ω) : variance 0 μ = 0 := by
   simp only [variance, evariance_zero, ENNReal.zero_toReal]
 #align probability_theory.variance_zero ProbabilityTheory.variance_zero
+-/
 
+#print ProbabilityTheory.variance_nonneg /-
 theorem variance_nonneg (X : Ω → ℝ) (μ : Measure Ω) : 0 ≤ variance X μ :=
   ENNReal.toReal_nonneg
 #align probability_theory.variance_nonneg ProbabilityTheory.variance_nonneg
+-/
 
+#print ProbabilityTheory.variance_mul /-
 theorem variance_mul (c : ℝ) (X : Ω → ℝ) (μ : Measure Ω) :
     variance (fun ω => c * X ω) μ = c ^ 2 * variance X μ :=
   by
   rw [variance, evariance_mul, ENNReal.toReal_mul, ENNReal.toReal_ofReal (sq_nonneg _)]
   rfl
 #align probability_theory.variance_mul ProbabilityTheory.variance_mul
+-/
 
+#print ProbabilityTheory.variance_smul /-
 theorem variance_smul (c : ℝ) (X : Ω → ℝ) (μ : Measure Ω) :
     variance (c • X) μ = c ^ 2 * variance X μ :=
   variance_mul c X μ
 #align probability_theory.variance_smul ProbabilityTheory.variance_smul
+-/
 
+#print ProbabilityTheory.variance_smul' /-
 theorem variance_smul' {A : Type _} [CommSemiring A] [Algebra A ℝ] (c : A) (X : Ω → ℝ)
     (μ : Measure Ω) : variance (c • X) μ = c ^ 2 • variance X μ :=
   by
@@ -203,11 +236,13 @@ theorem variance_smul' {A : Type _} [CommSemiring A] [Algebra A ℝ] (c : A) (X
   · ext1 x; simp only [algebraMap_smul]
   · simp only [Algebra.smul_def, map_pow]
 #align probability_theory.variance_smul' ProbabilityTheory.variance_smul'
+-/
 
 scoped notation "Var[" X "]" => ProbabilityTheory.variance X MeasureTheory.MeasureSpace.volume
 
 variable [MeasureSpace Ω]
 
+#print ProbabilityTheory.variance_def' /-
 theorem variance_def' [IsProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ} (hX : Memℒp X 2) :
     Var[X] = 𝔼[X ^ 2] - 𝔼[X] ^ 2 :=
   by
@@ -224,7 +259,9 @@ theorem variance_def' [IsProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}
     measure_univ, ENNReal.one_toReal, Algebra.id.smul_eq_mul]
   ring
 #align probability_theory.variance_def' ProbabilityTheory.variance_def'
+-/
 
+#print ProbabilityTheory.variance_le_expectation_sq /-
 theorem variance_le_expectation_sq [IsProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}
     (hm : AEStronglyMeasurable X ℙ) : Var[X] ≤ 𝔼[X ^ 2] :=
   by
@@ -248,7 +285,9 @@ theorem variance_le_expectation_sq [IsProbabilityMeasure (ℙ : Measure Ω)] {X
   · exact ae_of_all _ fun x => sq_nonneg _
   · exact (AEMeasurable.pow_const (hm.ae_measurable.sub_const _) _).AEStronglyMeasurable
 #align probability_theory.variance_le_expectation_sq ProbabilityTheory.variance_le_expectation_sq
+-/
 
+#print ProbabilityTheory.evariance_def' /-
 theorem evariance_def' [IsProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}
     (hX : AEStronglyMeasurable X ℙ) : eVar[X] = (∫⁻ ω, ‖X ω‖₊ ^ 2) - ENNReal.ofReal (𝔼[X] ^ 2) :=
   by
@@ -271,7 +310,9 @@ theorem evariance_def' [IsProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ
       not_and_or] at hℒ 
     exact hℒ fun _ => zero_le_two
 #align probability_theory.evariance_def' ProbabilityTheory.evariance_def'
+-/
 
+#print ProbabilityTheory.meas_ge_le_evariance_div_sq /-
 /-- *Chebyshev's inequality* for `ℝ≥0∞`-valued variance. -/
 theorem meas_ge_le_evariance_div_sq {X : Ω → ℝ} (hX : AEStronglyMeasurable X ℙ) {c : ℝ≥0}
     (hc : c ≠ 0) : ℙ {ω | ↑c ≤ |X ω - 𝔼[X]|} ≤ eVar[X] / c ^ 2 :=
@@ -289,7 +330,9 @@ theorem meas_ge_le_evariance_div_sq {X : Ω → ℝ} (hX : AEStronglyMeasurable
     simp_rw [← ENNReal.rpow_mul, inv_mul_cancel (two_ne_zero : (2 : ℝ) ≠ 0), ENNReal.rpow_two,
       ENNReal.rpow_one, evariance]
 #align probability_theory.meas_ge_le_evariance_div_sq ProbabilityTheory.meas_ge_le_evariance_div_sq
+-/
 
+#print ProbabilityTheory.meas_ge_le_variance_div_sq /-
 /-- *Chebyshev's inequality* : one can control the deviation probability of a real random variable
 from its expectation in terms of the variance. -/
 theorem meas_ge_le_variance_div_sq [IsFiniteMeasure (ℙ : Measure Ω)] {X : Ω → ℝ} (hX : Memℒp X 2)
@@ -301,7 +344,9 @@ theorem meas_ge_le_variance_div_sq [IsFiniteMeasure (ℙ : Measure Ω)] {X : Ω
   · rw [ENNReal.ofReal_pow hc.le]
     rfl
 #align probability_theory.meas_ge_le_variance_div_sq ProbabilityTheory.meas_ge_le_variance_div_sq
+-/
 
+#print ProbabilityTheory.IndepFun.variance_add /-
 /-- The variance of the sum of two independent random variables is the sum of the variances. -/
 theorem IndepFun.variance_add [IsProbabilityMeasure (ℙ : Measure Ω)] {X Y : Ω → ℝ} (hX : Memℒp X 2)
     (hY : Memℒp Y 2) (h : IndepFun X Y) : Var[X + Y] = Var[X] + Var[Y] :=
@@ -325,7 +370,9 @@ theorem IndepFun.variance_add [IsProbabilityMeasure (ℙ : Measure Ω)] {X Y : 
       exact h.integral_mul_of_integrable (hX.integrable one_le_two) (hY.integrable one_le_two)
     _ = Var[X] + Var[Y] := by simp only [variance_def', hX, hY, Pi.pow_apply]; ring
 #align probability_theory.indep_fun.variance_add ProbabilityTheory.IndepFun.variance_add
+-/
 
+#print ProbabilityTheory.IndepFun.variance_sum /-
 /-- The variance of a finite sum of pairwise independent random variables is the sum of the
 variances. -/
 theorem IndepFun.variance_sum [IsProbabilityMeasure (ℙ : Measure Ω)] {ι : Type _} {X : ι → Ω → ℝ}
@@ -393,6 +440,7 @@ theorem IndepFun.variance_sum [IsProbabilityMeasure (ℙ : Measure Ω)] {ι : Ty
       rw [IH (fun i hi => hs i (mem_insert_of_mem hi))
           (h.mono (by simp only [coe_insert, Set.subset_insert]))]
 #align probability_theory.indep_fun.variance_sum ProbabilityTheory.IndepFun.variance_sum
+-/
 
 end ProbabilityTheory
 

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

@@ -101,8 +101,6 @@ theorem MeasureTheory.Memℒp.ofReal_variance_eq [IsFiniteMeasure μ] (hX : Mem
   exact hX.evariance_lt_top.ne
 #align measure_theory.mem_ℒp.of_real_variance_eq MeasureTheory.Memℒp.ofReal_variance_eq
 
-include m
-
 theorem evariance_eq_lintegral_ofReal (X : Ω → ℝ) (μ : Measure Ω) :
     evariance X μ = ∫⁻ ω, ENNReal.ofReal ((X ω - μ[X]) ^ 2) ∂μ :=
   by
@@ -175,7 +173,6 @@ theorem evariance_mul (c : ℝ) (X : Ω → ℝ) (μ : Measure Ω) :
   simp_rw [← smul_eq_mul, ← integral_smul_const, smul_eq_mul, mul_comm]
 #align probability_theory.evariance_mul ProbabilityTheory.evariance_mul
 
--- mathport name: probability_theory.evariance
 scoped notation "eVar[" X "]" => ProbabilityTheory.evariance X MeasureTheory.MeasureSpace.volume
 
 @[simp]
@@ -207,11 +204,8 @@ theorem variance_smul' {A : Type _} [CommSemiring A] [Algebra A ℝ] (c : A) (X
   · simp only [Algebra.smul_def, map_pow]
 #align probability_theory.variance_smul' ProbabilityTheory.variance_smul'
 
--- mathport name: probability_theory.variance
 scoped notation "Var[" X "]" => ProbabilityTheory.variance X MeasureTheory.MeasureSpace.volume
 
-omit m
-
 variable [MeasureSpace Ω]
 
 theorem variance_def' [IsProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ} (hX : Memℒp X 2) :

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

@@ -330,7 +330,6 @@ theorem IndepFun.variance_add [IsProbabilityMeasure (ℙ : Measure Ω)] {X Y : 
       congr
       exact h.integral_mul_of_integrable (hX.integrable one_le_two) (hY.integrable one_le_two)
     _ = Var[X] + Var[Y] := by simp only [variance_def', hX, hY, Pi.pow_apply]; ring
-    
 #align probability_theory.indep_fun.variance_add ProbabilityTheory.IndepFun.variance_add
 
 /-- The variance of a finite sum of pairwise independent random variables is the sum of the
@@ -399,7 +398,6 @@ theorem IndepFun.variance_sum [IsProbabilityMeasure (ℙ : Measure Ω)] {ι : Ty
     _ = Var[X k] + ∑ i in s, Var[X i] := by
       rw [IH (fun i hi => hs i (mem_insert_of_mem hi))
           (h.mono (by simp only [coe_insert, Set.subset_insert]))]
-    
 #align probability_theory.indep_fun.variance_sum ProbabilityTheory.IndepFun.variance_sum
 
 end ProbabilityTheory

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

@@ -60,7 +60,7 @@ def variance {Ω : Type _} {m : MeasurableSpace Ω} (X : Ω → ℝ) (μ : Measu
 
 variable {Ω : Type _} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : Measure Ω}
 
-theorem MeasureTheory.Memℒp.evariance_lt_top [FiniteMeasure μ] (hX : Memℒp X 2 μ) :
+theorem MeasureTheory.Memℒp.evariance_lt_top [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
     evariance X μ < ∞ :=
   by
   have := ENNReal.pow_lt_top (hX.sub <| mem_ℒp_const <| μ[X]).2 2
@@ -71,7 +71,7 @@ theorem MeasureTheory.Memℒp.evariance_lt_top [FiniteMeasure μ] (hX : Memℒp
   exact this
 #align measure_theory.mem_ℒp.evariance_lt_top MeasureTheory.Memℒp.evariance_lt_top
 
-theorem evariance_eq_top [FiniteMeasure μ] (hXm : AEStronglyMeasurable X μ) (hX : ¬Memℒp X 2 μ) :
+theorem evariance_eq_top [IsFiniteMeasure μ] (hXm : AEStronglyMeasurable X μ) (hX : ¬Memℒp X 2 μ) :
     evariance X μ = ∞ := by
   by_contra h
   rw [← Ne.def, ← lt_top_iff_ne_top] at h 
@@ -87,7 +87,7 @@ theorem evariance_eq_top [FiniteMeasure μ] (hXm : AEStronglyMeasurable X μ) (h
   rw [Pi.add_apply, sub_add_cancel]
 #align probability_theory.evariance_eq_top ProbabilityTheory.evariance_eq_top
 
-theorem evariance_lt_top_iff_memℒp [FiniteMeasure μ] (hX : AEStronglyMeasurable X μ) :
+theorem evariance_lt_top_iff_memℒp [IsFiniteMeasure μ] (hX : AEStronglyMeasurable X μ) :
     evariance X μ < ∞ ↔ Memℒp X 2 μ :=
   by
   refine' ⟨_, MeasureTheory.Memℒp.evariance_lt_top⟩
@@ -96,7 +96,7 @@ theorem evariance_lt_top_iff_memℒp [FiniteMeasure μ] (hX : AEStronglyMeasurab
   exact evariance_eq_top hX
 #align probability_theory.evariance_lt_top_iff_mem_ℒp ProbabilityTheory.evariance_lt_top_iff_memℒp
 
-theorem MeasureTheory.Memℒp.ofReal_variance_eq [FiniteMeasure μ] (hX : Memℒp X 2 μ) :
+theorem MeasureTheory.Memℒp.ofReal_variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
     ENNReal.ofReal (variance X μ) = evariance X μ := by rw [variance, ENNReal.ofReal_toReal];
   exact hX.evariance_lt_top.ne
 #align measure_theory.mem_ℒp.of_real_variance_eq MeasureTheory.Memℒp.ofReal_variance_eq
@@ -129,14 +129,14 @@ theorem MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero (hX : Memℒp X 2
   · exact ae_of_all _ fun ω => pow_two_nonneg _
 #align measure_theory.mem_ℒp.variance_eq_of_integral_eq_zero MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero
 
