measure_theory.decomposition.radon_nikodym

Changes since the initial port

The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.

Changes in mathlib3

fc758559

(last sync)

Changes in mathlib3port

mathlib3

mathlib3port

7e5137f5

bump to nightly-2024-04-19-08

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

a9e81450

Diff

@@ -77,8 +77,8 @@ theorem absolutelyContinuous_iff_withDensity_rnDeriv_eq {μ ν : Measure α}
 #align measure_theory.measure.absolutely_continuous_iff_with_density_rn_deriv_eq MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq
 -/
 
-#print MeasureTheory.Measure.set_integral_toReal_rnDeriv' /-
-theorem set_integral_toReal_rnDeriv' {μ ν : Measure α} [IsFiniteMeasure μ]
+#print MeasureTheory.Measure.setIntegral_toReal_rnDeriv' /-
+theorem setIntegral_toReal_rnDeriv' {μ ν : Measure α} [IsFiniteMeasure μ]
     [HaveLebesgueDecomposition μ ν] (h : μ ≪ ν) {i : Set α} (hi : MeasurableSet i) :
     ∫ x in i, (μ.rnDeriv ν x).toReal ∂ν = (μ i).toReal :=
   by
@@ -88,7 +88,7 @@ theorem set_integral_toReal_rnDeriv' {μ ν : Measure α} [IsFiniteMeasure μ]
       ae_lt_top (μ.measurable_rn_deriv ν) (lt_of_le_of_lt (lintegral_mono_set i.subset_univ) _).Ne
     rw [← with_density_apply _ MeasurableSet.univ, with_density_rn_deriv_eq μ ν h]
     exact measure_lt_top _ _
-#align measure_theory.measure.with_density_rn_deriv_to_real_eq MeasureTheory.Measure.set_integral_toReal_rnDeriv'
+#align measure_theory.measure.with_density_rn_deriv_to_real_eq MeasureTheory.Measure.setIntegral_toReal_rnDeriv'
 -/
 
 end Measure

7e5137f5

bump to nightly-2024-03-17-08

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

22f01ce4

Diff

@@ -60,9 +60,9 @@ theorem withDensity_rnDeriv_eq (μ ν : Measure α) [HaveLebesgueDecomposition 
       exact measure_mono (Set.subset_univ _)
     rw [← measure_add_measure_compl hE₁, hE₂, zero_add]
     have : (singular_part μ ν + ν.with_density (rn_deriv μ ν)) (Eᶜ) = μ (Eᶜ) := by rw [← hadd]
-    rw [measure.coe_add, Pi.add_apply, h hE₃] at this 
+    rw [measure.coe_add, Pi.add_apply, h hE₃] at this
     exact (add_eq_zero_iff.1 this).1
-  rw [this, zero_add] at hadd 
+  rw [this, zero_add] at hadd
   exact hadd.symm
 #align measure_theory.measure.with_density_rn_deriv_eq MeasureTheory.Measure.withDensity_rnDeriv_eq
 -/
@@ -102,7 +102,7 @@ theorem withDensityᵥ_rnDeriv_eq (s : SignedMeasure α) (μ : Measure α) [Sigm
     (h : s ≪ᵥ μ.toENNRealVectorMeasure) : μ.withDensityᵥ (s.rnDeriv μ) = s :=
   by
   rw [absolutely_continuous_ennreal_iff, (_ : μ.to_ennreal_vector_measure.ennreal_to_measure = μ),
-    total_variation_absolutely_continuous_iff] at h 
+    total_variation_absolutely_continuous_iff] at h
   · ext1 i hi
     rw [with_densityᵥ_apply (integrable_rn_deriv _ _) hi, rn_deriv, integral_sub,
       with_density_rn_deriv_to_real_eq h.1 hi, with_density_rn_deriv_to_real_eq h.2 hi]

7e5137f5

bump to nightly-2023-11-29-11

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

049e2cad

Diff

@@ -77,8 +77,8 @@ theorem absolutelyContinuous_iff_withDensity_rnDeriv_eq {μ ν : Measure α}
 #align measure_theory.measure.absolutely_continuous_iff_with_density_rn_deriv_eq MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq
 -/
 
-#print MeasureTheory.Measure.set_integral_toReal_rnDeriv /-
-theorem set_integral_toReal_rnDeriv {μ ν : Measure α} [IsFiniteMeasure μ]
+#print MeasureTheory.Measure.set_integral_toReal_rnDeriv' /-
+theorem set_integral_toReal_rnDeriv' {μ ν : Measure α} [IsFiniteMeasure μ]
     [HaveLebesgueDecomposition μ ν] (h : μ ≪ ν) {i : Set α} (hi : MeasurableSet i) :
     ∫ x in i, (μ.rnDeriv ν x).toReal ∂ν = (μ i).toReal :=
   by
@@ -88,7 +88,7 @@ theorem set_integral_toReal_rnDeriv {μ ν : Measure α} [IsFiniteMeasure μ]
       ae_lt_top (μ.measurable_rn_deriv ν) (lt_of_le_of_lt (lintegral_mono_set i.subset_univ) _).Ne
     rw [← with_density_apply _ MeasurableSet.univ, with_density_rn_deriv_eq μ ν h]
     exact measure_lt_top _ _
-#align measure_theory.measure.with_density_rn_deriv_to_real_eq MeasureTheory.Measure.set_integral_toReal_rnDeriv
+#align measure_theory.measure.with_density_rn_deriv_to_real_eq MeasureTheory.Measure.set_integral_toReal_rnDeriv'
 -/
 
 end Measure

7e5137f5

bump to nightly-2023-11-15-06

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

38c8ceef

Diff

@@ -77,8 +77,8 @@ theorem absolutelyContinuous_iff_withDensity_rnDeriv_eq {μ ν : Measure α}
 #align measure_theory.measure.absolutely_continuous_iff_with_density_rn_deriv_eq MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq
 -/
 
-#print MeasureTheory.Measure.withDensity_rnDeriv_toReal_eq /-
-theorem withDensity_rnDeriv_toReal_eq {μ ν : Measure α} [IsFiniteMeasure μ]
+#print MeasureTheory.Measure.set_integral_toReal_rnDeriv /-
+theorem set_integral_toReal_rnDeriv {μ ν : Measure α} [IsFiniteMeasure μ]
     [HaveLebesgueDecomposition μ ν] (h : μ ≪ ν) {i : Set α} (hi : MeasurableSet i) :
     ∫ x in i, (μ.rnDeriv ν x).toReal ∂ν = (μ i).toReal :=
   by
@@ -88,7 +88,7 @@ theorem withDensity_rnDeriv_toReal_eq {μ ν : Measure α} [IsFiniteMeasure μ]
       ae_lt_top (μ.measurable_rn_deriv ν) (lt_of_le_of_lt (lintegral_mono_set i.subset_univ) _).Ne
     rw [← with_density_apply _ MeasurableSet.univ, with_density_rn_deriv_eq μ ν h]
     exact measure_lt_top _ _
-#align measure_theory.measure.with_density_rn_deriv_to_real_eq MeasureTheory.Measure.withDensity_rnDeriv_toReal_eq
+#align measure_theory.measure.with_density_rn_deriv_to_real_eq MeasureTheory.Measure.set_integral_toReal_rnDeriv
 -/
 
 end Measure

7e5137f5

bump to nightly-2023-09-24-06

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

5655435f

Diff

@@ -3,7 +3,7 @@ Copyright (c) 2021 Kexing Ying. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kexing Ying
 -/
-import Mathbin.MeasureTheory.Decomposition.Lebesgue
+import MeasureTheory.Decomposition.Lebesgue
 
 #align_import measure_theory.decomposition.radon_nikodym from "leanprover-community/mathlib"@"7e5137f579de09a059a5ce98f364a04e221aabf0"

7e5137f5

bump to nightly-2023-07-20-09

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

9c56d0dd

Diff

@@ -2,14 +2,11 @@
 Copyright (c) 2021 Kexing Ying. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kexing Ying
-
-! This file was ported from Lean 3 source module measure_theory.decomposition.radon_nikodym
-! leanprover-community/mathlib commit 7e5137f579de09a059a5ce98f364a04e221aabf0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.MeasureTheory.Decomposition.Lebesgue
 
+#align_import measure_theory.decomposition.radon_nikodym from "leanprover-community/mathlib"@"7e5137f579de09a059a5ce98f364a04e221aabf0"
+
 /-!
 # Radon-Nikodym theorem

7e5137f5

bump to nightly-2023-06-13-21

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

829520ac

Diff

@@ -49,8 +49,6 @@ namespace MeasureTheory
 
 namespace Measure
 
-include m
-
 #print MeasureTheory.Measure.withDensity_rnDeriv_eq /-
 theorem withDensity_rnDeriv_eq (μ ν : Measure α) [HaveLebesgueDecomposition μ ν] (h : μ ≪ ν) :
     ν.withDensity (rnDeriv μ ν) = μ :=
@@ -100,10 +98,9 @@ end Measure
 
 namespace SignedMeasure
 
-include m
-
 open Measure VectorMeasure
 
+#print MeasureTheory.SignedMeasure.withDensityᵥ_rnDeriv_eq /-
 theorem withDensityᵥ_rnDeriv_eq (s : SignedMeasure α) (μ : Measure α) [SigmaFinite μ]
     (h : s ≪ᵥ μ.toENNRealVectorMeasure) : μ.withDensityᵥ (s.rnDeriv μ) = s :=
   by
@@ -125,12 +122,15 @@ theorem withDensityᵥ_rnDeriv_eq (s : SignedMeasure α) (μ : Measure α) [Sigm
         exact (lintegral_rn_deriv_lt_top _ _).Ne
   · exact equiv_measure.right_inv μ
 #align measure_theory.signed_measure.with_densityᵥ_rn_deriv_eq MeasureTheory.SignedMeasure.withDensityᵥ_rnDeriv_eq
+-/
 
+#print MeasureTheory.SignedMeasure.absolutelyContinuous_iff_withDensityᵥ_rnDeriv_eq /-
 /-- The Radon-Nikodym theorem for signed measures. -/
 theorem absolutelyContinuous_iff_withDensityᵥ_rnDeriv_eq (s : SignedMeasure α) (μ : Measure α)
     [SigmaFinite μ] : s ≪ᵥ μ.toENNRealVectorMeasure ↔ μ.withDensityᵥ (s.rnDeriv μ) = s :=
   ⟨withDensityᵥ_rnDeriv_eq s μ, fun h => h ▸ withDensityᵥ_absolutelyContinuous _ _⟩
 #align measure_theory.signed_measure.absolutely_continuous_iff_with_densityᵥ_rn_deriv_eq MeasureTheory.SignedMeasure.absolutelyContinuous_iff_withDensityᵥ_rnDeriv_eq
+-/
 
 end SignedMeasure

7e5137f5

bump to nightly-2023-06-10-07

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

3fdc57bc

Diff

@@ -85,7 +85,7 @@ theorem absolutelyContinuous_iff_withDensity_rnDeriv_eq {μ ν : Measure α}
 #print MeasureTheory.Measure.withDensity_rnDeriv_toReal_eq /-
 theorem withDensity_rnDeriv_toReal_eq {μ ν : Measure α} [IsFiniteMeasure μ]
     [HaveLebesgueDecomposition μ ν] (h : μ ≪ ν) {i : Set α} (hi : MeasurableSet i) :
-    (∫ x in i, (μ.rnDeriv ν x).toReal ∂ν) = (μ i).toReal :=
+    ∫ x in i, (μ.rnDeriv ν x).toReal ∂ν = (μ i).toReal :=
   by
   rw [integral_to_real, ← with_density_apply _ hi, with_density_rn_deriv_eq μ ν h]
   · measurability

7e5137f5

bump to nightly-2023-06-09-03

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

fe176e9b

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kexing Ying
 
 ! This file was ported from Lean 3 source module measure_theory.decomposition.radon_nikodym
-! leanprover-community/mathlib commit fc75855907eaa8ff39791039710f567f37d4556f
+! leanprover-community/mathlib commit 7e5137f579de09a059a5ce98f364a04e221aabf0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -13,6 +13,9 @@ import Mathbin.MeasureTheory.Decomposition.Lebesgue
 /-!
 # Radon-Nikodym theorem
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file proves the Radon-Nikodym theorem. The Radon-Nikodym theorem states that, given measures
 `μ, ν`, if `have_lebesgue_decomposition μ ν`, then `μ` is absolutely continuous with respect to
 `ν` if and only if there exists a measurable function `f : α → ℝ≥0∞` such that `μ = fν`.

fc758559

bump to nightly-2023-06-07-09

mathlib commit https://github.com/leanprover-community/mathlib/commit/58a272265b5e05f258161260dd2c5d247213cbd3

13516651

Diff

@@ -48,6 +48,7 @@ namespace Measure
 
 include m
 
+#print MeasureTheory.Measure.withDensity_rnDeriv_eq /-
 theorem withDensity_rnDeriv_eq (μ ν : Measure α) [HaveLebesgueDecomposition μ ν] (h : μ ≪ ν) :
     ν.withDensity (rnDeriv μ ν) = μ :=
   by
@@ -66,7 +67,9 @@ theorem withDensity_rnDeriv_eq (μ ν : Measure α) [HaveLebesgueDecomposition 
   rw [this, zero_add] at hadd 
   exact hadd.symm
 #align measure_theory.measure.with_density_rn_deriv_eq MeasureTheory.Measure.withDensity_rnDeriv_eq
+-/
 
+#print MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq /-
 /-- **The Radon-Nikodym theorem**: Given two measures `μ` and `ν`, if
 `have_lebesgue_decomposition μ ν`, then `μ` is absolutely continuous to `ν` if and only if
 `ν.with_density (rn_deriv μ ν) = μ`. -/
@@ -74,8 +77,10 @@ theorem absolutelyContinuous_iff_withDensity_rnDeriv_eq {μ ν : Measure α}
     [HaveLebesgueDecomposition μ ν] : μ ≪ ν ↔ ν.withDensity (rnDeriv μ ν) = μ :=
   ⟨withDensity_rnDeriv_eq μ ν, fun h => h ▸ withDensity_absolutelyContinuous _ _⟩
 #align measure_theory.measure.absolutely_continuous_iff_with_density_rn_deriv_eq MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq
+-/
 
-theorem with_density_rnDeriv_toReal_eq {μ ν : Measure α} [IsFiniteMeasure μ]
+#print MeasureTheory.Measure.withDensity_rnDeriv_toReal_eq /-
+theorem withDensity_rnDeriv_toReal_eq {μ ν : Measure α} [IsFiniteMeasure μ]
     [HaveLebesgueDecomposition μ ν] (h : μ ≪ ν) {i : Set α} (hi : MeasurableSet i) :
     (∫ x in i, (μ.rnDeriv ν x).toReal ∂ν) = (μ i).toReal :=
   by
@@ -85,7 +90,8 @@ theorem with_density_rnDeriv_toReal_eq {μ ν : Measure α} [IsFiniteMeasure μ]
       ae_lt_top (μ.measurable_rn_deriv ν) (lt_of_le_of_lt (lintegral_mono_set i.subset_univ) _).Ne
     rw [← with_density_apply _ MeasurableSet.univ, with_density_rn_deriv_eq μ ν h]
     exact measure_lt_top _ _
-#align measure_theory.measure.with_density_rn_deriv_to_real_eq MeasureTheory.Measure.with_density_rnDeriv_toReal_eq
+#align measure_theory.measure.with_density_rn_deriv_to_real_eq MeasureTheory.Measure.withDensity_rnDeriv_toReal_eq
+-/
 
 end Measure

fc758559

bump to nightly-2023-06-06-09

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3209ddf94136d36e5e5c624b10b2a347cc9d090