-theorem MeasureTheory.Memℒp.variance_eq [FiniteMeasure μ] (hX : Memℒp X 2 μ) :
+theorem MeasureTheory.Memℒp.variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
     variance X μ = μ[(X - fun ω => μ[X]) ^ 2] :=
   by
   rw [variance, evariance_eq_lintegral_of_real, ← of_real_integral_eq_lintegral_of_real,
     ENNReal.toReal_ofReal]
   · rfl
   · exact integral_nonneg fun ω => pow_two_nonneg _
-  · convert(hX.sub <| mem_ℒp_const (μ[X])).integrable_norm_rpow two_ne_zero ENNReal.two_ne_top
+  · convert (hX.sub <| mem_ℒp_const (μ[X])).integrable_norm_rpow two_ne_zero ENNReal.two_ne_top
     ext ω
     simp only [Pi.sub_apply, Real.norm_eq_abs, ENNReal.toReal_bit0, ENNReal.one_toReal,
       Real.rpow_two, pow_bit0_abs]
@@ -150,7 +150,7 @@ theorem evariance_zero : evariance 0 μ = 0 := by simp [evariance]
 theorem evariance_eq_zero_iff (hX : AEMeasurable X μ) : evariance X μ = 0 ↔ X =ᵐ[μ] fun ω => μ[X] :=
   by
   rw [evariance, lintegral_eq_zero_iff']
-  constructor <;> intro hX <;> filter_upwards [hX]with ω hω
+  constructor <;> intro hX <;> filter_upwards [hX] with ω hω
   · simp only [Pi.zero_apply, pow_eq_zero_iff, Nat.succ_pos', ENNReal.coe_eq_zero, nnnorm_eq_zero,
       sub_eq_zero] at hω 
     exact hω
@@ -214,7 +214,7 @@ omit m
 
 variable [MeasureSpace Ω]
 
-theorem variance_def' [ProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ} (hX : Memℒp X 2) :
+theorem variance_def' [IsProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ} (hX : Memℒp X 2) :
     Var[X] = 𝔼[X ^ 2] - 𝔼[X] ^ 2 :=
   by
   rw [hX.variance_eq, sub_sq', integral_sub', integral_add']; rotate_left
@@ -231,7 +231,7 @@ theorem variance_def' [ProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ} (
   ring
 #align probability_theory.variance_def' ProbabilityTheory.variance_def'
 
-theorem variance_le_expectation_sq [ProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}
+theorem variance_le_expectation_sq [IsProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}
     (hm : AEStronglyMeasurable X ℙ) : Var[X] ≤ 𝔼[X ^ 2] :=
   by
   by_cases hX : mem_ℒp X 2
@@ -255,7 +255,7 @@ theorem variance_le_expectation_sq [ProbabilityMeasure (ℙ : Measure Ω)] {X :
   · exact (AEMeasurable.pow_const (hm.ae_measurable.sub_const _) _).AEStronglyMeasurable
 #align probability_theory.variance_le_expectation_sq ProbabilityTheory.variance_le_expectation_sq
 
-theorem evariance_def' [ProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}
+theorem evariance_def' [IsProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}
     (hX : AEStronglyMeasurable X ℙ) : eVar[X] = (∫⁻ ω, ‖X ω‖₊ ^ 2) - ENNReal.ofReal (𝔼[X] ^ 2) :=
   by
   by_cases hℒ : mem_ℒp X 2
@@ -280,7 +280,7 @@ theorem evariance_def' [ProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}
 
 /-- *Chebyshev's inequality* for `ℝ≥0∞`-valued variance. -/
 theorem meas_ge_le_evariance_div_sq {X : Ω → ℝ} (hX : AEStronglyMeasurable X ℙ) {c : ℝ≥0}
-    (hc : c ≠ 0) : ℙ { ω | ↑c ≤ |X ω - 𝔼[X]| } ≤ eVar[X] / c ^ 2 :=
+    (hc : c ≠ 0) : ℙ {ω | ↑c ≤ |X ω - 𝔼[X]|} ≤ eVar[X] / c ^ 2 :=
   by
   have A : (c : ℝ≥0∞) ≠ 0 := by rwa [Ne.def, ENNReal.coe_eq_zero]
   have B : ae_strongly_measurable (fun ω : Ω => 𝔼[X]) ℙ := ae_strongly_measurable_const
@@ -298,19 +298,19 @@ theorem meas_ge_le_evariance_div_sq {X : Ω → ℝ} (hX : AEStronglyMeasurable
 
 /-- *Chebyshev's inequality* : one can control the deviation probability of a real random variable
 from its expectation in terms of the variance. -/
-theorem meas_ge_le_variance_div_sq [FiniteMeasure (ℙ : Measure Ω)] {X : Ω → ℝ} (hX : Memℒp X 2)
-    {c : ℝ} (hc : 0 < c) : ℙ { ω | c ≤ |X ω - 𝔼[X]| } ≤ ENNReal.ofReal (Var[X] / c ^ 2) :=
+theorem meas_ge_le_variance_div_sq [IsFiniteMeasure (ℙ : Measure Ω)] {X : Ω → ℝ} (hX : Memℒp X 2)
+    {c : ℝ} (hc : 0 < c) : ℙ {ω | c ≤ |X ω - 𝔼[X]|} ≤ ENNReal.ofReal (Var[X] / c ^ 2) :=
   by
   rw [ENNReal.ofReal_div_of_pos (sq_pos_of_ne_zero _ hc.ne.symm), hX.of_real_variance_eq]
-  convert@meas_ge_le_evariance_div_sq _ _ _ hX.1 c.to_nnreal (by simp [hc])
+  convert @meas_ge_le_evariance_div_sq _ _ _ hX.1 c.to_nnreal (by simp [hc])
   · simp only [Real.coe_toNNReal', max_le_iff, abs_nonneg, and_true_iff]
   · rw [ENNReal.ofReal_pow hc.le]
     rfl
 #align probability_theory.meas_ge_le_variance_div_sq ProbabilityTheory.meas_ge_le_variance_div_sq
 
 /-- The variance of the sum of two independent random variables is the sum of the variances. -/
-theorem IndepFunCat.variance_add [ProbabilityMeasure (ℙ : Measure Ω)] {X Y : Ω → ℝ} (hX : Memℒp X 2)
-    (hY : Memℒp Y 2) (h : IndepFunCat X Y) : Var[X + Y] = Var[X] + Var[Y] :=
+theorem IndepFun.variance_add [IsProbabilityMeasure (ℙ : Measure Ω)] {X Y : Ω → ℝ} (hX : Memℒp X 2)
+    (hY : Memℒp Y 2) (h : IndepFun X Y) : Var[X + Y] = Var[X] + Var[Y] :=
   calc
     Var[X + Y] = 𝔼[fun a => X a ^ 2 + Y a ^ 2 + 2 * X a * Y a] - 𝔼[X + Y] ^ 2 := by
       simp [variance_def' (hX.add hY), add_sq']
@@ -331,76 +331,76 @@ theorem IndepFunCat.variance_add [ProbabilityMeasure (ℙ : Measure Ω)] {X Y :
       exact h.integral_mul_of_integrable (hX.integrable one_le_two) (hY.integrable one_le_two)
     _ = Var[X] + Var[Y] := by simp only [variance_def', hX, hY, Pi.pow_apply]; ring
     
-#align probability_theory.indep_fun.variance_add ProbabilityTheory.IndepFunCat.variance_add
+#align probability_theory.indep_fun.variance_add ProbabilityTheory.IndepFun.variance_add
 
 /-- The variance of a finite sum of pairwise independent random variables is the sum of the
 variances. -/
-theorem IndepFunCat.variance_sum [ProbabilityMeasure (ℙ : Measure Ω)] {ι : Type _} {X : ι → Ω → ℝ}
+theorem IndepFun.variance_sum [IsProbabilityMeasure (ℙ : Measure Ω)] {ι : Type _} {X : ι → Ω → ℝ}
     {s : Finset ι} (hs : ∀ i ∈ s, Memℒp (X i) 2)
-    (h : Set.Pairwise ↑s fun i j => IndepFunCat (X i) (X j)) :
+    (h : Set.Pairwise ↑s fun i j => IndepFun (X i) (X j)) :
     Var[∑ i in s, X i] = ∑ i in s, Var[X i] := by
   classical
-    induction' s using Finset.induction_on with k s ks IH
-    · simp only [Finset.sum_empty, variance_zero]
-    rw [variance_def' (mem_ℒp_finset_sum' _ hs), sum_insert ks, sum_insert ks]
-    simp only [add_sq']
-    calc
-      𝔼[X k ^ 2 + (∑ i in s, X i) ^ 2 + 2 * X k * ∑ i in s, X i] - 𝔼[X k + ∑ i in s, X i] ^ 2 =
-          𝔼[X k ^ 2] + 𝔼[(∑ i in s, X i) ^ 2] + 𝔼[2 * X k * ∑ i in s, X i] -
-            (𝔼[X k] + 𝔼[∑ i in s, X i]) ^ 2 :=
-        by
-        rw [integral_add', integral_add', integral_add']
-        · exact mem_ℒp.integrable one_le_two (hs _ (mem_insert_self _ _))
-        · apply integrable_finset_sum' _ fun i hi => _
-          exact mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi))
+  induction' s using Finset.induction_on with k s ks IH
+  · simp only [Finset.sum_empty, variance_zero]
+  rw [variance_def' (mem_ℒp_finset_sum' _ hs), sum_insert ks, sum_insert ks]
+  simp only [add_sq']
+  calc
+    𝔼[X k ^ 2 + (∑ i in s, X i) ^ 2 + 2 * X k * ∑ i in s, X i] - 𝔼[X k + ∑ i in s, X i] ^ 2 =
+        𝔼[X k ^ 2] + 𝔼[(∑ i in s, X i) ^ 2] + 𝔼[2 * X k * ∑ i in s, X i] -
+          (𝔼[X k] + 𝔼[∑ i in s, X i]) ^ 2 :=
+      by
+      rw [integral_add', integral_add', integral_add']
+      · exact mem_ℒp.integrable one_le_two (hs _ (mem_insert_self _ _))
+      · apply integrable_finset_sum' _ fun i hi => _
+        exact mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi))
+      · exact mem_ℒp.integrable_sq (hs _ (mem_insert_self _ _))
+      · apply mem_ℒp.integrable_sq
+        exact mem_ℒp_finset_sum' _ fun i hi => hs _ (mem_insert_of_mem hi)
+      · apply integrable.add
         · exact mem_ℒp.integrable_sq (hs _ (mem_insert_self _ _))
         · apply mem_ℒp.integrable_sq
           exact mem_ℒp_finset_sum' _ fun i hi => hs _ (mem_insert_of_mem hi)
-        · apply integrable.add
-          · exact mem_ℒp.integrable_sq (hs _ (mem_insert_self _ _))
-          · apply mem_ℒp.integrable_sq
-            exact mem_ℒp_finset_sum' _ fun i hi => hs _ (mem_insert_of_mem hi)
-        · rw [mul_assoc]
-          apply integrable.const_mul _ (2 : ℝ)
-          simp only [mul_sum, sum_apply, Pi.mul_apply]
-          apply integrable_finset_sum _ fun i hi => _
-          apply
-            indep_fun.integrable_mul _ (mem_ℒp.integrable one_le_two (hs _ (mem_insert_self _ _)))
-              (mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi)))
-          apply h (mem_insert_self _ _) (mem_insert_of_mem hi)
-          exact fun hki => ks (hki.symm ▸ hi)
-      _ =
-          Var[X k] + Var[∑ i in s, X i] +
-            (𝔼[2 * X k * ∑ i in s, X i] - 2 * 𝔼[X k] * 𝔼[∑ i in s, X i]) :=
-        by
-        rw [variance_def' (hs _ (mem_insert_self _ _)),
-          variance_def' (mem_ℒp_finset_sum' _ fun i hi => hs _ (mem_insert_of_mem hi))]
-        ring
-      _ = Var[X k] + Var[∑ i in s, X i] :=
-        by
-        simp only [mul_assoc, integral_mul_left, Pi.mul_apply, Pi.bit0_apply, Pi.one_apply,
-          sum_apply, add_right_eq_self, mul_sum]
-        rw [integral_finset_sum s fun i hi => _]; swap
-        · apply integrable.const_mul _ (2 : ℝ)
-          apply
-            indep_fun.integrable_mul _ (mem_ℒp.integrable one_le_two (hs _ (mem_insert_self _ _)))
-              (mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi)))
-          apply h (mem_insert_self _ _) (mem_insert_of_mem hi)
-          exact fun hki => ks (hki.symm ▸ hi)
-        rw [integral_finset_sum s fun i hi =>
-            mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi)),
-          mul_sum, mul_sum, ← sum_sub_distrib]
-        apply Finset.sum_eq_zero fun i hi => _
-        rw [integral_mul_left, indep_fun.integral_mul', sub_self]
-        · apply h (mem_insert_self _ _) (mem_insert_of_mem hi)
-          exact fun hki => ks (hki.symm ▸ hi)
-        · exact mem_ℒp.ae_strongly_measurable (hs _ (mem_insert_self _ _))
-        · exact mem_ℒp.ae_strongly_measurable (hs _ (mem_insert_of_mem hi))
-      _ = Var[X k] + ∑ i in s, Var[X i] := by
-        rw [IH (fun i hi => hs i (mem_insert_of_mem hi))
-            (h.mono (by simp only [coe_insert, Set.subset_insert]))]
-      
-#align probability_theory.indep_fun.variance_sum ProbabilityTheory.IndepFunCat.variance_sum
+      · rw [mul_assoc]
+        apply integrable.const_mul _ (2 : ℝ)
+        simp only [mul_sum, sum_apply, Pi.mul_apply]
+        apply integrable_finset_sum _ fun i hi => _
+        apply
+          indep_fun.integrable_mul _ (mem_ℒp.integrable one_le_two (hs _ (mem_insert_self _ _)))
+            (mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi)))
+        apply h (mem_insert_self _ _) (mem_insert_of_mem hi)
+        exact fun hki => ks (hki.symm ▸ hi)
+    _ =
+        Var[X k] + Var[∑ i in s, X i] +
+          (𝔼[2 * X k * ∑ i in s, X i] - 2 * 𝔼[X k] * 𝔼[∑ i in s, X i]) :=
+      by
+      rw [variance_def' (hs _ (mem_insert_self _ _)),
+        variance_def' (mem_ℒp_finset_sum' _ fun i hi => hs _ (mem_insert_of_mem hi))]
+      ring
+    _ = Var[X k] + Var[∑ i in s, X i] :=
+      by
+      simp only [mul_assoc, integral_mul_left, Pi.mul_apply, Pi.bit0_apply, Pi.one_apply, sum_apply,
+        add_right_eq_self, mul_sum]
+      rw [integral_finset_sum s fun i hi => _]; swap
+      · apply integrable.const_mul _ (2 : ℝ)
+        apply
+          indep_fun.integrable_mul _ (mem_ℒp.integrable one_le_two (hs _ (mem_insert_self _ _)))
+            (mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi)))
+        apply h (mem_insert_self _ _) (mem_insert_of_mem hi)
+        exact fun hki => ks (hki.symm ▸ hi)
+      rw [integral_finset_sum s fun i hi =>
+          mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi)),
+        mul_sum, mul_sum, ← sum_sub_distrib]
+      apply Finset.sum_eq_zero fun i hi => _
+      rw [integral_mul_left, indep_fun.integral_mul', sub_self]
+      · apply h (mem_insert_self _ _) (mem_insert_of_mem hi)
+        exact fun hki => ks (hki.symm ▸ hi)
+      · exact mem_ℒp.ae_strongly_measurable (hs _ (mem_insert_self _ _))
+      · exact mem_ℒp.ae_strongly_measurable (hs _ (mem_insert_of_mem hi))
+    _ = Var[X k] + ∑ i in s, Var[X i] := by
+      rw [IH (fun i hi => hs i (mem_insert_of_mem hi))
+          (h.mono (by simp only [coe_insert, Set.subset_insert]))]
+    
+#align probability_theory.indep_fun.variance_sum ProbabilityTheory.IndepFun.variance_sum
 
 end ProbabilityTheory
 

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

@@ -64,17 +64,17 @@ theorem MeasureTheory.Memℒp.evariance_lt_top [FiniteMeasure μ] (hX : Memℒp
     evariance X μ < ∞ :=
   by
   have := ENNReal.pow_lt_top (hX.sub <| mem_ℒp_const <| μ[X]).2 2
-  rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, ← ENNReal.rpow_two] at this
-  simp only [Pi.sub_apply, ENNReal.toReal_bit0, ENNReal.one_toReal, one_div] at this
-  rw [← ENNReal.rpow_mul, inv_mul_cancel (two_ne_zero : (2 : ℝ) ≠ 0), ENNReal.rpow_one] at this
-  simp_rw [ENNReal.rpow_two] at this
+  rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, ← ENNReal.rpow_two] at this 
+  simp only [Pi.sub_apply, ENNReal.toReal_bit0, ENNReal.one_toReal, one_div] at this 
+  rw [← ENNReal.rpow_mul, inv_mul_cancel (two_ne_zero : (2 : ℝ) ≠ 0), ENNReal.rpow_one] at this 
+  simp_rw [ENNReal.rpow_two] at this 
   exact this
 #align measure_theory.mem_ℒp.evariance_lt_top MeasureTheory.Memℒp.evariance_lt_top
 
 theorem evariance_eq_top [FiniteMeasure μ] (hXm : AEStronglyMeasurable X μ) (hX : ¬Memℒp X 2 μ) :
     evariance X μ = ∞ := by
   by_contra h
-  rw [← Ne.def, ← lt_top_iff_ne_top] at h
+  rw [← Ne.def, ← lt_top_iff_ne_top] at h 
   have : mem_ℒp (fun ω => X ω - μ[X]) 2 μ :=
     by
     refine' ⟨hXm.sub ae_strongly_measurable_const, _⟩
@@ -152,7 +152,7 @@ theorem evariance_eq_zero_iff (hX : AEMeasurable X μ) : evariance X μ = 0 ↔
   rw [evariance, lintegral_eq_zero_iff']
   constructor <;> intro hX <;> filter_upwards [hX]with ω hω
   · simp only [Pi.zero_apply, pow_eq_zero_iff, Nat.succ_pos', ENNReal.coe_eq_zero, nnnorm_eq_zero,
-      sub_eq_zero] at hω
+      sub_eq_zero] at hω 
     exact hω
   · rw [hω]
     simp
@@ -269,12 +269,12 @@ theorem evariance_def' [ProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}
   · symm
     rw [evariance_eq_top hX hℒ, ENNReal.sub_eq_top_iff]
     refine' ⟨_, ENNReal.ofReal_ne_top⟩
-    rw [mem_ℒp, not_and] at hℒ
+    rw [mem_ℒp, not_and] at hℒ 
     specialize hℒ hX
     simp only [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, not_lt, top_le_iff,
       ENNReal.toReal_bit0, ENNReal.one_toReal, ENNReal.rpow_two, one_div, ENNReal.rpow_eq_top_iff,
       inv_lt_zero, inv_pos, zero_lt_bit0, zero_lt_one, and_true_iff, or_iff_not_imp_left,
-      not_and_or] at hℒ
+      not_and_or] at hℒ 
     exact hℒ fun _ => zero_le_two
 #align probability_theory.evariance_def' ProbabilityTheory.evariance_def'
 

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

@@ -65,7 +65,7 @@ theorem MeasureTheory.Memℒp.evariance_lt_top [FiniteMeasure μ] (hX : Memℒp
   by
   have := ENNReal.pow_lt_top (hX.sub <| mem_ℒp_const <| μ[X]).2 2
   rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, ← ENNReal.rpow_two] at this
-  simp only [Pi.sub_apply, [anonymous], ENNReal.one_toReal, one_div] at this
+  simp only [Pi.sub_apply, ENNReal.toReal_bit0, ENNReal.one_toReal, one_div] at this
   rw [← ENNReal.rpow_mul, inv_mul_cancel (two_ne_zero : (2 : ℝ) ≠ 0), ENNReal.rpow_one] at this
   simp_rw [ENNReal.rpow_two] at this
   exact this
@@ -79,7 +79,7 @@ theorem evariance_eq_top [FiniteMeasure μ] (hXm : AEStronglyMeasurable X μ) (h
     by
     refine' ⟨hXm.sub ae_strongly_measurable_const, _⟩
     rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top]
-    simp only [[anonymous], ENNReal.one_toReal, ENNReal.rpow_two, Ne.def]
+    simp only [ENNReal.toReal_bit0, ENNReal.one_toReal, ENNReal.rpow_two, Ne.def]
     exact ENNReal.rpow_lt_top_of_nonneg (by simp) h.ne
   refine' hX _
   convert this.add (mem_ℒp_const <| μ[X])
@@ -124,8 +124,8 @@ theorem MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero (hX : Memℒp X 2
   · exact integral_nonneg fun ω => pow_two_nonneg _
   · convert hX.integrable_norm_rpow two_ne_zero ENNReal.two_ne_top
     ext ω
-    simp only [Pi.sub_apply, Real.norm_eq_abs, [anonymous], ENNReal.one_toReal, Real.rpow_two,
-      pow_bit0_abs]
+    simp only [Pi.sub_apply, Real.norm_eq_abs, ENNReal.toReal_bit0, ENNReal.one_toReal,
+      Real.rpow_two, pow_bit0_abs]
   · exact ae_of_all _ fun ω => pow_two_nonneg _
 #align measure_theory.mem_ℒp.variance_eq_of_integral_eq_zero MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero
 
@@ -138,8 +138,8 @@ theorem MeasureTheory.Memℒp.variance_eq [FiniteMeasure μ] (hX : Memℒp X 2 
   · exact integral_nonneg fun ω => pow_two_nonneg _
   · convert(hX.sub <| mem_ℒp_const (μ[X])).integrable_norm_rpow two_ne_zero ENNReal.two_ne_top
     ext ω
-    simp only [Pi.sub_apply, Real.norm_eq_abs, [anonymous], ENNReal.one_toReal, Real.rpow_two,
-      pow_bit0_abs]
+    simp only [Pi.sub_apply, Real.norm_eq_abs, ENNReal.toReal_bit0, ENNReal.one_toReal,
+      Real.rpow_two, pow_bit0_abs]
   · exact ae_of_all _ fun ω => pow_two_nonneg _
 #align measure_theory.mem_ℒp.variance_eq MeasureTheory.Memℒp.variance_eq
 
@@ -272,7 +272,7 @@ theorem evariance_def' [ProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}
     rw [mem_ℒp, not_and] at hℒ
     specialize hℒ hX
     simp only [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, not_lt, top_le_iff,
-      [anonymous], ENNReal.one_toReal, ENNReal.rpow_two, one_div, ENNReal.rpow_eq_top_iff,
+      ENNReal.toReal_bit0, ENNReal.one_toReal, ENNReal.rpow_two, one_div, ENNReal.rpow_eq_top_iff,
       inv_lt_zero, inv_pos, zero_lt_bit0, zero_lt_one, and_true_iff, or_iff_not_imp_left,
       not_and_or] at hℒ
     exact hℒ fun _ => zero_le_two
@@ -289,7 +289,7 @@ theorem meas_ge_le_evariance_div_sq {X : Ω → ℝ} (hX : AEStronglyMeasurable
     simp only [Pi.sub_apply, ENNReal.coe_le_coe, ← Real.norm_eq_abs, ← coe_nnnorm,
       NNReal.coe_le_coe, ENNReal.ofReal_coe_nnreal]
   · rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top]
-    simp only [[anonymous], ENNReal.one_toReal, Pi.sub_apply, one_div]
+    simp only [ENNReal.toReal_bit0, ENNReal.one_toReal, Pi.sub_apply, one_div]
     rw [div_eq_mul_inv, ENNReal.inv_pow, mul_comm, ENNReal.rpow_two]
     congr
     simp_rw [← ENNReal.rpow_mul, inv_mul_cancel (two_ne_zero : (2 : ℝ) ≠ 0), ENNReal.rpow_two,

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

@@ -42,7 +42,7 @@ open MeasureTheory Filter Finset
 
 noncomputable section
 
-open BigOperators MeasureTheory ProbabilityTheory ENNReal NNReal
+open scoped BigOperators MeasureTheory ProbabilityTheory ENNReal NNReal
 
 namespace ProbabilityTheory
 

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

@@ -97,9 +97,7 @@ theorem evariance_lt_top_iff_memℒp [FiniteMeasure μ] (hX : AEStronglyMeasurab
 #align probability_theory.evariance_lt_top_iff_mem_ℒp ProbabilityTheory.evariance_lt_top_iff_memℒp
 
 theorem MeasureTheory.Memℒp.ofReal_variance_eq [FiniteMeasure μ] (hX : Memℒp X 2 μ) :
-    ENNReal.ofReal (variance X μ) = evariance X μ :=
-  by
-  rw [variance, ENNReal.ofReal_toReal]
+    ENNReal.ofReal (variance X μ) = evariance X μ := by rw [variance, ENNReal.ofReal_toReal];
   exact hX.evariance_lt_top.ne
 #align measure_theory.mem_ℒp.of_real_variance_eq MeasureTheory.Memℒp.ofReal_variance_eq
 
@@ -205,8 +203,7 @@ theorem variance_smul' {A : Type _} [CommSemiring A] [Algebra A ℝ] (c : A) (X
     (μ : Measure Ω) : variance (c • X) μ = c ^ 2 • variance X μ :=
   by
   convert variance_smul (algebraMap A ℝ c) X μ
-  · ext1 x
-    simp only [algebraMap_smul]
+  · ext1 x; simp only [algebraMap_smul]
   · simp only [Algebra.smul_def, map_pow]
 #align probability_theory.variance_smul' ProbabilityTheory.variance_smul'
 
@@ -332,9 +329,7 @@ theorem IndepFunCat.variance_add [ProbabilityMeasure (ℙ : Measure Ω)] {X Y :
       by
       congr
       exact h.integral_mul_of_integrable (hX.integrable one_le_two) (hY.integrable one_le_two)
-    _ = Var[X] + Var[Y] := by
-      simp only [variance_def', hX, hY, Pi.pow_apply]
-      ring
+    _ = Var[X] + Var[Y] := by simp only [variance_def', hX, hY, Pi.pow_apply]; ring
     
 #align probability_theory.indep_fun.variance_add ProbabilityTheory.IndepFunCat.variance_add
 

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

@@ -71,7 +71,7 @@ theorem MeasureTheory.Memℒp.evariance_lt_top [FiniteMeasure μ] (hX : Memℒp
   exact this
 #align measure_theory.mem_ℒp.evariance_lt_top MeasureTheory.Memℒp.evariance_lt_top
 
-theorem evariance_eq_top [FiniteMeasure μ] (hXm : AeStronglyMeasurable X μ) (hX : ¬Memℒp X 2 μ) :
+theorem evariance_eq_top [FiniteMeasure μ] (hXm : AEStronglyMeasurable X μ) (hX : ¬Memℒp X 2 μ) :
     evariance X μ = ∞ := by
   by_contra h
   rw [← Ne.def, ← lt_top_iff_ne_top] at h
@@ -87,7 +87,7 @@ theorem evariance_eq_top [FiniteMeasure μ] (hXm : AeStronglyMeasurable X μ) (h
   rw [Pi.add_apply, sub_add_cancel]
 #align probability_theory.evariance_eq_top ProbabilityTheory.evariance_eq_top
 