7f33669e

Diff

@@ -96,7 +96,7 @@ include m
 open Measure VectorMeasure
 
 theorem withDensityᵥ_rnDeriv_eq (s : SignedMeasure α) (μ : Measure α) [SigmaFinite μ]
-    (h : s ≪ᵥ μ.toEnnrealVectorMeasure) : μ.withDensityᵥ (s.rnDeriv μ) = s :=
+    (h : s ≪ᵥ μ.toENNRealVectorMeasure) : μ.withDensityᵥ (s.rnDeriv μ) = s :=
   by
   rw [absolutely_continuous_ennreal_iff, (_ : μ.to_ennreal_vector_measure.ennreal_to_measure = μ),
     total_variation_absolutely_continuous_iff] at h 
@@ -119,7 +119,7 @@ theorem withDensityᵥ_rnDeriv_eq (s : SignedMeasure α) (μ : Measure α) [Sigm
 
 /-- The Radon-Nikodym theorem for signed measures. -/
 theorem absolutelyContinuous_iff_withDensityᵥ_rnDeriv_eq (s : SignedMeasure α) (μ : Measure α)
-    [SigmaFinite μ] : s ≪ᵥ μ.toEnnrealVectorMeasure ↔ μ.withDensityᵥ (s.rnDeriv μ) = s :=
+    [SigmaFinite μ] : s ≪ᵥ μ.toENNRealVectorMeasure ↔ μ.withDensityᵥ (s.rnDeriv μ) = s :=
   ⟨withDensityᵥ_rnDeriv_eq s μ, fun h => h ▸ withDensityᵥ_absolutelyContinuous _ _⟩
 #align measure_theory.signed_measure.absolutely_continuous_iff_with_densityᵥ_rn_deriv_eq MeasureTheory.SignedMeasure.absolutelyContinuous_iff_withDensityᵥ_rnDeriv_eq

fc758559

bump to nightly-2023-06-04-18

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

3fb4268b

Diff

@@ -75,7 +75,7 @@ theorem absolutelyContinuous_iff_withDensity_rnDeriv_eq {μ ν : Measure α}
   ⟨withDensity_rnDeriv_eq μ ν, fun h => h ▸ withDensity_absolutelyContinuous _ _⟩
 #align measure_theory.measure.absolutely_continuous_iff_with_density_rn_deriv_eq MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq
 
-theorem with_density_rnDeriv_toReal_eq {μ ν : Measure α} [FiniteMeasure μ]
+theorem with_density_rnDeriv_toReal_eq {μ ν : Measure α} [IsFiniteMeasure μ]
     [HaveLebesgueDecomposition μ ν] (h : μ ≪ ν) {i : Set α} (hi : MeasurableSet i) :
     (∫ x in i, (μ.rnDeriv ν x).toReal ∂ν) = (μ i).toReal :=
   by

fc758559

bump to nightly-2023-06-01-16

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

b9c87c70

Diff

@@ -61,9 +61,9 @@ theorem withDensity_rnDeriv_eq (μ ν : Measure α) [HaveLebesgueDecomposition 
       exact measure_mono (Set.subset_univ _)
     rw [← measure_add_measure_compl hE₁, hE₂, zero_add]
     have : (singular_part μ ν + ν.with_density (rn_deriv μ ν)) (Eᶜ) = μ (Eᶜ) := by rw [← hadd]
-    rw [measure.coe_add, Pi.add_apply, h hE₃] at this
+    rw [measure.coe_add, Pi.add_apply, h hE₃] at this 
     exact (add_eq_zero_iff.1 this).1
-  rw [this, zero_add] at hadd
+  rw [this, zero_add] at hadd 
   exact hadd.symm
 #align measure_theory.measure.with_density_rn_deriv_eq MeasureTheory.Measure.withDensity_rnDeriv_eq
 
@@ -99,7 +99,7 @@ theorem withDensityᵥ_rnDeriv_eq (s : SignedMeasure α) (μ : Measure α) [Sigm
     (h : s ≪ᵥ μ.toEnnrealVectorMeasure) : μ.withDensityᵥ (s.rnDeriv μ) = s :=
   by
   rw [absolutely_continuous_ennreal_iff, (_ : μ.to_ennreal_vector_measure.ennreal_to_measure = μ),
-    total_variation_absolutely_continuous_iff] at h
+    total_variation_absolutely_continuous_iff] at h 
   · ext1 i hi
     rw [with_densityᵥ_apply (integrable_rn_deriv _ _) hi, rn_deriv, integral_sub,
       with_density_rn_deriv_to_real_eq h.1 hi, with_density_rn_deriv_to_real_eq h.2 hi]

fc758559

bump to nightly-2023-06-01-04

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

4d9eaa85

Diff

@@ -120,7 +120,7 @@ theorem withDensityᵥ_rnDeriv_eq (s : SignedMeasure α) (μ : Measure α) [Sigm
 /-- The Radon-Nikodym theorem for signed measures. -/
 theorem absolutelyContinuous_iff_withDensityᵥ_rnDeriv_eq (s : SignedMeasure α) (μ : Measure α)
     [SigmaFinite μ] : s ≪ᵥ μ.toEnnrealVectorMeasure ↔ μ.withDensityᵥ (s.rnDeriv μ) = s :=
-  ⟨withDensityᵥ_rnDeriv_eq s μ, fun h => h ▸ withDensityᵥAbsolutelyContinuous _ _⟩
+  ⟨withDensityᵥ_rnDeriv_eq s μ, fun h => h ▸ withDensityᵥ_absolutelyContinuous _ _⟩
 #align measure_theory.signed_measure.absolutely_continuous_iff_with_densityᵥ_rn_deriv_eq MeasureTheory.SignedMeasure.absolutelyContinuous_iff_withDensityᵥ_rnDeriv_eq
 
 end SignedMeasure

fc758559

bump to nightly-2023-05-28-18

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

8d353152

Diff

@@ -38,7 +38,7 @@ Radon-Nikodym theorem
 
 noncomputable section
 
-open Classical MeasureTheory NNReal ENNReal
+open scoped Classical MeasureTheory NNReal ENNReal
 
 variable {α β : Type _} {m : MeasurableSpace α}

fc758559

bump to nightly-2023-05-23-20

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

4d066d3b

Diff

@@ -110,7 +110,7 @@ theorem withDensityᵥ_rnDeriv_eq (s : SignedMeasure α) (μ : Measure α) [Sigm
       rw [← integrable_on_univ]
       refine' integrable_on.restrict _ MeasurableSet.univ
       refine' ⟨_, has_finite_integral_to_real_of_lintegral_ne_top _⟩
-      · apply Measurable.aeStronglyMeasurable
+      · apply Measurable.aestronglyMeasurable
         measurability
       · rw [set_lintegral_univ]
         exact (lintegral_rn_deriv_lt_top _ _).Ne

fc758559

bump to nightly-2023-05-09-08

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

244524af

Diff

@@ -75,7 +75,7 @@ theorem absolutelyContinuous_iff_withDensity_rnDeriv_eq {μ ν : Measure α}
   ⟨withDensity_rnDeriv_eq μ ν, fun h => h ▸ withDensity_absolutelyContinuous _ _⟩
 #align measure_theory.measure.absolutely_continuous_iff_with_density_rn_deriv_eq MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq
 