-theorem evariance_lt_top_iff_memℒp [FiniteMeasure μ] (hX : AeStronglyMeasurable X μ) :
+theorem evariance_lt_top_iff_memℒp [FiniteMeasure μ] (hX : AEStronglyMeasurable X μ) :
     evariance X μ < ∞ ↔ Memℒp X 2 μ :=
   by
   refine' ⟨_, MeasureTheory.Memℒp.evariance_lt_top⟩
@@ -235,7 +235,7 @@ theorem variance_def' [ProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ} (
 #align probability_theory.variance_def' ProbabilityTheory.variance_def'
 
 theorem variance_le_expectation_sq [ProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}
-    (hm : AeStronglyMeasurable X ℙ) : Var[X] ≤ 𝔼[X ^ 2] :=
+    (hm : AEStronglyMeasurable X ℙ) : Var[X] ≤ 𝔼[X ^ 2] :=
   by
   by_cases hX : mem_ℒp X 2
   · rw [variance_def' hX]
@@ -255,11 +255,11 @@ theorem variance_le_expectation_sq [ProbabilityMeasure (ℙ : Measure Ω)] {X :
       convert A.add B
       simp
   · exact ae_of_all _ fun x => sq_nonneg _
-  · exact (AEMeasurable.pow_const (hm.ae_measurable.sub_const _) _).AeStronglyMeasurable
+  · exact (AEMeasurable.pow_const (hm.ae_measurable.sub_const _) _).AEStronglyMeasurable
 #align probability_theory.variance_le_expectation_sq ProbabilityTheory.variance_le_expectation_sq
 
 theorem evariance_def' [ProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}
-    (hX : AeStronglyMeasurable X ℙ) : eVar[X] = (∫⁻ ω, ‖X ω‖₊ ^ 2) - ENNReal.ofReal (𝔼[X] ^ 2) :=
+    (hX : AEStronglyMeasurable X ℙ) : eVar[X] = (∫⁻ ω, ‖X ω‖₊ ^ 2) - ENNReal.ofReal (𝔼[X] ^ 2) :=
   by
   by_cases hℒ : mem_ℒp X 2
   · rw [← hℒ.of_real_variance_eq, variance_def' hℒ, ENNReal.ofReal_sub _ (sq_nonneg _)]
@@ -282,7 +282,7 @@ theorem evariance_def' [ProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}
 #align probability_theory.evariance_def' ProbabilityTheory.evariance_def'
 
 /-- *Chebyshev's inequality* for `ℝ≥0∞`-valued variance. -/
-theorem meas_ge_le_evariance_div_sq {X : Ω → ℝ} (hX : AeStronglyMeasurable X ℙ) {c : ℝ≥0}
+theorem meas_ge_le_evariance_div_sq {X : Ω → ℝ} (hX : AEStronglyMeasurable X ℙ) {c : ℝ≥0}
     (hc : c ≠ 0) : ℙ { ω | ↑c ≤ |X ω - 𝔼[X]| } ≤ eVar[X] / c ^ 2 :=
   by
   have A : (c : ℝ≥0∞) ≠ 0 := by rwa [Ne.def, ENNReal.coe_eq_zero]

mathlib commit https://github.com/leanprover-community/mathlib/commit/d4437c68c8d350fc9d4e95e1e174409db35e30d7

@@ -60,7 +60,7 @@ def variance {Ω : Type _} {m : MeasurableSpace Ω} (X : Ω → ℝ) (μ : Measu
 
 variable {Ω : Type _} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : Measure Ω}
 
-theorem MeasureTheory.Memℒp.evariance_lt_top [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
+theorem MeasureTheory.Memℒp.evariance_lt_top [FiniteMeasure μ] (hX : Memℒp X 2 μ) :
     evariance X μ < ∞ :=
   by
   have := ENNReal.pow_lt_top (hX.sub <| mem_ℒp_const <| μ[X]).2 2
@@ -71,7 +71,7 @@ theorem MeasureTheory.Memℒp.evariance_lt_top [IsFiniteMeasure μ] (hX : Memℒ
   exact this
 #align measure_theory.mem_ℒp.evariance_lt_top MeasureTheory.Memℒp.evariance_lt_top
 
-theorem evariance_eq_top [IsFiniteMeasure μ] (hXm : AeStronglyMeasurable X μ) (hX : ¬Memℒp X 2 μ) :
+theorem evariance_eq_top [FiniteMeasure μ] (hXm : AeStronglyMeasurable X μ) (hX : ¬Memℒp X 2 μ) :
     evariance X μ = ∞ := by
   by_contra h
   rw [← Ne.def, ← lt_top_iff_ne_top] at h
@@ -87,7 +87,7 @@ theorem evariance_eq_top [IsFiniteMeasure μ] (hXm : AeStronglyMeasurable X μ)
   rw [Pi.add_apply, sub_add_cancel]
 #align probability_theory.evariance_eq_top ProbabilityTheory.evariance_eq_top
 
-theorem evariance_lt_top_iff_memℒp [IsFiniteMeasure μ] (hX : AeStronglyMeasurable X μ) :
+theorem evariance_lt_top_iff_memℒp [FiniteMeasure μ] (hX : AeStronglyMeasurable X μ) :
     evariance X μ < ∞ ↔ Memℒp X 2 μ :=
   by
   refine' ⟨_, MeasureTheory.Memℒp.evariance_lt_top⟩
@@ -96,7 +96,7 @@ theorem evariance_lt_top_iff_memℒp [IsFiniteMeasure μ] (hX : AeStronglyMeasur
   exact evariance_eq_top hX
 #align probability_theory.evariance_lt_top_iff_mem_ℒp ProbabilityTheory.evariance_lt_top_iff_memℒp
 
-theorem MeasureTheory.Memℒp.ofReal_variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
+theorem MeasureTheory.Memℒp.ofReal_variance_eq [FiniteMeasure μ] (hX : Memℒp X 2 μ) :
     ENNReal.ofReal (variance X μ) = evariance X μ :=
   by
   rw [variance, ENNReal.ofReal_toReal]
@@ -131,7 +131,7 @@ theorem MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero (hX : Memℒp X 2
   · exact ae_of_all _ fun ω => pow_two_nonneg _
 #align measure_theory.mem_ℒp.variance_eq_of_integral_eq_zero MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero
 
-theorem MeasureTheory.Memℒp.variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
+theorem MeasureTheory.Memℒp.variance_eq [FiniteMeasure μ] (hX : Memℒp X 2 μ) :
     variance X μ = μ[(X - fun ω => μ[X]) ^ 2] :=
   by
   rw [variance, evariance_eq_lintegral_of_real, ← of_real_integral_eq_lintegral_of_real,
@@ -217,7 +217,7 @@ omit m
 
 variable [MeasureSpace Ω]
 
-theorem variance_def' [IsProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ} (hX : Memℒp X 2) :
+theorem variance_def' [ProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ} (hX : Memℒp X 2) :
     Var[X] = 𝔼[X ^ 2] - 𝔼[X] ^ 2 :=
   by
   rw [hX.variance_eq, sub_sq', integral_sub', integral_add']; rotate_left
@@ -234,7 +234,7 @@ theorem variance_def' [IsProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}
   ring
 #align probability_theory.variance_def' ProbabilityTheory.variance_def'
 
-theorem variance_le_expectation_sq [IsProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}
+theorem variance_le_expectation_sq [ProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}
     (hm : AeStronglyMeasurable X ℙ) : Var[X] ≤ 𝔼[X ^ 2] :=
   by
   by_cases hX : mem_ℒp X 2
@@ -258,7 +258,7 @@ theorem variance_le_expectation_sq [IsProbabilityMeasure (ℙ : Measure Ω)] {X
   · exact (AEMeasurable.pow_const (hm.ae_measurable.sub_const _) _).AeStronglyMeasurable
 #align probability_theory.variance_le_expectation_sq ProbabilityTheory.variance_le_expectation_sq
 
-theorem evariance_def' [IsProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}
+theorem evariance_def' [ProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}
     (hX : AeStronglyMeasurable X ℙ) : eVar[X] = (∫⁻ ω, ‖X ω‖₊ ^ 2) - ENNReal.ofReal (𝔼[X] ^ 2) :=
   by
   by_cases hℒ : mem_ℒp X 2
@@ -301,7 +301,7 @@ theorem meas_ge_le_evariance_div_sq {X : Ω → ℝ} (hX : AeStronglyMeasurable
 
 /-- *Chebyshev's inequality* : one can control the deviation probability of a real random variable
 from its expectation in terms of the variance. -/
-theorem meas_ge_le_variance_div_sq [IsFiniteMeasure (ℙ : Measure Ω)] {X : Ω → ℝ} (hX : Memℒp X 2)
+theorem meas_ge_le_variance_div_sq [FiniteMeasure (ℙ : Measure Ω)] {X : Ω → ℝ} (hX : Memℒp X 2)
     {c : ℝ} (hc : 0 < c) : ℙ { ω | c ≤ |X ω - 𝔼[X]| } ≤ ENNReal.ofReal (Var[X] / c ^ 2) :=
   by
   rw [ENNReal.ofReal_div_of_pos (sq_pos_of_ne_zero _ hc.ne.symm), hX.of_real_variance_eq]
@@ -312,8 +312,8 @@ theorem meas_ge_le_variance_div_sq [IsFiniteMeasure (ℙ : Measure Ω)] {X : Ω
 #align probability_theory.meas_ge_le_variance_div_sq ProbabilityTheory.meas_ge_le_variance_div_sq
 
 /-- The variance of the sum of two independent random variables is the sum of the variances. -/
-theorem IndepFunCat.variance_add [IsProbabilityMeasure (ℙ : Measure Ω)] {X Y : Ω → ℝ}
-    (hX : Memℒp X 2) (hY : Memℒp Y 2) (h : IndepFunCat X Y) : Var[X + Y] = Var[X] + Var[Y] :=
+theorem IndepFunCat.variance_add [ProbabilityMeasure (ℙ : Measure Ω)] {X Y : Ω → ℝ} (hX : Memℒp X 2)
+    (hY : Memℒp Y 2) (h : IndepFunCat X Y) : Var[X + Y] = Var[X] + Var[Y] :=
   calc
     Var[X + Y] = 𝔼[fun a => X a ^ 2 + Y a ^ 2 + 2 * X a * Y a] - 𝔼[X + Y] ^ 2 := by
       simp [variance_def' (hX.add hY), add_sq']
@@ -340,7 +340,7 @@ theorem IndepFunCat.variance_add [IsProbabilityMeasure (ℙ : Measure Ω)] {X Y
 
 /-- The variance of a finite sum of pairwise independent random variables is the sum of the
 variances. -/
-theorem IndepFunCat.variance_sum [IsProbabilityMeasure (ℙ : Measure Ω)] {ι : Type _} {X : ι → Ω → ℝ}
+theorem IndepFunCat.variance_sum [ProbabilityMeasure (ℙ : Measure Ω)] {ι : Type _} {X : ι → Ω → ℝ}
     {s : Finset ι} (hs : ∀ i ∈ s, Memℒp (X i) 2)
     (h : Set.Pairwise ↑s fun i j => IndepFunCat (X i) (X j)) :
     Var[∑ i in s, X i] = ∑ i in s, Var[X i] := by

mathlib commit https://github.com/leanprover-community/mathlib/commit/738054fa93d43512da144ec45ce799d18fd44248

@@ -4,12 +4,13 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Kexing Ying
 
 ! This file was ported from Lean 3 source module probability.variance
-! leanprover-community/mathlib commit 57ac39bd365c2f80589a700f9fbb664d3a1a30c2
+! leanprover-community/mathlib commit f0c8bf9245297a541f468be517f1bde6195105e9
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.Probability.Notation
 import Mathbin.Probability.Integration
+import Mathbin.MeasureTheory.Function.L2Space
 
 /-!
 # Variance of random variables

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

@@ -137,7 +137,7 @@ theorem MeasureTheory.Memℒp.variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2
     ENNReal.toReal_ofReal]
   · rfl
   · exact integral_nonneg fun ω => pow_two_nonneg _
-  · convert(hX.sub <| mem_ℒp_const (μ[X])).integrableNormRpow two_ne_zero ENNReal.two_ne_top
+  · convert(hX.sub <| mem_ℒp_const (μ[X])).integrable_norm_rpow two_ne_zero ENNReal.two_ne_top
     ext ω
     simp only [Pi.sub_apply, Real.norm_eq_abs, [anonymous], ENNReal.one_toReal, Real.rpow_two,
       pow_bit0_abs]
@@ -148,7 +148,7 @@ theorem MeasureTheory.Memℒp.variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2
 theorem evariance_zero : evariance 0 μ = 0 := by simp [evariance]
 #align probability_theory.evariance_zero ProbabilityTheory.evariance_zero
 
-theorem evariance_eq_zero_iff (hX : AeMeasurable X μ) : evariance X μ = 0 ↔ X =ᵐ[μ] fun ω => μ[X] :=
+theorem evariance_eq_zero_iff (hX : AEMeasurable X μ) : evariance X μ = 0 ↔ X =ᵐ[μ] fun ω => μ[X] :=
   by
   rw [evariance, lintegral_eq_zero_iff']
   constructor <;> intro hX <;> filter_upwards [hX]with ω hω
@@ -254,7 +254,7 @@ theorem variance_le_expectation_sq [IsProbabilityMeasure (ℙ : Measure Ω)] {X
       convert A.add B
       simp
   · exact ae_of_all _ fun x => sq_nonneg _
-  · exact (AeMeasurable.powConst (hm.ae_measurable.sub_const _) _).AeStronglyMeasurable
+  · exact (AEMeasurable.pow_const (hm.ae_measurable.sub_const _) _).AeStronglyMeasurable
 #align probability_theory.variance_le_expectation_sq ProbabilityTheory.variance_le_expectation_sq
 
 theorem evariance_def' [IsProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d