-theorem with_density_rnDeriv_toReal_eq {μ ν : Measure α} [IsFiniteMeasure μ]
+theorem with_density_rnDeriv_toReal_eq {μ ν : Measure α} [FiniteMeasure μ]
     [HaveLebesgueDecomposition μ ν] (h : μ ≪ ν) {i : Set α} (hi : MeasurableSet i) :
     (∫ x in i, (μ.rnDeriv ν x).toReal ∂ν) = (μ i).toReal :=
   by

fc758559

bump to nightly-2023-05-04-10

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

3d8894bd

Diff

@@ -72,7 +72,7 @@ theorem withDensity_rnDeriv_eq (μ ν : Measure α) [HaveLebesgueDecomposition 
 `ν.with_density (rn_deriv μ ν) = μ`. -/
 theorem absolutelyContinuous_iff_withDensity_rnDeriv_eq {μ ν : Measure α}
     [HaveLebesgueDecomposition μ ν] : μ ≪ ν ↔ ν.withDensity (rnDeriv μ ν) = μ :=
-  ⟨withDensity_rnDeriv_eq μ ν, fun h => h ▸ withDensityAbsolutelyContinuous _ _⟩
+  ⟨withDensity_rnDeriv_eq μ ν, fun h => h ▸ withDensity_absolutelyContinuous _ _⟩
 #align measure_theory.measure.absolutely_continuous_iff_with_density_rn_deriv_eq MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq
 
 theorem with_density_rnDeriv_toReal_eq {μ ν : Measure α} [IsFiniteMeasure μ]

fc758559

bump to nightly-2023-02-27-10

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

49ff026a

Diff

@@ -38,7 +38,7 @@ Radon-Nikodym theorem
 
 noncomputable section
 
-open Classical MeasureTheory NNReal Ennreal
+open Classical MeasureTheory NNReal ENNReal
 
 variable {α β : Type _} {m : MeasurableSpace α}

fc758559

bump to nightly-2023-02-21-16

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

1107b979

Changes in mathlib4

mathlib3

mathlib4

fc758559

chore: adapt to multiple goal linter 2 (#12361)

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

fe96aa45

Diff

@@ -255,14 +255,14 @@ lemma rnDeriv_add_right_of_mutuallySingular {ν' : Measure α}
   have h₂ := rnDeriv_add' (μ.singularPart ν') (ν'.withDensity (μ.rnDeriv ν')) ν
   refine (Filter.EventuallyEq.trans (h_ac.ae_le h₁) ?_).trans h₂.symm
   have h₃ := rnDeriv_add_right_of_mutuallySingular' (?_ : μ.singularPart ν' ⟂ₘ ν') hνν'
-  have h₄ : (ν'.withDensity (rnDeriv μ ν')).rnDeriv (ν + ν') =ᵐ[ν] 0 := by
-    refine rnDeriv_eq_zero_of_mutuallySingular ?_ h_ac
-    exact hνν'.symm.withDensity
-  have h₅ : (ν'.withDensity (rnDeriv μ ν')).rnDeriv ν =ᵐ[ν] 0 := by
-    rw [rnDeriv_eq_zero]
-    exact hνν'.symm.withDensity
-  filter_upwards [h₃, h₄, h₅] with x hx₃ hx₄ hx₅
-  rw [Pi.add_apply, Pi.add_apply, hx₃, hx₄, hx₅]
+  · have h₄ : (ν'.withDensity (rnDeriv μ ν')).rnDeriv (ν + ν') =ᵐ[ν] 0 := by
+      refine rnDeriv_eq_zero_of_mutuallySingular ?_ h_ac
+      exact hνν'.symm.withDensity
+    have h₅ : (ν'.withDensity (rnDeriv μ ν')).rnDeriv ν =ᵐ[ν] 0 := by
+      rw [rnDeriv_eq_zero]
+      exact hνν'.symm.withDensity
+    filter_upwards [h₃, h₄, h₅] with x hx₃ hx₄ hx₅
+    rw [Pi.add_apply, Pi.add_apply, hx₃, hx₄, hx₅]
   exact mutuallySingular_singularPart μ ν'
 
 lemma rnDeriv_withDensity_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :

fc758559

chore: replace set_integral with setIntegral (#12215)

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

f4b4298b

Diff

@@ -329,7 +329,7 @@ lemma integrableOn_toReal_rnDeriv {s : Set α} (hμs : μ s ≠ ∞) :
   refine integrable_toReal_of_lintegral_ne_top (Measure.measurable_rnDeriv _ _).aemeasurable ?_
   exact ((set_lintegral_rnDeriv_le _).trans_lt hμs.lt_top).ne
 
-lemma set_integral_toReal_rnDeriv_eq_withDensity' [SigmaFinite μ]
+lemma setIntegral_toReal_rnDeriv_eq_withDensity' [SigmaFinite μ]
     {s : Set α} (hs : MeasurableSet s) :
     ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (ν.withDensity (μ.rnDeriv ν) s).toReal := by
   rw [integral_toReal (Measure.measurable_rnDeriv _ _).aemeasurable]
@@ -337,43 +337,63 @@ lemma set_integral_toReal_rnDeriv_eq_withDensity' [SigmaFinite μ]
     simp
   · exact ae_restrict_of_ae (Measure.rnDeriv_lt_top _ _)
 
-lemma set_integral_toReal_rnDeriv_eq_withDensity [SigmaFinite μ] [SFinite ν] (s : Set α) :
+@[deprecated]
+alias set_integral_toReal_rnDeriv_eq_withDensity' :=
+  setIntegral_toReal_rnDeriv_eq_withDensity' -- deprecated on 2024-04-17
+
+lemma setIntegral_toReal_rnDeriv_eq_withDensity [SigmaFinite μ] [SFinite ν] (s : Set α) :
     ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (ν.withDensity (μ.rnDeriv ν) s).toReal := by
   rw [integral_toReal (Measure.measurable_rnDeriv _ _).aemeasurable]
   · rw [ENNReal.toReal_eq_toReal_iff, ← withDensity_apply' _ s]
     simp
   · exact ae_restrict_of_ae (Measure.rnDeriv_lt_top _ _)
 
-lemma set_integral_toReal_rnDeriv_le [SigmaFinite μ] {s : Set α} (hμs : μ s ≠ ∞) :
+@[deprecated]
+alias set_integral_toReal_rnDeriv_eq_withDensity :=
+  setIntegral_toReal_rnDeriv_eq_withDensity -- deprecated on 2024-04-17
+
+lemma setIntegral_toReal_rnDeriv_le [SigmaFinite μ] {s : Set α} (hμs : μ s ≠ ∞) :
     ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν ≤ (μ s).toReal := by
   set t := toMeasurable μ s with ht
   have ht_m : MeasurableSet t := measurableSet_toMeasurable μ s
   have hμt : μ t ≠ ∞ := by rwa [ht, measure_toMeasurable s]
   calc ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν
     ≤ ∫ x in t, (μ.rnDeriv ν x).toReal ∂ν := by
-        refine set_integral_mono_set ?_ ?_ (HasSubset.Subset.eventuallyLE (subset_toMeasurable _ _))
+        refine setIntegral_mono_set ?_ ?_ (HasSubset.Subset.eventuallyLE (subset_toMeasurable _ _))
         · exact integrableOn_toReal_rnDeriv hμt
         · exact ae_of_all _ (by simp)
-  _ = (withDensity ν (rnDeriv μ ν) t).toReal := set_integral_toReal_rnDeriv_eq_withDensity' ht_m
+  _ = (withDensity ν (rnDeriv μ ν) t).toReal := setIntegral_toReal_rnDeriv_eq_withDensity' ht_m
   _ ≤ (μ t).toReal := by
         gcongr
         · exact hμt
         · apply withDensity_rnDeriv_le
   _ = (μ s).toReal := by rw [measure_toMeasurable s]
 
-lemma set_integral_toReal_rnDeriv' [SigmaFinite μ] [HaveLebesgueDecomposition μ ν]
+@[deprecated]
+alias set_integral_toReal_rnDeriv_le :=
+  setIntegral_toReal_rnDeriv_le -- deprecated on 2024-04-17
+
+lemma setIntegral_toReal_rnDeriv' [SigmaFinite μ] [HaveLebesgueDecomposition μ ν]
     (hμν : μ ≪ ν) {s : Set α} (hs : MeasurableSet s) :
     ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (μ s).toReal := by
-  rw [set_integral_toReal_rnDeriv_eq_withDensity' hs, Measure.withDensity_rnDeriv_eq _ _ hμν]
-#align measure_theory.measure.with_density_rn_deriv_to_real_eq MeasureTheory.Measure.set_integral_toReal_rnDeriv'
+  rw [setIntegral_toReal_rnDeriv_eq_withDensity' hs, Measure.withDensity_rnDeriv_eq _ _ hμν]
+#align measure_theory.measure.with_density_rn_deriv_to_real_eq MeasureTheory.Measure.setIntegral_toReal_rnDeriv'
 
-lemma set_integral_toReal_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) (s : Set α) :
+@[deprecated]
+alias set_integral_toReal_rnDeriv' :=
+  setIntegral_toReal_rnDeriv' -- deprecated on 2024-04-17
+
+lemma setIntegral_toReal_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) (s : Set α) :
     ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (μ s).toReal := by
-  rw [set_integral_toReal_rnDeriv_eq_withDensity s, Measure.withDensity_rnDeriv_eq _ _ hμν]
+  rw [setIntegral_toReal_rnDeriv_eq_withDensity s, Measure.withDensity_rnDeriv_eq _ _ hμν]
+
+@[deprecated]
+alias set_integral_toReal_rnDeriv :=
+  setIntegral_toReal_rnDeriv -- deprecated on 2024-04-17
 
 lemma integral_toReal_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
     ∫ x, (μ.rnDeriv ν x).toReal ∂ν = (μ Set.univ).toReal := by
-  rw [← integral_univ, set_integral_toReal_rnDeriv hμν Set.univ]
+  rw [← integral_univ, setIntegral_toReal_rnDeriv hμν Set.univ]
 
 end integral
 
@@ -398,7 +418,7 @@ theorem withDensityᵥ_rnDeriv_eq (s : SignedMeasure α) (μ : Measure α) [Sigm
     totalVariation_absolutelyContinuous_iff] at h
   · ext1 i hi
     rw [withDensityᵥ_apply (integrable_rnDeriv _ _) hi, rnDeriv_def, integral_sub,
-      set_integral_toReal_rnDeriv h.1 i, set_integral_toReal_rnDeriv h.2 i]
+      setIntegral_toReal_rnDeriv h.1 i, setIntegral_toReal_rnDeriv h.2 i]
     · conv_rhs => rw [← s.toSignedMeasure_toJordanDecomposition]
       erw [VectorMeasure.sub_apply]
       rw [toSignedMeasure_apply_measurable hi, toSignedMeasure_apply_measurable hi]

fc758559

chore: tidy various files (#12042)

8856dfd0

Diff

@@ -437,9 +437,8 @@ lemma set_lintegral_rnDeriv_mul [HaveLebesgueDecomposition μ ν] (hμν : μ 
     (hf : AEMeasurable f ν) {s : Set α} (hs : MeasurableSet s) :
     ∫⁻ x in s, μ.rnDeriv ν x * f x ∂ν = ∫⁻ x in s, f x ∂μ := by
   nth_rw 2 [← Measure.withDensity_rnDeriv_eq μ ν hμν]
-  rw [set_lintegral_withDensity_eq_lintegral_mul₀ (Measure.measurable_rnDeriv μ ν).aemeasurable hf
-    hs]
-  rfl
+  rw [set_lintegral_withDensity_eq_lintegral_mul₀ (measurable_rnDeriv μ ν).aemeasurable hf hs]
+  simp only [Pi.mul_apply]
 
 variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]

fc758559

feat(MeasureTheory): add singularPart and rnDeriv lemmas (#11883)

Also golf and move rnDeriv_restrict.

b8fc37d6

Diff

@@ -189,18 +189,6 @@ lemma rnDeriv_withDensity_right (μ ν : Measure α) [SigmaFinite μ] [SigmaFini
 
 end rnDeriv_withDensity_leftRight
 
-theorem rnDeriv_restrict (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν]
-    {s : Set α} (hs : MeasurableSet s) :
-    (μ.restrict s).rnDeriv ν =ᵐ[ν] s.indicator (μ.rnDeriv ν) := by
-  rw [← withDensity_indicator_one hs]
-  refine (rnDeriv_withDensity_left ?_ ?_ ?_).trans (ae_of_all _ (fun x ↦ ?_))
-  · exact measurable_one.aemeasurable.indicator hs
-  · exact measurable_one.aemeasurable.indicator hs
-  · filter_upwards with x
-    simp only [Set.indicator_apply, Pi.one_apply, ne_eq]
-    split_ifs <;> simp [ENNReal.zero_ne_top]
-  · simp [Set.indicator_apply]
-
 lemma rnDeriv_eq_zero_of_mutuallySingular [SigmaFinite μ] {ν' : Measure α}
     [SigmaFinite ν'] (h : μ ⟂ₘ ν) (hνν' : ν ≪ ν') :
     μ.rnDeriv ν' =ᵐ[ν] 0 := by

fc758559

chore: remove unnecessary @[eqns] attributes (#11460)

These attributes are unused in Mathlib.

Many of them were workarounds for the now-resolved leanprover/lean4#2243; this also allows the lemmas themselves (hasFiniteIntegral_def, integrable_def, memℒp_def, and integrableOn_def) to be deleted.

We are currently experiencing problems with the @[eqns] attribute on the Lean nightlies. I'm uncertain yet what the outcome is going to be there, but it seems prudent to reduce our unnecessary exposure to a language feature added in Mathlib.

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

31188bc4

Diff

@@ -409,7 +409,7 @@ theorem withDensityᵥ_rnDeriv_eq (s : SignedMeasure α) (μ : Measure α) [Sigm
   rw [absolutelyContinuous_ennreal_iff, (_ : μ.toENNRealVectorMeasure.ennrealToMeasure = μ),
     totalVariation_absolutelyContinuous_iff] at h
   · ext1 i hi
-    rw [withDensityᵥ_apply (integrable_rnDeriv _ _) hi, rnDeriv, integral_sub,
+    rw [withDensityᵥ_apply (integrable_rnDeriv _ _) hi, rnDeriv_def, integral_sub,
       set_integral_toReal_rnDeriv h.1 i, set_integral_toReal_rnDeriv h.2 i]
     · conv_rhs => rw [← s.toSignedMeasure_toJordanDecomposition]
       erw [VectorMeasure.sub_apply]

fc758559

chore: golf using filter_upwards (#11208)

This is presumably not exhaustive, but covers about a hundred instances.

Style opinions (e.g., why a particular change is great/not a good idea) are very welcome; I'm still forming my own.

f637899e

Diff

@@ -196,7 +196,7 @@ theorem rnDeriv_restrict (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν]
   refine (rnDeriv_withDensity_left ?_ ?_ ?_).trans (ae_of_all _ (fun x ↦ ?_))
   · exact measurable_one.aemeasurable.indicator hs
   · exact measurable_one.aemeasurable.indicator hs
-  · refine ae_of_all _ (fun x ↦ ?_)
+  · filter_upwards with x
     simp only [Set.indicator_apply, Pi.one_apply, ne_eq]
     split_ifs <;> simp [ENNReal.zero_ne_top]
   · simp [Set.indicator_apply]

fc758559

feat: add set_lintegral variants of lintegral lemmas (#11274)

2731f819

Diff

@@ -445,6 +445,14 @@ theorem lintegral_rnDeriv_mul [HaveLebesgueDecomposition μ ν] (hμν : μ ≪
   rw [lintegral_withDensity_eq_lintegral_mul₀ (measurable_rnDeriv μ ν).aemeasurable hf]
   rfl
 
+lemma set_lintegral_rnDeriv_mul [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) {f : α → ℝ≥0∞}
+    (hf : AEMeasurable f ν) {s : Set α} (hs : MeasurableSet s) :
+    ∫⁻ x in s, μ.rnDeriv ν x * f x ∂ν = ∫⁻ x in s, f x ∂μ := by
+  nth_rw 2 [← Measure.withDensity_rnDeriv_eq μ ν hμν]
+  rw [set_lintegral_withDensity_eq_lintegral_mul₀ (Measure.measurable_rnDeriv μ ν).aemeasurable hf
+    hs]
+  rfl
+
 variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
 
 theorem integrable_rnDeriv_smul_iff [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν)

fc758559

chore: more backporting of simp changes from #10995 (#11001)