@@ -137,7 +137,7 @@ theorem MeasureTheory.Memℒp.variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2
     ENNReal.toReal_ofReal]
   · rfl
   · exact integral_nonneg fun ω => pow_two_nonneg _
-  · convert (hX.sub <| mem_ℒp_const (μ[X])).integrableNormRpow two_ne_zero ENNReal.two_ne_top
+  · convert(hX.sub <| mem_ℒp_const (μ[X])).integrableNormRpow two_ne_zero ENNReal.two_ne_top
     ext ω
     simp only [Pi.sub_apply, Real.norm_eq_abs, [anonymous], ENNReal.one_toReal, Real.rpow_two,
       pow_bit0_abs]
@@ -304,7 +304,7 @@ theorem meas_ge_le_variance_div_sq [IsFiniteMeasure (ℙ : Measure Ω)] {X : Ω
     {c : ℝ} (hc : 0 < c) : ℙ { ω | c ≤ |X ω - 𝔼[X]| } ≤ ENNReal.ofReal (Var[X] / c ^ 2) :=
   by
   rw [ENNReal.ofReal_div_of_pos (sq_pos_of_ne_zero _ hc.ne.symm), hX.of_real_variance_eq]
-  convert @meas_ge_le_evariance_div_sq _ _ _ hX.1 c.to_nnreal (by simp [hc])
+  convert@meas_ge_le_evariance_div_sq _ _ _ hX.1 c.to_nnreal (by simp [hc])
   · simp only [Real.coe_toNNReal', max_le_iff, abs_nonneg, and_true_iff]
   · rw [ENNReal.ofReal_pow hc.le]
     rfl

mathlib commit https://github.com/leanprover-community/mathlib/commit/9da1b3534b65d9661eb8f42443598a92bbb49211

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Kexing Ying
 
 ! This file was ported from Lean 3 source module probability.variance
-! leanprover-community/mathlib commit 7c60702a4145fa8de521d53b93e602463f555cd7
+! leanprover-community/mathlib commit 57ac39bd365c2f80589a700f9fbb664d3a1a30c2
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -63,8 +63,7 @@ theorem MeasureTheory.Memℒp.evariance_lt_top [IsFiniteMeasure μ] (hX : Memℒ
     evariance X μ < ∞ :=
   by
   have := ENNReal.pow_lt_top (hX.sub <| mem_ℒp_const <| μ[X]).2 2
-  rw [snorm_eq_lintegral_rpow_nnnorm ENNReal.two_ne_zero ENNReal.two_ne_top, ← ENNReal.rpow_two] at
-    this
+  rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, ← ENNReal.rpow_two] at this
   simp only [Pi.sub_apply, [anonymous], ENNReal.one_toReal, one_div] at this
   rw [← ENNReal.rpow_mul, inv_mul_cancel (two_ne_zero : (2 : ℝ) ≠ 0), ENNReal.rpow_one] at this
   simp_rw [ENNReal.rpow_two] at this
@@ -78,7 +77,7 @@ theorem evariance_eq_top [IsFiniteMeasure μ] (hXm : AeStronglyMeasurable X μ)
   have : mem_ℒp (fun ω => X ω - μ[X]) 2 μ :=
     by
     refine' ⟨hXm.sub ae_strongly_measurable_const, _⟩
-    rw [snorm_eq_lintegral_rpow_nnnorm ENNReal.two_ne_zero ENNReal.two_ne_top]
+    rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top]
     simp only [[anonymous], ENNReal.one_toReal, ENNReal.rpow_two, Ne.def]
     exact ENNReal.rpow_lt_top_of_nonneg (by simp) h.ne
   refine' hX _
@@ -124,7 +123,7 @@ theorem MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero (hX : Memℒp X 2
     simp_rw [hXint, sub_zero]
   · rfl
   · exact integral_nonneg fun ω => pow_two_nonneg _
-  · convert hX.integrable_norm_rpow ENNReal.two_ne_zero ENNReal.two_ne_top
+  · convert hX.integrable_norm_rpow two_ne_zero ENNReal.two_ne_top
     ext ω
     simp only [Pi.sub_apply, Real.norm_eq_abs, [anonymous], ENNReal.one_toReal, Real.rpow_two,
       pow_bit0_abs]
@@ -138,8 +137,7 @@ theorem MeasureTheory.Memℒp.variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2
     ENNReal.toReal_ofReal]
   · rfl
   · exact integral_nonneg fun ω => pow_two_nonneg _
-  · convert
-      (hX.sub <| mem_ℒp_const (μ[X])).integrableNormRpow ENNReal.two_ne_zero ENNReal.two_ne_top
+  · convert (hX.sub <| mem_ℒp_const (μ[X])).integrableNormRpow two_ne_zero ENNReal.two_ne_top
     ext ω
     simp only [Pi.sub_apply, Real.norm_eq_abs, [anonymous], ENNReal.one_toReal, Real.rpow_two,
       pow_bit0_abs]
@@ -275,10 +273,10 @@ theorem evariance_def' [IsProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ
     refine' ⟨_, ENNReal.ofReal_ne_top⟩
     rw [mem_ℒp, not_and] at hℒ
     specialize hℒ hX
-    simp only [snorm_eq_lintegral_rpow_nnnorm ENNReal.two_ne_zero ENNReal.two_ne_top, not_lt,
-      top_le_iff, [anonymous], ENNReal.one_toReal, ENNReal.rpow_two, one_div,
-      ENNReal.rpow_eq_top_iff, inv_lt_zero, inv_pos, zero_lt_bit0, zero_lt_one, and_true_iff,
-      or_iff_not_imp_left, not_and_or] at hℒ
+    simp only [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, not_lt, top_le_iff,
+      [anonymous], ENNReal.one_toReal, ENNReal.rpow_two, one_div, ENNReal.rpow_eq_top_iff,
+      inv_lt_zero, inv_pos, zero_lt_bit0, zero_lt_one, and_true_iff, or_iff_not_imp_left,
+      not_and_or] at hℒ
     exact hℒ fun _ => zero_le_two
 #align probability_theory.evariance_def' ProbabilityTheory.evariance_def'
 
@@ -288,11 +286,11 @@ theorem meas_ge_le_evariance_div_sq {X : Ω → ℝ} (hX : AeStronglyMeasurable
   by
   have A : (c : ℝ≥0∞) ≠ 0 := by rwa [Ne.def, ENNReal.coe_eq_zero]
   have B : ae_strongly_measurable (fun ω : Ω => 𝔼[X]) ℙ := ae_strongly_measurable_const
-  convert meas_ge_le_mul_pow_snorm ℙ ENNReal.two_ne_zero ENNReal.two_ne_top (hX.sub B) A
+  convert meas_ge_le_mul_pow_snorm ℙ two_ne_zero ENNReal.two_ne_top (hX.sub B) A
   · ext ω
     simp only [Pi.sub_apply, ENNReal.coe_le_coe, ← Real.norm_eq_abs, ← coe_nnnorm,
       NNReal.coe_le_coe, ENNReal.ofReal_coe_nnreal]
-  · rw [snorm_eq_lintegral_rpow_nnnorm ENNReal.two_ne_zero ENNReal.two_ne_top]
+  · rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top]
     simp only [[anonymous], ENNReal.one_toReal, Pi.sub_apply, one_div]
     rw [div_eq_mul_inv, ENNReal.inv_pow, mul_comm, ENNReal.rpow_two]
     congr

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

@@ -41,7 +41,7 @@ open MeasureTheory Filter Finset
 
 noncomputable section
 
-open BigOperators MeasureTheory ProbabilityTheory Ennreal NNReal
+open BigOperators MeasureTheory ProbabilityTheory ENNReal NNReal
 
 namespace ProbabilityTheory
 
@@ -62,12 +62,12 @@ variable {Ω : Type _} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : Measure 
 theorem MeasureTheory.Memℒp.evariance_lt_top [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
     evariance X μ < ∞ :=
   by
-  have := Ennreal.pow_lt_top (hX.sub <| mem_ℒp_const <| μ[X]).2 2
-  rw [snorm_eq_lintegral_rpow_nnnorm Ennreal.two_ne_zero Ennreal.two_ne_top, ← Ennreal.rpow_two] at
+  have := ENNReal.pow_lt_top (hX.sub <| mem_ℒp_const <| μ[X]).2 2
+  rw [snorm_eq_lintegral_rpow_nnnorm ENNReal.two_ne_zero ENNReal.two_ne_top, ← ENNReal.rpow_two] at
     this
-  simp only [Pi.sub_apply, Ennreal.toReal_bit0, Ennreal.one_toReal, one_div] at this
-  rw [← Ennreal.rpow_mul, inv_mul_cancel (two_ne_zero : (2 : ℝ) ≠ 0), Ennreal.rpow_one] at this
-  simp_rw [Ennreal.rpow_two] at this
+  simp only [Pi.sub_apply, [anonymous], ENNReal.one_toReal, one_div] at this
+  rw [← ENNReal.rpow_mul, inv_mul_cancel (two_ne_zero : (2 : ℝ) ≠ 0), ENNReal.rpow_one] at this
+  simp_rw [ENNReal.rpow_two] at this
   exact this
 #align measure_theory.mem_ℒp.evariance_lt_top MeasureTheory.Memℒp.evariance_lt_top
 
@@ -78,9 +78,9 @@ theorem evariance_eq_top [IsFiniteMeasure μ] (hXm : AeStronglyMeasurable X μ)
   have : mem_ℒp (fun ω => X ω - μ[X]) 2 μ :=
     by
     refine' ⟨hXm.sub ae_strongly_measurable_const, _⟩
-    rw [snorm_eq_lintegral_rpow_nnnorm Ennreal.two_ne_zero Ennreal.two_ne_top]
-    simp only [Ennreal.toReal_bit0, Ennreal.one_toReal, Ennreal.rpow_two, Ne.def]
-    exact Ennreal.rpow_lt_top_of_nonneg (by simp) h.ne
+    rw [snorm_eq_lintegral_rpow_nnnorm ENNReal.two_ne_zero ENNReal.two_ne_top]
+    simp only [[anonymous], ENNReal.one_toReal, ENNReal.rpow_two, Ne.def]
+    exact ENNReal.rpow_lt_top_of_nonneg (by simp) h.ne
   refine' hX _
   convert this.add (mem_ℒp_const <| μ[X])
   ext ω
@@ -97,21 +97,21 @@ theorem evariance_lt_top_iff_memℒp [IsFiniteMeasure μ] (hX : AeStronglyMeasur
 #align probability_theory.evariance_lt_top_iff_mem_ℒp ProbabilityTheory.evariance_lt_top_iff_memℒp
 
 theorem MeasureTheory.Memℒp.ofReal_variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
-    Ennreal.ofReal (variance X μ) = evariance X μ :=
+    ENNReal.ofReal (variance X μ) = evariance X μ :=
   by
-  rw [variance, Ennreal.ofReal_toReal]
+  rw [variance, ENNReal.ofReal_toReal]
   exact hX.evariance_lt_top.ne
 #align measure_theory.mem_ℒp.of_real_variance_eq MeasureTheory.Memℒp.ofReal_variance_eq
 
 include m
 
 theorem evariance_eq_lintegral_ofReal (X : Ω → ℝ) (μ : Measure Ω) :
-    evariance X μ = ∫⁻ ω, Ennreal.ofReal ((X ω - μ[X]) ^ 2) ∂μ :=
+    evariance X μ = ∫⁻ ω, ENNReal.ofReal ((X ω - μ[X]) ^ 2) ∂μ :=
   by
   rw [evariance]
   congr
   ext1 ω
-  rw [pow_two, ← Ennreal.coe_mul, ← nnnorm_mul, ← pow_two]
+  rw [pow_two, ← ENNReal.coe_mul, ← nnnorm_mul, ← pow_two]
   congr
   exact (Real.toNNReal_eq_nnnorm_of_nonneg <| sq_nonneg _).symm
 #align probability_theory.evariance_eq_lintegral_of_real ProbabilityTheory.evariance_eq_lintegral_ofReal
@@ -120,14 +120,14 @@ theorem MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero (hX : Memℒp X 2
     variance X μ = μ[X ^ 2] :=
   by
   rw [variance, evariance_eq_lintegral_of_real, ← of_real_integral_eq_lintegral_of_real,
-      Ennreal.toReal_ofReal] <;>
+      ENNReal.toReal_ofReal] <;>
     simp_rw [hXint, sub_zero]
   · rfl
   · exact integral_nonneg fun ω => pow_two_nonneg _
-  · convert hX.integrable_norm_rpow Ennreal.two_ne_zero Ennreal.two_ne_top
+  · convert hX.integrable_norm_rpow ENNReal.two_ne_zero ENNReal.two_ne_top
     ext ω
-    simp only [Pi.sub_apply, Real.norm_eq_abs, Ennreal.toReal_bit0, Ennreal.one_toReal,
-      Real.rpow_two, pow_bit0_abs]
+    simp only [Pi.sub_apply, Real.norm_eq_abs, [anonymous], ENNReal.one_toReal, Real.rpow_two,
+      pow_bit0_abs]
   · exact ae_of_all _ fun ω => pow_two_nonneg _
 #align measure_theory.mem_ℒp.variance_eq_of_integral_eq_zero MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero
 
@@ -135,14 +135,14 @@ theorem MeasureTheory.Memℒp.variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2
     variance X μ = μ[(X - fun ω => μ[X]) ^ 2] :=
   by
   rw [variance, evariance_eq_lintegral_of_real, ← of_real_integral_eq_lintegral_of_real,
-    Ennreal.toReal_ofReal]
+    ENNReal.toReal_ofReal]
   · rfl
   · exact integral_nonneg fun ω => pow_two_nonneg _
   · convert
-      (hX.sub <| mem_ℒp_const (μ[X])).integrableNormRpow Ennreal.two_ne_zero Ennreal.two_ne_top
+      (hX.sub <| mem_ℒp_const (μ[X])).integrableNormRpow ENNReal.two_ne_zero ENNReal.two_ne_top
     ext ω
-    simp only [Pi.sub_apply, Real.norm_eq_abs, Ennreal.toReal_bit0, Ennreal.one_toReal,
-      Real.rpow_two, pow_bit0_abs]
+    simp only [Pi.sub_apply, Real.norm_eq_abs, [anonymous], ENNReal.one_toReal, Real.rpow_two,
+      pow_bit0_abs]
   · exact ae_of_all _ fun ω => pow_two_nonneg _
 #align measure_theory.mem_ℒp.variance_eq MeasureTheory.Memℒp.variance_eq
 
@@ -154,7 +154,7 @@ theorem evariance_eq_zero_iff (hX : AeMeasurable X μ) : evariance X μ = 0 ↔
   by
   rw [evariance, lintegral_eq_zero_iff']
   constructor <;> intro hX <;> filter_upwards [hX]with ω hω
-  · simp only [Pi.zero_apply, pow_eq_zero_iff, Nat.succ_pos', Ennreal.coe_eq_zero, nnnorm_eq_zero,
+  · simp only [Pi.zero_apply, pow_eq_zero_iff, Nat.succ_pos', ENNReal.coe_eq_zero, nnnorm_eq_zero,
       sub_eq_zero] at hω
     exact hω
   · rw [hω]
@@ -163,12 +163,12 @@ theorem evariance_eq_zero_iff (hX : AeMeasurable X μ) : evariance X μ = 0 ↔
 #align probability_theory.evariance_eq_zero_iff ProbabilityTheory.evariance_eq_zero_iff
 
 theorem evariance_mul (c : ℝ) (X : Ω → ℝ) (μ : Measure Ω) :
-    evariance (fun ω => c * X ω) μ = Ennreal.ofReal (c ^ 2) * evariance X μ :=
+    evariance (fun ω => c * X ω) μ = ENNReal.ofReal (c ^ 2) * evariance X μ :=
   by
   rw [evariance, evariance, ← lintegral_const_mul' _ _ ennreal.of_real_lt_top.ne]
   congr
   ext1 ω
-  rw [Ennreal.ofReal, ← Ennreal.coe_pow, ← Ennreal.coe_pow, ← Ennreal.coe_mul]
+  rw [ENNReal.ofReal, ← ENNReal.coe_pow, ← ENNReal.coe_pow, ← ENNReal.coe_mul]
   congr
   rw [← sq_abs, ← Real.rpow_two, Real.toNNReal_rpow_of_nonneg (abs_nonneg _), NNReal.rpow_two, ←
     mul_pow, Real.toNNReal_mul_nnnorm _ (abs_nonneg _)]
@@ -183,17 +183,17 @@ scoped notation "eVar[" X "]" => ProbabilityTheory.evariance X MeasureTheory.Mea
 
 @[simp]
 theorem variance_zero (μ : Measure Ω) : variance 0 μ = 0 := by
-  simp only [variance, evariance_zero, Ennreal.zero_toReal]
+  simp only [variance, evariance_zero, ENNReal.zero_toReal]
 #align probability_theory.variance_zero ProbabilityTheory.variance_zero
 
 theorem variance_nonneg (X : Ω → ℝ) (μ : Measure Ω) : 0 ≤ variance X μ :=
-  Ennreal.toReal_nonneg
+  ENNReal.toReal_nonneg
 #align probability_theory.variance_nonneg ProbabilityTheory.variance_nonneg
 
 theorem variance_mul (c : ℝ) (X : Ω → ℝ) (μ : Measure Ω) :
     variance (fun ω => c * X ω) μ = c ^ 2 * variance X μ :=
   by
-  rw [variance, evariance_mul, Ennreal.toReal_mul, Ennreal.toReal_ofReal (sq_nonneg _)]
+  rw [variance, evariance_mul, ENNReal.toReal_mul, ENNReal.toReal_ofReal (sq_nonneg _)]
   rfl
 #align probability_theory.variance_mul ProbabilityTheory.variance_mul
 
@@ -231,7 +231,7 @@ theorem variance_def' [IsProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}
   · exact ((hX.integrable one_le_two).const_mul 2).mul_const' _
   simp only [integral_mul_right, Pi.pow_apply, Pi.mul_apply, Pi.bit0_apply, Pi.one_apply,
     integral_const (integral ℙ X ^ 2), integral_mul_left (2 : ℝ), one_mul, variance, Pi.pow_apply,
-    measure_univ, Ennreal.one_toReal, Algebra.id.smul_eq_mul]
+    measure_univ, ENNReal.one_toReal, Algebra.id.smul_eq_mul]
   ring
 #align probability_theory.variance_def' ProbabilityTheory.variance_def'
 
@@ -260,24 +260,24 @@ theorem variance_le_expectation_sq [IsProbabilityMeasure (ℙ : Measure Ω)] {X
 #align probability_theory.variance_le_expectation_sq ProbabilityTheory.variance_le_expectation_sq
 
 theorem evariance_def' [IsProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ}
-    (hX : AeStronglyMeasurable X ℙ) : eVar[X] = (∫⁻ ω, ‖X ω‖₊ ^ 2) - Ennreal.ofReal (𝔼[X] ^ 2) :=
+    (hX : AeStronglyMeasurable X ℙ) : eVar[X] = (∫⁻ ω, ‖X ω‖₊ ^ 2) - ENNReal.ofReal (𝔼[X] ^ 2) :=
   by
   by_cases hℒ : mem_ℒp X 2
-  · rw [← hℒ.of_real_variance_eq, variance_def' hℒ, Ennreal.ofReal_sub _ (sq_nonneg _)]
+  · rw [← hℒ.of_real_variance_eq, variance_def' hℒ, ENNReal.ofReal_sub _ (sq_nonneg _)]
     congr
-    simp_rw [← Ennreal.coe_pow]
+    simp_rw [← ENNReal.coe_pow]
     rw [lintegral_coe_eq_integral]
     · congr 2 with ω
       simp only [Pi.pow_apply, NNReal.coe_pow, coe_nnnorm, Real.norm_eq_abs, pow_bit0_abs]
     · exact hℒ.abs.integrable_sq
   · symm
-    rw [evariance_eq_top hX hℒ, Ennreal.sub_eq_top_iff]
-    refine' ⟨_, Ennreal.ofReal_ne_top⟩
+    rw [evariance_eq_top hX hℒ, ENNReal.sub_eq_top_iff]
+    refine' ⟨_, ENNReal.ofReal_ne_top⟩
     rw [mem_ℒp, not_and] at hℒ
     specialize hℒ hX
-    simp only [snorm_eq_lintegral_rpow_nnnorm Ennreal.two_ne_zero Ennreal.two_ne_top, not_lt,
-      top_le_iff, Ennreal.toReal_bit0, Ennreal.one_toReal, Ennreal.rpow_two, one_div,
-      Ennreal.rpow_eq_top_iff, inv_lt_zero, inv_pos, zero_lt_bit0, zero_lt_one, and_true_iff,
+    simp only [snorm_eq_lintegral_rpow_nnnorm ENNReal.two_ne_zero ENNReal.two_ne_top, not_lt,
+      top_le_iff, [anonymous], ENNReal.one_toReal, ENNReal.rpow_two, one_div,
+      ENNReal.rpow_eq_top_iff, inv_lt_zero, inv_pos, zero_lt_bit0, zero_lt_one, and_true_iff,
       or_iff_not_imp_left, not_and_or] at hℒ
     exact hℒ fun _ => zero_le_two
 #align probability_theory.evariance_def' ProbabilityTheory.evariance_def'
@@ -286,29 +286,29 @@ theorem evariance_def' [IsProbabilityMeasure (ℙ : Measure Ω)] {X : Ω → ℝ
 theorem meas_ge_le_evariance_div_sq {X : Ω → ℝ} (hX : AeStronglyMeasurable X ℙ) {c : ℝ≥0}
     (hc : c ≠ 0) : ℙ { ω | ↑c ≤ |X ω - 𝔼[X]| } ≤ eVar[X] / c ^ 2 :=
   by
-  have A : (c : ℝ≥0∞) ≠ 0 := by rwa [Ne.def, Ennreal.coe_eq_zero]
+  have A : (c : ℝ≥0∞) ≠ 0 := by rwa [Ne.def, ENNReal.coe_eq_zero]
   have B : ae_strongly_measurable (fun ω : Ω => 𝔼[X]) ℙ := ae_strongly_measurable_const
-  convert meas_ge_le_mul_pow_snorm ℙ Ennreal.two_ne_zero Ennreal.two_ne_top (hX.sub B) A
+  convert meas_ge_le_mul_pow_snorm ℙ ENNReal.two_ne_zero ENNReal.two_ne_top (hX.sub B) A
   · ext ω
-    simp only [Pi.sub_apply, Ennreal.coe_le_coe, ← Real.norm_eq_abs, ← coe_nnnorm,
-      NNReal.coe_le_coe, Ennreal.ofReal_coe_nNReal]
-  · rw [snorm_eq_lintegral_rpow_nnnorm Ennreal.two_ne_zero Ennreal.two_ne_top]
-    simp only [Ennreal.toReal_bit0, Ennreal.one_toReal, Pi.sub_apply, one_div]
-    rw [div_eq_mul_inv, Ennreal.inv_pow, mul_comm, Ennreal.rpow_two]
+    simp only [Pi.sub_apply, ENNReal.coe_le_coe, ← Real.norm_eq_abs, ← coe_nnnorm,
+      NNReal.coe_le_coe, ENNReal.ofReal_coe_nnreal]
+  · rw [snorm_eq_lintegral_rpow_nnnorm ENNReal.two_ne_zero ENNReal.two_ne_top]
+    simp only [[anonymous], ENNReal.one_toReal, Pi.sub_apply, one_div]
+    rw [div_eq_mul_inv, ENNReal.inv_pow, mul_comm, ENNReal.rpow_two]
     congr
-    simp_rw [← Ennreal.rpow_mul, inv_mul_cancel (two_ne_zero : (2 : ℝ) ≠ 0), Ennreal.rpow_two,
-      Ennreal.rpow_one, evariance]
+    simp_rw [← ENNReal.rpow_mul, inv_mul_cancel (two_ne_zero : (2 : ℝ) ≠ 0), ENNReal.rpow_two,
+      ENNReal.rpow_one, evariance]
 #align probability_theory.meas_ge_le_evariance_div_sq ProbabilityTheory.meas_ge_le_evariance_div_sq
 
 /-- *Chebyshev's inequality* : one can control the deviation probability of a real random variable
 from its expectation in terms of the variance. -/
 theorem meas_ge_le_variance_div_sq [IsFiniteMeasure (ℙ : Measure Ω)] {X : Ω → ℝ} (hX : Memℒp X 2)
-    {c : ℝ} (hc : 0 < c) : ℙ { ω | c ≤ |X ω - 𝔼[X]| } ≤ Ennreal.ofReal (Var[X] / c ^ 2) :=
+    {c : ℝ} (hc : 0 < c) : ℙ { ω | c ≤ |X ω - 𝔼[X]| } ≤ ENNReal.ofReal (Var[X] / c ^ 2) :=
   by
-  rw [Ennreal.ofReal_div_of_pos (sq_pos_of_ne_zero _ hc.ne.symm), hX.of_real_variance_eq]
+  rw [ENNReal.ofReal_div_of_pos (sq_pos_of_ne_zero _ hc.ne.symm), hX.of_real_variance_eq]
   convert @meas_ge_le_evariance_div_sq _ _ _ hX.1 c.to_nnreal (by simp [hc])
   · simp only [Real.coe_toNNReal', max_le_iff, abs_nonneg, and_true_iff]
-  · rw [Ennreal.ofReal_pow hc.le]
+  · rw [ENNReal.ofReal_pow hc.le]
     rfl
 #align probability_theory.meas_ge_le_variance_div_sq ProbabilityTheory.meas_ge_le_variance_div_sq
 

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

A PR analogous to #12338: reformatting proofs following the multiple goals linter of #12339.

@@ -221,12 +221,12 @@ theorem variance_le_expectation_sq [@IsProbabilityMeasure Ω _ ℙ] {X : Ω →
   · rw [variance_def' hX]
     simp only [sq_nonneg, sub_le_self_iff]
   rw [variance, evariance_eq_lintegral_ofReal, ← integral_eq_lintegral_of_nonneg_ae]
-  by_cases hint : Integrable X; swap
-  · simp only [integral_undef hint, Pi.pow_apply, Pi.sub_apply, sub_zero]
-    exact le_rfl
-  · rw [integral_undef]
-    · exact integral_nonneg fun a => sq_nonneg _
-    · intro h
+  · by_cases hint : Integrable X; swap
+    · simp only [integral_undef hint, Pi.pow_apply, Pi.sub_apply, sub_zero]
+      exact le_rfl
+    · rw [integral_undef]
+      · exact integral_nonneg fun a => sq_nonneg _
+      intro h
       have A : Memℒp (X - fun ω : Ω => 𝔼[X]) 2 ℙ :=
         (memℒp_two_iff_integrable_sq (hint.aestronglyMeasurable.sub aestronglyMeasurable_const)).2 h
       have B : Memℒp (fun _ : Ω => 𝔼[X]) 2 ℙ := memℒp_const _

This matches our general policy and zpow_two_pos_of_ne_zero.