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

d9589c14

Diff

@@ -206,7 +206,7 @@ lemma rnDeriv_eq_zero_of_mutuallySingular [SigmaFinite μ] {ν' : Measure α}
     μ.rnDeriv ν' =ᵐ[ν] 0 := by
   let t := h.nullSet
   have ht : MeasurableSet t := h.measurableSet_nullSet
-  refine ae_of_ae_restrict_of_ae_restrict_compl t ?_ (by simp)
+  refine ae_of_ae_restrict_of_ae_restrict_compl t ?_ (by simp [t])
   change μ.rnDeriv ν' =ᵐ[ν.restrict t] 0
   have : μ.rnDeriv ν' =ᵐ[ν.restrict t] (μ.restrict t).rnDeriv ν' := by
     have h : (μ.restrict t).rnDeriv ν' =ᵐ[ν] t.indicator (μ.rnDeriv ν') :=
@@ -215,7 +215,7 @@ lemma rnDeriv_eq_zero_of_mutuallySingular [SigmaFinite μ] {ν' : Measure α}
     filter_upwards [h] with x hx hxt
     rw [hx, Set.indicator_of_mem hxt]
   refine this.trans ?_
-  simp only [MutuallySingular.restrict_nullSet]
+  simp only [t, MutuallySingular.restrict_nullSet]
   suffices (0 : Measure α).rnDeriv ν' =ᵐ[ν'] 0 by
     have h_ac' : ν.restrict t ≪ ν' := restrict_le_self.absolutelyContinuous.trans hνν'
     exact h_ac'.ae_le this
@@ -228,12 +228,12 @@ lemma rnDeriv_add_right_of_absolutelyContinuous_of_mutuallySingular {ν' : Measu
     μ.rnDeriv (ν + ν') =ᵐ[ν] μ.rnDeriv ν := by
   let t := hνν'.nullSet
   have ht : MeasurableSet t := hνν'.measurableSet_nullSet
-  refine ae_of_ae_restrict_of_ae_restrict_compl t (by simp) ?_
+  refine ae_of_ae_restrict_of_ae_restrict_compl t (by simp [t]) ?_
   change μ.rnDeriv (ν + ν') =ᵐ[ν.restrict tᶜ] μ.rnDeriv ν
   rw [← withDensity_eq_iff_of_sigmaFinite (μ := ν.restrict tᶜ)
     (Measure.measurable_rnDeriv _ _).aemeasurable (Measure.measurable_rnDeriv _ _).aemeasurable]
   have : (ν.restrict tᶜ).withDensity (μ.rnDeriv (ν + ν'))
-      = ((ν + ν').restrict tᶜ).withDensity (μ.rnDeriv (ν + ν')) := by simp
+      = ((ν + ν').restrict tᶜ).withDensity (μ.rnDeriv (ν + ν')) := by simp [t]
   rw [this, ← restrict_withDensity ht.compl, ← restrict_withDensity ht.compl,
       Measure.withDensity_rnDeriv_eq _ _ (hμν.add_right ν'), Measure.withDensity_rnDeriv_eq _ _ hμν]

fc758559

refactor(MeasureTheory): redefine ≤ on measures (#10714)

Redefine ≤ on MeasureTheory.Measure so that μ ≤ ν ↔ ∀ s, μ s ≤ ν s by definition instead of ∀ s, MeasurableSet s → μ s ≤ ν s.

Reasons

this way it is defeq to ≤ on outer measures;
if we decide to introduce an order on all DFunLike types and migrate measures to FunLike, then this is unavoidable;
the reasoning for the old definition was "it's slightly easier to prove μ ≤ ν this way"; the counter-argument is "it's slightly harder to apply μ ≤ ν this way".

Other changes

golf some proofs broken by this change;
add @[gcongr] tags to some ENNReal lemmas;
fix the name ENNReal.coe_lt_coe_of_le -> ENNReal.ENNReal.coe_lt_coe_of_lt;
drop an unneeded MeasurableSet assumption in set_lintegral_pdf_le_map

f17e2324

Diff

@@ -62,9 +62,8 @@ theorem withDensity_rnDeriv_eq (μ ν : Measure α) [HaveLebesgueDecomposition 
   rw [← measure_add_measure_compl h_sing.measurableSet_nullSet]
   simp only [MutuallySingular.measure_nullSet, zero_add]
   refine le_antisymm ?_ (zero_le _)
-  refine (singularPart_le μ ν ?_ ?_).trans_eq ?_
-  · exact h_sing.measurableSet_nullSet.compl
-  · exact h h_sing.measure_compl_nullSet
+  refine (singularPart_le μ ν ?_ ).trans_eq ?_
+  exact h h_sing.measure_compl_nullSet
 #align measure_theory.measure.with_density_rn_deriv_eq MeasureTheory.Measure.withDensity_rnDeriv_eq
 
 variable {μ ν : Measure α}
@@ -367,11 +366,12 @@ lemma set_integral_toReal_rnDeriv_le [SigmaFinite μ] {s : Set α} (hμs : μ s
         refine set_integral_mono_set ?_ ?_ (HasSubset.Subset.eventuallyLE (subset_toMeasurable _ _))
         · exact integrableOn_toReal_rnDeriv hμt
         · exact ae_of_all _ (by simp)
+  _ = (withDensity ν (rnDeriv μ ν) t).toReal := set_integral_toReal_rnDeriv_eq_withDensity' ht_m
   _ ≤ (μ t).toReal := by
-        rw [set_integral_toReal_rnDeriv_eq_withDensity' ht_m, ENNReal.toReal_le_toReal _ hμt]
-        · exact withDensity_rnDeriv_le _ _ _ ht_m
-        · exact ((withDensity_rnDeriv_le _ _ _ ht_m).trans_lt hμt.lt_top).ne
-  _ = (μ s).toReal := by rw [← measure_toMeasurable s]
+        gcongr
+        · exact hμt
+        · apply withDensity_rnDeriv_le
+  _ = (μ s).toReal := by rw [measure_toMeasurable s]
 
 lemma set_integral_toReal_rnDeriv' [SigmaFinite μ] [HaveLebesgueDecomposition μ ν]
     (hμν : μ ≪ ν) {s : Set α} (hs : MeasurableSet s) :

fc758559

feat: exponentially tilted measures (#7676)

Define exponentially tilted measures. The exponential tilting of a measure μ on α by a function f : α → ℝ is the measure with density x ↦ exp (f x) / ∫ y, exp (f y) ∂μ with respect to μ.

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

5522fc37

Diff

@@ -317,6 +317,8 @@ lemma inv_rnDeriv' [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
     (ν.rnDeriv μ)⁻¹ =ᵐ[μ] μ.rnDeriv ν := by
   filter_upwards [inv_rnDeriv hμν] with x hx; simp only [Pi.inv_apply, ← hx, inv_inv]
 
+section integral
+
 lemma set_lintegral_rnDeriv_le (s : Set α) :
     ∫⁻ x in s, μ.rnDeriv ν x ∂ν ≤ μ s :=
   (withDensity_apply_le _ _).trans (Measure.le_iff'.1 (withDensity_rnDeriv_le μ ν) s)
@@ -385,6 +387,17 @@ lemma integral_toReal_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪
     ∫ x, (μ.rnDeriv ν x).toReal ∂ν = (μ Set.univ).toReal := by
   rw [← integral_univ, set_integral_toReal_rnDeriv hμν Set.univ]
 
+end integral
+
+lemma rnDeriv_mul_rnDeriv {μ ν κ : Measure α} [SigmaFinite μ] [SigmaFinite ν] [SigmaFinite κ]
+    (hμν : μ ≪ ν) :
+    μ.rnDeriv ν * ν.rnDeriv κ =ᵐ[κ] μ.rnDeriv κ := by
+  refine (rnDeriv_withDensity_left ?_ ?_ ?_).symm.trans ?_
+  · exact (Measure.measurable_rnDeriv _ _).aemeasurable
+  · exact (Measure.measurable_rnDeriv _ _).aemeasurable
+  · exact rnDeriv_ne_top _ _
+  · rw [Measure.withDensity_rnDeriv_eq _ _ hμν]
+
 end Measure
 
 namespace SignedMeasure

fc758559

refactor(Probability/Density): define pdf using rnDeriv (#8544)

Defines pdf in terms of rnDeriv.

Main definition change:

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

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

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

Zulip thread

3f88a472

Diff

@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kexing Ying, Rémy Degenne
 -/
 import Mathlib.MeasureTheory.Decomposition.SignedLebesgue
+import Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure
 
 #align_import measure_theory.decomposition.radon_nikodym from "leanprover-community/mathlib"@"fc75855907eaa8ff39791039710f567f37d4556f"
 
@@ -419,4 +420,43 @@ theorem absolutelyContinuous_iff_withDensityᵥ_rnDeriv_eq (s : SignedMeasure α
 
 end SignedMeasure
 
+section IntegralRNDerivMul
+
+open Measure
+
+variable {α : Type*} {m : MeasurableSpace α} {μ ν : Measure α}
+
+theorem lintegral_rnDeriv_mul [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) {f : α → ℝ≥0∞}
+    (hf : AEMeasurable f ν) : ∫⁻ x, μ.rnDeriv ν x * f x ∂ν = ∫⁻ x, f x ∂μ := by
+  nth_rw 2 [← withDensity_rnDeriv_eq μ ν hμν]
+  rw [lintegral_withDensity_eq_lintegral_mul₀ (measurable_rnDeriv μ ν).aemeasurable hf]
+  rfl
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+
+theorem integrable_rnDeriv_smul_iff [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν)
+    [SigmaFinite μ] {f : α → E} :
+    Integrable (fun x ↦ (μ.rnDeriv ν x).toReal • f x) ν ↔ Integrable f μ := by
+  nth_rw 2 [← withDensity_rnDeriv_eq μ ν hμν]
+  rw [← integrable_withDensity_iff_integrable_smul' (E := E)
+    (measurable_rnDeriv μ ν) (rnDeriv_lt_top μ ν)]
+
+theorem withDensityᵥ_rnDeriv_smul [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν)
+    [SigmaFinite μ] {f : α → E} (hf : Integrable f μ) :
+    ν.withDensityᵥ (fun x ↦ (rnDeriv μ ν x).toReal • f x) = μ.withDensityᵥ f := by
+  rw [withDensityᵥ_smul_eq_withDensityᵥ_withDensity' (measurable_rnDeriv μ ν).aemeasurable
+    (rnDeriv_lt_top μ ν) ((integrable_rnDeriv_smul_iff hμν).mpr hf), withDensity_rnDeriv_eq μ ν hμν]
+
+theorem integral_rnDeriv_smul [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν)
+    [SigmaFinite μ] {f : α → E} :
+    ∫ x, (μ.rnDeriv ν x).toReal • f x ∂ν = ∫ x, f x ∂μ := by
+  by_cases hf : Integrable f μ
+  · rw [← integral_univ, ← withDensityᵥ_apply ((integrable_rnDeriv_smul_iff hμν).mpr hf) .univ,
+      ← integral_univ, ← withDensityᵥ_apply hf .univ, withDensityᵥ_rnDeriv_smul hμν hf]
+  · rw [integral_undef hf, integral_undef]
+    contrapose! hf
+    exact (integrable_rnDeriv_smul_iff hμν).mp hf
+
+end IntegralRNDerivMul
+
 end MeasureTheory

fc758559

feat: generalize some lemmas using withDensity_apply' (#8383)

@sgouezel added a version of withDensity_apply that does not require measurability of the set if the measure is s-finite. This PR uses that result in other files of the library.

For results about rnDeriv, I put a prime on the version that assumes measurability of the set and no prime on the version for s-finite measures, as the second one should be the main use case.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: RemyDegenne <Remydegenne@gmail.com>

21b59396

Diff

@@ -320,21 +320,26 @@ lemma set_lintegral_rnDeriv_le (s : Set α) :
     ∫⁻ x in s, μ.rnDeriv ν x ∂ν ≤ μ s :=
   (withDensity_apply_le _ _).trans (Measure.le_iff'.1 (withDensity_rnDeriv_le μ ν) s)
 
-lemma set_lintegral_rnDeriv [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) {s : Set α}
+lemma set_lintegral_rnDeriv' [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) {s : Set α}
     (hs : MeasurableSet s) :
     ∫⁻ x in s, μ.rnDeriv ν x ∂ν = μ s := by
   rw [← withDensity_apply _ hs, Measure.withDensity_rnDeriv_eq _ _ hμν]
 
+lemma set_lintegral_rnDeriv [HaveLebesgueDecomposition μ ν] [SFinite ν]
+    (hμν : μ ≪ ν) (s : Set α) :
+    ∫⁻ x in s, μ.rnDeriv ν x ∂ν = μ s := by
+  rw [← withDensity_apply' _ s, Measure.withDensity_rnDeriv_eq _ _ hμν]
+
 lemma lintegral_rnDeriv [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) :
     ∫⁻ x, μ.rnDeriv ν x ∂ν = μ Set.univ := by
-  rw [← set_lintegral_univ, set_lintegral_rnDeriv hμν MeasurableSet.univ]
+  rw [← set_lintegral_univ, set_lintegral_rnDeriv' hμν MeasurableSet.univ]
 
 lemma integrableOn_toReal_rnDeriv {s : Set α} (hμs : μ s ≠ ∞) :
     IntegrableOn (fun x ↦ (μ.rnDeriv ν x).toReal) s ν := by
   refine integrable_toReal_of_lintegral_ne_top (Measure.measurable_rnDeriv _ _).aemeasurable ?_
   exact ((set_lintegral_rnDeriv_le _).trans_lt hμs.lt_top).ne
 
-lemma set_integral_toReal_rnDeriv_eq_withDensity [SigmaFinite μ]
+lemma set_integral_toReal_rnDeriv_eq_withDensity' [SigmaFinite μ]
     {s : Set α} (hs : MeasurableSet s) :
     ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (ν.withDensity (μ.rnDeriv ν) s).toReal := by
   rw [integral_toReal (Measure.measurable_rnDeriv _ _).aemeasurable]
@@ -342,6 +347,13 @@ lemma set_integral_toReal_rnDeriv_eq_withDensity [SigmaFinite μ]
     simp
   · exact ae_restrict_of_ae (Measure.rnDeriv_lt_top _ _)
 
+lemma set_integral_toReal_rnDeriv_eq_withDensity [SigmaFinite μ] [SFinite ν] (s : Set α) :
+    ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (ν.withDensity (μ.rnDeriv ν) s).toReal := by
+  rw [integral_toReal (Measure.measurable_rnDeriv _ _).aemeasurable]
+  · rw [ENNReal.toReal_eq_toReal_iff, ← withDensity_apply' _ s]
+    simp
+  · exact ae_restrict_of_ae (Measure.rnDeriv_lt_top _ _)
+
 lemma set_integral_toReal_rnDeriv_le [SigmaFinite μ] {s : Set α} (hμs : μ s ≠ ∞) :
     ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν ≤ (μ s).toReal := by
   set t := toMeasurable μ s with ht
@@ -353,20 +365,24 @@ lemma set_integral_toReal_rnDeriv_le [SigmaFinite μ] {s : Set α} (hμs : μ s
         · exact integrableOn_toReal_rnDeriv hμt
         · exact ae_of_all _ (by simp)
   _ ≤ (μ t).toReal := by
-        rw [set_integral_toReal_rnDeriv_eq_withDensity ht_m, ENNReal.toReal_le_toReal _ hμt]
+        rw [set_integral_toReal_rnDeriv_eq_withDensity' ht_m, ENNReal.toReal_le_toReal _ hμt]
         · exact withDensity_rnDeriv_le _ _ _ ht_m
         · exact ((withDensity_rnDeriv_le _ _ _ ht_m).trans_lt hμt.lt_top).ne
   _ = (μ s).toReal := by rw [← measure_toMeasurable s]
 