Also define sq_pos_of_ne_zero as an alias.

@@ -279,7 +279,7 @@ theorem meas_ge_le_evariance_div_sq {X : Ω → ℝ} (hX : AEStronglyMeasurable
 from its expectation in terms of the variance. -/
 theorem meas_ge_le_variance_div_sq [@IsFiniteMeasure Ω _ ℙ] {X : Ω → ℝ} (hX : Memℒp X 2) {c : ℝ}
     (hc : 0 < c) : ℙ {ω | c ≤ |X ω - 𝔼[X]|} ≤ ENNReal.ofReal (Var[X] / c ^ 2) := by
-  rw [ENNReal.ofReal_div_of_pos (sq_pos_of_ne_zero _ hc.ne.symm), hX.ofReal_variance_eq]
+  rw [ENNReal.ofReal_div_of_pos (sq_pos_of_ne_zero hc.ne.symm), hX.ofReal_variance_eq]
   convert @meas_ge_le_evariance_div_sq _ _ _ hX.1 c.toNNReal (by simp [hc]) using 1
   · simp only [Real.coe_toNNReal', max_le_iff, abs_nonneg, and_true_iff]
   · rw [ENNReal.ofReal_pow hc.le]

These are changes from #11997, the latest adaptation PR for nightly-2024-04-07, which can be made directly on master.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com>

@@ -75,11 +75,11 @@ theorem _root_.MeasureTheory.Memℒp.evariance_lt_top [IsFiniteMeasure μ] (hX :
 theorem evariance_eq_top [IsFiniteMeasure μ] (hXm : AEStronglyMeasurable X μ) (hX : ¬Memℒp X 2 μ) :
     evariance X μ = ∞ := by
   by_contra h
-  rw [← Ne.def, ← lt_top_iff_ne_top] at h
+  rw [← Ne, ← lt_top_iff_ne_top] at h
   have : Memℒp (fun ω => X ω - μ[X]) 2 μ := by
     refine' ⟨hXm.sub aestronglyMeasurable_const, _⟩
     rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top]
-    simp only [coe_two, ENNReal.one_toReal, ENNReal.rpow_two, Ne.def]
+    simp only [coe_two, ENNReal.one_toReal, ENNReal.rpow_two, Ne]
     exact ENNReal.rpow_lt_top_of_nonneg (by linarith) h.ne
   refine' hX _
   -- Porting note: `μ[X]` without whitespace is ambiguous as it could be GetElem,

@@ -260,7 +260,7 @@ theorem evariance_def' [@IsProbabilityMeasure Ω _ ℙ] {X : Ω → ℝ} (hX : A
 /-- **Chebyshev's inequality** for `ℝ≥0∞`-valued variance. -/
 theorem meas_ge_le_evariance_div_sq {X : Ω → ℝ} (hX : AEStronglyMeasurable X ℙ) {c : ℝ≥0}
     (hc : c ≠ 0) : ℙ {ω | ↑c ≤ |X ω - 𝔼[X]|} ≤ eVar[X] / c ^ 2 := by
-  have A : (c : ℝ≥0∞) ≠ 0 := by rwa [Ne.def, ENNReal.coe_eq_zero]
+  have A : (c : ℝ≥0∞) ≠ 0 := by rwa [Ne, ENNReal.coe_eq_zero]
   have B : AEStronglyMeasurable (fun _ : Ω => 𝔼[X]) ℙ := aestronglyMeasurable_const
   convert meas_ge_le_mul_pow_snorm ℙ two_ne_zero ENNReal.two_ne_top (hX.sub B) A using 1
   · congr

Inspired by #10538 add a positivity extension for Bochner integrals.

@@ -116,10 +116,9 @@ theorem evariance_eq_lintegral_ofReal (X : Ω → ℝ) (μ : Measure Ω) :
 theorem _root_.MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero (hX : Memℒp X 2 μ)
     (hXint : μ[X] = 0) : variance X μ = μ[X ^ (2 : Nat)] := by
   rw [variance, evariance_eq_lintegral_ofReal, ← ofReal_integral_eq_lintegral_ofReal,
-      ENNReal.toReal_ofReal] <;>
+      ENNReal.toReal_ofReal (by positivity)] <;>
     simp_rw [hXint, sub_zero]
   · rfl
-  · exact integral_nonneg fun ω => pow_two_nonneg _
   · convert hX.integrable_norm_rpow two_ne_zero ENNReal.two_ne_top with ω
     simp only [Pi.sub_apply, Real.norm_eq_abs, coe_two, ENNReal.one_toReal,
       Real.rpow_two, sq_abs, abs_pow]
@@ -129,9 +128,8 @@ theorem _root_.MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero (hX : Mem
 theorem _root_.MeasureTheory.Memℒp.variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
     variance X μ = μ[(X - fun _ => μ[X] :) ^ (2 : Nat)] := by
   rw [variance, evariance_eq_lintegral_ofReal, ← ofReal_integral_eq_lintegral_ofReal,
-    ENNReal.toReal_ofReal]
+    ENNReal.toReal_ofReal (by positivity)]
   · rfl
-  · exact integral_nonneg fun ω => pow_two_nonneg _
   · -- Porting note: `μ[X]` without whitespace is ambiguous as it could be GetElem,
     -- and `convert` cannot disambiguate based on typeclass inference failure.
     convert (hX.sub <| memℒp_const (μ [X])).integrable_norm_rpow two_ne_zero ENNReal.two_ne_top

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Joachim Breitner <mail@joachim-breitner.de>

@@ -364,7 +364,7 @@ theorem IndepFun.variance_sum [@IsProbabilityMeasure Ω _ ℙ] {ι : Type*} {X :
           Memℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi)),
         mul_sum, mul_sum, ← sum_sub_distrib]
       apply Finset.sum_eq_zero fun i hi => ?_
-      have : ∀ (a : Ω), @OfNat.ofNat (Ω → ℝ) 2 instOfNat a = (2 : ℝ) := fun a => rfl
+      have : ∀ (a : Ω), @OfNat.ofNat (Ω → ℝ) 2 instOfNatAtLeastTwo a = (2 : ℝ) := fun a => rfl
       conv_lhs => enter [1, 2, a]; rw [this]
       rw [integral_mul_left, IndepFun.integral_mul', sub_self]
       · apply h (mem_insert_self _ _) (mem_insert_of_mem hi)

This involves moving lemmas from Algebra.GroupPower.Ring to Algebra.GroupWithZero.Basic and changing some 0 < n assumptions to n ≠ 0.

From LeanAPAP

@@ -149,9 +149,8 @@ theorem evariance_eq_zero_iff (hX : AEMeasurable X μ) :
     evariance X μ = 0 ↔ X =ᵐ[μ] fun _ => μ[X] := by
   rw [evariance, lintegral_eq_zero_iff']
   constructor <;> intro hX <;> filter_upwards [hX] with ω hω
-  · simp only [Pi.zero_apply, pow_eq_zero_iff, Nat.succ_pos', ENNReal.coe_eq_zero, nnnorm_eq_zero,
-      sub_eq_zero] at hω
-    exact hω
+  · simpa only [Pi.zero_apply, sq_eq_zero_iff, ENNReal.coe_eq_zero, nnnorm_eq_zero, sub_eq_zero]
+      using hω
   · rw [hω]
     simp
   · measurability

@@ -260,7 +260,7 @@ theorem evariance_def' [@IsProbabilityMeasure Ω _ ℙ] {X : Ω → ℝ} (hX : A
     exact mod_cast hℒ fun _ => zero_le_two
 #align probability_theory.evariance_def' ProbabilityTheory.evariance_def'
 
-/-- *Chebyshev's inequality* for `ℝ≥0∞`-valued variance. -/
+/-- **Chebyshev's inequality** for `ℝ≥0∞`-valued variance. -/
 theorem meas_ge_le_evariance_div_sq {X : Ω → ℝ} (hX : AEStronglyMeasurable X ℙ) {c : ℝ≥0}
     (hc : c ≠ 0) : ℙ {ω | ↑c ≤ |X ω - 𝔼[X]|} ≤ eVar[X] / c ^ 2 := by
   have A : (c : ℝ≥0∞) ≠ 0 := by rwa [Ne.def, ENNReal.coe_eq_zero]
@@ -278,7 +278,7 @@ theorem meas_ge_le_evariance_div_sq {X : Ω → ℝ} (hX : AEStronglyMeasurable
       ENNReal.rpow_one, evariance]
 #align probability_theory.meas_ge_le_evariance_div_sq ProbabilityTheory.meas_ge_le_evariance_div_sq
 
-/-- *Chebyshev's inequality* : one can control the deviation probability of a real random variable
+/-- **Chebyshev's inequality**: one can control the deviation probability of a real random variable
 from its expectation in terms of the variance. -/
 theorem meas_ge_le_variance_div_sq [@IsFiniteMeasure Ω _ ℙ] {X : Ω → ℝ} (hX : Memℒp X 2) {c : ℝ}
     (hc : 0 < c) : ℙ {ω | c ≤ |X ω - 𝔼[X]|} ≤ ENNReal.ofReal (Var[X] / c ^ 2) := by

We still have the exact_mod_cast tactic, used in a few places, which somehow (?) works a little bit harder to prevent the expected type influencing the elaboration of the term. I would like to get to the bottom of this, and it will be easier once the only usages of exact_mod_cast are the ones that don't work using the term elaborator by itself.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -257,7 +257,7 @@ theorem evariance_def' [@IsProbabilityMeasure Ω _ ℙ] {X : Ω → ℝ} (hX : A
     simp only [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, not_lt, top_le_iff,
       coe_two, one_div, ENNReal.rpow_eq_top_iff, inv_lt_zero, inv_pos, and_true_iff,
       or_iff_not_imp_left, not_and_or, zero_lt_two] at hℒ
-    exact_mod_cast hℒ fun _ => zero_le_two
+    exact mod_cast hℒ fun _ => zero_le_two
 #align probability_theory.evariance_def' ProbabilityTheory.evariance_def'
 
 /-- *Chebyshev's inequality* for `ℝ≥0∞`-valued variance. -/

PR contents

This is the supremum of

• #8284
• #8056
• #8023
• #8332
• #8226 (already approved)
• #7834 (already approved)

along with some minor fixes from failures on nightly-testing as Mathlib master is merged into it.

Note that some PRs for changes that are already compatible with the current toolchain and will be necessary have already been split out: #8380.

I am hopeful that in future we will be able to progressively merge adaptation PRs into a bump/v4.X.0 branch, so we never end up with a "big merge" like this. However one of these adaptation PRs (#8056) predates my new scheme for combined CI, and it wasn't possible to keep that PR viable in the meantime.

Lean PRs involved in this bump

In particular this includes adjustments for the Lean PRs

• leanprover/lean4#2778
• leanprover/lean4#2790
• leanprover/lean4#2783
• leanprover/lean4#2825
• leanprover/lean4#2722

leanprover/lean4#2778

We can get rid of all the

local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue [lean4#2220](https://github.com/leanprover/lean4/pull/2220)

macros across Mathlib (and in any projects that want to write natural number powers of reals).

leanprover/lean4#2722

Changes the default behaviour of simp to (config := {decide := false}). This makes simp (and consequentially norm_num) less powerful, but also more consistent, and less likely to blow up in long failures. This requires a variety of changes: changing some previously by simp or norm_num to decide or rfl, or adding (config := {decide := true}).

leanprover/lean4#2783

This changed the behaviour of simp so that simp [f] will only unfold "fully applied" occurrences of f. The old behaviour can be recovered with simp (config := { unfoldPartialApp := true }). We may in future add a syntax for this, e.g. simp [!f]; please provide feedback! In the meantime, we have made the following changes:

• switching to using explicit lemmas that have the intended level of application
• (config := { unfoldPartialApp := true }) in some places, to recover the old behaviour
• Using @[eqns] to manually adjust the equation lemmas for a particular definition, recovering the old behaviour just for that definition. See #8371, where we do this for Function.comp and Function.flip.

This change in Lean may require further changes down the line (e.g. adding the !f syntax, and/or upstreaming the special treatment for Function.comp and Function.flip, and/or removing this special treatment). Please keep an open and skeptical mind about these changes!

Co-authored-by: leanprover-community-mathlib4-bot <leanprover-community-mathlib4-bot@users.noreply.github.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Mauricio Collares <mauricio@collares.org>

@@ -43,8 +43,6 @@ open scoped BigOperators MeasureTheory ProbabilityTheory ENNReal NNReal
 
 namespace ProbabilityTheory
 
-local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue lean4#2220
-
 -- Porting note: this lemma replaces `ENNReal.toReal_bit0`, which does not exist in Lean 4
 private lemma coe_two : ENNReal.toReal 2 = (2 : ℝ) := rfl
 
@@ -116,7 +114,7 @@ theorem evariance_eq_lintegral_ofReal (X : Ω → ℝ) (μ : Measure Ω) :
 #align probability_theory.evariance_eq_lintegral_of_real ProbabilityTheory.evariance_eq_lintegral_ofReal
 
 theorem _root_.MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero (hX : Memℒp X 2 μ)
-    (hXint : μ[X] = 0) : variance X μ = μ[X ^ 2] := by
+    (hXint : μ[X] = 0) : variance X μ = μ[X ^ (2 : Nat)] := by
   rw [variance, evariance_eq_lintegral_ofReal, ← ofReal_integral_eq_lintegral_ofReal,
       ENNReal.toReal_ofReal] <;>
     simp_rw [hXint, sub_zero]
@@ -129,7 +127,7 @@ theorem _root_.MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero (hX : Mem
 #align measure_theory.mem_ℒp.variance_eq_of_integral_eq_zero MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero
 
 theorem _root_.MeasureTheory.Memℒp.variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
-    variance X μ = μ[(X - fun _ => μ[X]) ^ 2] := by
+    variance X μ = μ[(X - fun _ => μ[X] :) ^ (2 : Nat)] := by
   rw [variance, evariance_eq_lintegral_ofReal, ← ofReal_integral_eq_lintegral_ofReal,
     ENNReal.toReal_ofReal]
   · rfl
@@ -243,7 +241,7 @@ theorem variance_le_expectation_sq [@IsProbabilityMeasure Ω _ ℙ] {X : Ω →
 #align probability_theory.variance_le_expectation_sq ProbabilityTheory.variance_le_expectation_sq
 
 theorem evariance_def' [@IsProbabilityMeasure Ω _ ℙ] {X : Ω → ℝ} (hX : AEStronglyMeasurable X ℙ) :
-    eVar[X] = (∫⁻ ω, ‖X ω‖₊ ^ 2) - ENNReal.ofReal (𝔼[X] ^ 2) := by
+    eVar[X] = (∫⁻ ω, (‖X ω‖₊ ^ 2 :)) - ENNReal.ofReal (𝔼[X] ^ 2) := by
   by_cases hℒ : Memℒp X 2
   · rw [← hℒ.ofReal_variance_eq, variance_def' hℒ, ENNReal.ofReal_sub _ (sq_nonneg _)]
     congr
@@ -287,7 +285,7 @@ theorem meas_ge_le_variance_div_sq [@IsFiniteMeasure Ω _ ℙ] {X : Ω → ℝ}
   rw [ENNReal.ofReal_div_of_pos (sq_pos_of_ne_zero _ hc.ne.symm), hX.ofReal_variance_eq]
   convert @meas_ge_le_evariance_div_sq _ _ _ hX.1 c.toNNReal (by simp [hc]) using 1
   · simp only [Real.coe_toNNReal', max_le_iff, abs_nonneg, and_true_iff]
-  · rw [ENNReal.ofReal_pow hc.le, ENNReal.coe_pow]
+  · rw [ENNReal.ofReal_pow hc.le]
     rfl
 #align probability_theory.meas_ge_le_variance_div_sq ProbabilityTheory.meas_ge_le_variance_div_sq
 

The some name feels like an implementation detail.

@@ -272,8 +272,8 @@ theorem meas_ge_le_evariance_div_sq {X : Ω → ℝ} (hX : AEStronglyMeasurable
     simp only [Pi.sub_apply, ENNReal.coe_le_coe, ← Real.norm_eq_abs, ← coe_nnnorm,
       NNReal.coe_le_coe, ENNReal.ofReal_coe_nnreal]
   · rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top]
-    simp only [show ENNReal.some (c ^ 2) = (ENNReal.some c) ^ 2 by norm_cast, coe_two, one_div,
-      Pi.sub_apply]
+    simp only [show ENNReal.ofNNReal (c ^ 2) = (ENNReal.ofNNReal c) ^ 2 by norm_cast, coe_two,
+      one_div, Pi.sub_apply]
     rw [div_eq_mul_inv, ENNReal.inv_pow, mul_comm, ENNReal.rpow_two]
     congr
     simp_rw [← ENNReal.rpow_mul, inv_mul_cancel (two_ne_zero : (2 : ℝ) ≠ 0), ENNReal.rpow_two,

Removes nonterminal simps on lines looking like simp [...]

@@ -215,7 +215,8 @@ theorem variance_def' [@IsProbabilityMeasure Ω _ ℙ] {X : Ω → ℝ} (hX : Me
   · apply hX.integrable_sq.add
     convert @integrable_const Ω ℝ (_) ℙ _ _ (𝔼[X] ^ 2)
   · exact ((hX.integrable one_le_two).const_mul 2).mul_const' _
-  simp [integral_mul_right, integral_mul_left]
+  simp only [Pi.pow_apply, integral_const, measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul,
+    Pi.mul_apply, Pi.ofNat_apply, Nat.cast_ofNat, integral_mul_right, integral_mul_left]
   ring
 #align probability_theory.variance_def' ProbabilityTheory.variance_def'
 

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

@@ -246,7 +246,6 @@ theorem evariance_def' [@IsProbabilityMeasure Ω _ ℙ] {X : Ω → ℝ} (hX : A
   by_cases hℒ : Memℒp X 2
   · rw [← hℒ.ofReal_variance_eq, variance_def' hℒ, ENNReal.ofReal_sub _ (sq_nonneg _)]
     congr
-    simp_rw [← ENNReal.coe_pow]
     rw [lintegral_coe_eq_integral]
     · congr 2 with ω
       simp only [Pi.pow_apply, NNReal.coe_pow, coe_nnnorm, Real.norm_eq_abs, Even.pow_abs even_two]

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

This has nice performance benefits.

@@ -52,17 +52,17 @@ private lemma coe_two : ENNReal.toReal 2 = (2 : ℝ) := rfl
 -- consider `eVariationOn` in `Mathlib.Analysis.BoundedVariation`.
 /-- The `ℝ≥0∞`-valued variance of a real-valued random variable defined as the Lebesgue integral of
 `(X - 𝔼[X])^2`. -/
-def evariance {Ω : Type _} {_ : MeasurableSpace Ω} (X : Ω → ℝ) (μ : Measure Ω) : ℝ≥0∞ :=
+def evariance {Ω : Type*} {_ : MeasurableSpace Ω} (X : Ω → ℝ) (μ : Measure Ω) : ℝ≥0∞ :=
   ∫⁻ ω, (‖X ω - μ[X]‖₊ : ℝ≥0∞) ^ 2 ∂μ
 #align probability_theory.evariance ProbabilityTheory.evariance
 
 /-- The `ℝ`-valued variance of a real-valued random variable defined by applying `ENNReal.toReal`
 to `evariance`. -/
-def variance {Ω : Type _} {_ : MeasurableSpace Ω} (X : Ω → ℝ) (μ : Measure Ω) : ℝ :=
+def variance {Ω : Type*} {_ : MeasurableSpace Ω} (X : Ω → ℝ) (μ : Measure Ω) : ℝ :=
   (evariance X μ).toReal
 #align probability_theory.variance ProbabilityTheory.variance
 
-variable {Ω : Type _} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : Measure Ω}
+variable {Ω : Type*} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : Measure Ω}
 
 theorem _root_.MeasureTheory.Memℒp.evariance_lt_top [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
     evariance X μ < ∞ := by
@@ -196,7 +196,7 @@ theorem variance_smul (c : ℝ) (X : Ω → ℝ) (μ : Measure Ω) :
   variance_mul c X μ
 #align probability_theory.variance_smul ProbabilityTheory.variance_smul
 
-theorem variance_smul' {A : Type _} [CommSemiring A] [Algebra A ℝ] (c : A) (X : Ω → ℝ)
+theorem variance_smul' {A : Type*} [CommSemiring A] [Algebra A ℝ] (c : A) (X : Ω → ℝ)
     (μ : Measure Ω) : variance (c • X) μ = c ^ 2 • variance X μ := by
   convert variance_smul (algebraMap A ℝ c) X μ using 1
   · congr; simp only [algebraMap_smul]
@@ -317,7 +317,7 @@ theorem IndepFun.variance_add [@IsProbabilityMeasure Ω _ ℙ] {X Y : Ω → ℝ
 -- Porting note: supplied `MeasurableSpace Ω` argument of `hs`, `h` by unification
 /-- The variance of a finite sum of pairwise independent random variables is the sum of the
 variances. -/
-theorem IndepFun.variance_sum [@IsProbabilityMeasure Ω _ ℙ] {ι : Type _} {X : ι → Ω → ℝ}
+theorem IndepFun.variance_sum [@IsProbabilityMeasure Ω _ ℙ] {ι : Type*} {X : ι → Ω → ℝ}
     {s : Finset ι} (hs : ∀ i ∈ s, @Memℒp _ _ _ (_) (X i) 2 ℙ)
     (h : Set.Pairwise ↑s fun i j => @IndepFun _ _ _ (_) _ _ (X i) (X j) ℙ) :
     Var[∑ i in s, X i] = ∑ i in s, Var[X i] := by

@@ -43,7 +43,7 @@ open scoped BigOperators MeasureTheory ProbabilityTheory ENNReal NNReal
 
 namespace ProbabilityTheory
 
-local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue #2220
+local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue lean4#2220
 
 -- Porting note: this lemma replaces `ENNReal.toReal_bit0`, which does not exist in Lean 4
 private lemma coe_two : ENNReal.toReal 2 = (2 : ℝ) := rfl

Changes the way the term is elaborated so that typeclass inference is tolerated and so that instances are allowed to come from the goal without re-inferring them.

@@ -84,7 +84,9 @@ theorem evariance_eq_top [IsFiniteMeasure μ] (hXm : AEStronglyMeasurable X μ)
     simp only [coe_two, ENNReal.one_toReal, ENNReal.rpow_two, Ne.def]
     exact ENNReal.rpow_lt_top_of_nonneg (by linarith) h.ne
   refine' hX _
-  convert this.add (memℒp_const <| μ[X])
+  -- Porting note: `μ[X]` without whitespace is ambiguous as it could be GetElem,
+  -- and `convert` cannot disambiguate based on typeclass inference failure.
+  convert this.add (memℒp_const <| μ [X])
   ext ω
   rw [Pi.add_apply, sub_add_cancel]
 #align probability_theory.evariance_eq_top ProbabilityTheory.evariance_eq_top
@@ -132,7 +134,9 @@ theorem _root_.MeasureTheory.Memℒp.variance_eq [IsFiniteMeasure μ] (hX : Mem
     ENNReal.toReal_ofReal]
   · rfl
   · exact integral_nonneg fun ω => pow_two_nonneg _
-  · convert (hX.sub <| memℒp_const (μ[X])).integrable_norm_rpow two_ne_zero ENNReal.two_ne_top
+  · -- Porting note: `μ[X]` without whitespace is ambiguous as it could be GetElem,
+    -- and `convert` cannot disambiguate based on typeclass inference failure.
+    convert (hX.sub <| memℒp_const (μ [X])).integrable_norm_rpow two_ne_zero ENNReal.two_ne_top
       with ω
     simp only [Pi.sub_apply, Real.norm_eq_abs, coe_two, ENNReal.one_toReal,
       Real.rpow_two, sq_abs, abs_pow]

• depends on: #5966

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

@@ -2,16 +2,13 @@
 Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Kexing Ying
-
-! This file was ported from Lean 3 source module probability.variance
-! leanprover-community/mathlib commit f0c8bf9245297a541f468be517f1bde6195105e9
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Probability.Notation
 import Mathlib.Probability.Integration
 import Mathlib.MeasureTheory.Function.L2Space
 
+#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
+
 /-!
 # Variance of random variables
 

@@ -214,11 +214,8 @@ theorem variance_def' [@IsProbabilityMeasure Ω _ ℙ] {X : Ω → ℝ} (hX : Me
   · apply hX.integrable_sq.add
     convert @integrable_const Ω ℝ (_) ℙ _ _ (𝔼[X] ^ 2)
   · exact ((hX.integrable one_le_two).const_mul 2).mul_const' _
-  simp [integral_mul_right]
-  have : ∀ (a : Ω), @OfNat.ofNat (Ω → ℝ) 2 instOfNat a = (2 : ℝ) := fun a => rfl
-  conv_lhs => enter [2, 1, 2, a]; rw [this]
-  simp only [integral_mul_left (2 : ℝ)]
-  ring_nf
+  simp [integral_mul_right, integral_mul_left]
+  ring
 #align probability_theory.variance_def' ProbabilityTheory.variance_def'
 
 theorem variance_le_expectation_sq [@IsProbabilityMeasure Ω _ ℙ] {X : Ω → ℝ}

@@ -20,7 +20,7 @@ We define the variance of a real-valued random variable as `Var[X] = 𝔼[(X - 
 
 ## Main definitions
 
-* `ProbabilityTheory.evariance`: the variance of a real-valued random variable as a extended
+* `ProbabilityTheory.evariance`: the variance of a real-valued random variable as an extended
   non-negative real.
 * `ProbabilityTheory.variance`: the variance of a real-valued random variable as a real number.
 

@@ -123,8 +123,7 @@ theorem _root_.MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero (hX : Mem
     simp_rw [hXint, sub_zero]
   · rfl
   · exact integral_nonneg fun ω => pow_two_nonneg _
-  · convert hX.integrable_norm_rpow two_ne_zero ENNReal.two_ne_top
-    rename_i ω
+  · convert hX.integrable_norm_rpow two_ne_zero ENNReal.two_ne_top with ω
     simp only [Pi.sub_apply, Real.norm_eq_abs, coe_two, ENNReal.one_toReal,
       Real.rpow_two, sq_abs, abs_pow]
   · exact ae_of_all _ fun ω => pow_two_nonneg _
@@ -137,7 +136,7 @@ theorem _root_.MeasureTheory.Memℒp.variance_eq [IsFiniteMeasure μ] (hX : Mem
   · rfl
   · exact integral_nonneg fun ω => pow_two_nonneg _
   · convert (hX.sub <| memℒp_const (μ[X])).integrable_norm_rpow two_ne_zero ENNReal.two_ne_top
-    rename_i ω
+      with ω
     simp only [Pi.sub_apply, Real.norm_eq_abs, coe_two, ENNReal.one_toReal,
       Real.rpow_two, sq_abs, abs_pow]
   · exact ae_of_all _ fun ω => pow_two_nonneg _

Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>