-lemma set_integral_toReal_rnDeriv [SigmaFinite μ] [HaveLebesgueDecomposition μ ν]
+lemma set_integral_toReal_rnDeriv' [SigmaFinite μ] [HaveLebesgueDecomposition μ ν]
     (hμν : μ ≪ ν) {s : Set α} (hs : MeasurableSet s) :
     ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (μ s).toReal := by
-  rw [set_integral_toReal_rnDeriv_eq_withDensity hs, Measure.withDensity_rnDeriv_eq _ _ hμν]
-#align measure_theory.measure.with_density_rn_deriv_to_real_eq MeasureTheory.Measure.set_integral_toReal_rnDeriv
+  rw [set_integral_toReal_rnDeriv_eq_withDensity' hs, Measure.withDensity_rnDeriv_eq _ _ hμν]
+#align measure_theory.measure.with_density_rn_deriv_to_real_eq MeasureTheory.Measure.set_integral_toReal_rnDeriv'
+
+lemma set_integral_toReal_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) (s : Set α) :
+    ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (μ s).toReal := by
+  rw [set_integral_toReal_rnDeriv_eq_withDensity s, Measure.withDensity_rnDeriv_eq _ _ hμν]
 
 lemma integral_toReal_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
     ∫ x, (μ.rnDeriv ν x).toReal ∂ν = (μ Set.univ).toReal := by
-  rw [← integral_univ, set_integral_toReal_rnDeriv hμν MeasurableSet.univ]
+  rw [← integral_univ, set_integral_toReal_rnDeriv hμν Set.univ]
 
 end Measure
 
@@ -380,7 +396,7 @@ theorem withDensityᵥ_rnDeriv_eq (s : SignedMeasure α) (μ : Measure α) [Sigm
     totalVariation_absolutelyContinuous_iff] at h
   · ext1 i hi
     rw [withDensityᵥ_apply (integrable_rnDeriv _ _) hi, rnDeriv, integral_sub,
-      set_integral_toReal_rnDeriv h.1 hi, set_integral_toReal_rnDeriv h.2 hi]
+      set_integral_toReal_rnDeriv h.1 i, set_integral_toReal_rnDeriv h.2 i]
     · conv_rhs => rw [← s.toSignedMeasure_toJordanDecomposition]
       erw [VectorMeasure.sub_apply]
       rw [toSignedMeasure_apply_measurable hi, toSignedMeasure_apply_measurable hi]

fc758559

feat: in a sigma-finite space, withDensity f s is the integral of f over s even if s is not measurable (#8377)

295f071b

Diff

@@ -317,14 +317,8 @@ lemma inv_rnDeriv' [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
   filter_upwards [inv_rnDeriv hμν] with x hx; simp only [Pi.inv_apply, ← hx, inv_inv]
 
 lemma set_lintegral_rnDeriv_le (s : Set α) :
-    ∫⁻ x in s, μ.rnDeriv ν x ∂ν ≤ μ s := by
-  let t := toMeasurable μ s
-  calc ∫⁻ x in s, μ.rnDeriv ν x ∂ν
-    ≤ ∫⁻ x in t, μ.rnDeriv ν x ∂ν := lintegral_mono_set (subset_toMeasurable μ s)
-  _ ≤ μ t := by
-        rw [← withDensity_apply _ (measurableSet_toMeasurable μ s)]
-        exact withDensity_rnDeriv_le _ _ _ (measurableSet_toMeasurable μ s)
-  _ = μ s := by rw [← measure_toMeasurable s]
+    ∫⁻ x in s, μ.rnDeriv ν x ∂ν ≤ μ s :=
+  (withDensity_apply_le _ _).trans (Measure.le_iff'.1 (withDensity_rnDeriv_le μ ν) s)
 
 lemma set_lintegral_rnDeriv [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) {s : Set α}
     (hs : MeasurableSet s) :

fc758559

feat(MeasureTheory): remove an AbsolutelyContinuous hypothesis from inv_rnDeriv (#8351)

In order to remove that hypothesis, we also:

add some basic lemmas about absolute continuity and mutually singular measures.
add HaveLebesgueDecomposition instances
rewrite the proof of withDensity_rnDeriv_eq to use the new API instead of unfolding the definitions
generalize rnDeriv_restrict and rnDeriv_withDensity to possibly different measures

daf40d2c

Diff

@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2021 Kexing Ying. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kexing Ying
+Authors: Kexing Ying, Rémy Degenne
 -/
 import Mathlib.MeasureTheory.Decomposition.SignedLebesgue
 
@@ -27,6 +27,15 @@ expectation of a real function, but we take a different approach (see the file
 * `MeasureTheory.SignedMeasure.absolutelyContinuous_iff_withDensityᵥ_rnDeriv_eq` :
   the Radon-Nikodym theorem for signed measures
 
+The file also contains properties of `rnDeriv` that use the Radon-Nikodym theorem, notably
+* `MeasureTheory.Measure.rnDeriv_withDensity_left`: the Radon-Nikodym derivative of
+  `μ.withDensity f` with respect to `ν` is `f * μ.rnDeriv ν`.
+* `MeasureTheory.Measure.rnDeriv_withDensity_right`: the Radon-Nikodym derivative of
+  `μ` with respect to `ν.withDensity f` is `f⁻¹ * μ.rnDeriv ν`.
+* `MeasureTheory.Measure.inv_rnDeriv`: `(μ.rnDeriv ν)⁻¹ =ᵐ[μ] ν.rnDeriv μ`.
+* `MeasureTheory.Measure.set_lintegral_rnDeriv`: `∫⁻ x in s, μ.rnDeriv ν x ∂ν = μ s` if `μ ≪ ν`.
+  There is also a version of this result for the Bochner integral.
+
 ## Tags
 
 Radon-Nikodym theorem
@@ -45,18 +54,16 @@ namespace Measure
 
 theorem withDensity_rnDeriv_eq (μ ν : Measure α) [HaveLebesgueDecomposition μ ν] (h : μ ≪ ν) :
     ν.withDensity (rnDeriv μ ν) = μ := by
-  obtain ⟨_, ⟨E, hE₁, hE₂, hE₃⟩, hadd⟩ := haveLebesgueDecomposition_spec μ ν
-  have : singularPart μ ν = 0 := by
-    refine' le_antisymm (fun A (_ : MeasurableSet A) => _) (Measure.zero_le _)
-    suffices singularPart μ ν Set.univ = 0 by
-      rw [Measure.coe_zero, Pi.zero_apply, ← this]
-      exact measure_mono (Set.subset_univ _)
-    rw [← measure_add_measure_compl hE₁, hE₂, zero_add]
-    have : (singularPart μ ν + ν.withDensity (rnDeriv μ ν)) Eᶜ = μ Eᶜ := by rw [← hadd]
-    rw [Measure.coe_add, Pi.add_apply, h hE₃] at this
-    exact (add_eq_zero_iff.1 this).1
-  rw [this, zero_add] at hadd
-  exact hadd.symm
+  suffices μ.singularPart ν = 0 by
+    conv_rhs => rw [haveLebesgueDecomposition_add μ ν, this, zero_add]
+  suffices μ.singularPart ν Set.univ = 0 by simpa using this
+  have h_sing := mutuallySingular_singularPart μ ν
+  rw [← measure_add_measure_compl h_sing.measurableSet_nullSet]
+  simp only [MutuallySingular.measure_nullSet, zero_add]
+  refine le_antisymm ?_ (zero_le _)
+  refine (singularPart_le μ ν ?_ ?_).trans_eq ?_
+  · exact h_sing.measurableSet_nullSet.compl
+  · exact h h_sing.measure_compl_nullSet
 #align measure_theory.measure.with_density_rn_deriv_eq MeasureTheory.Measure.withDensity_rnDeriv_eq
 
 variable {μ ν : Measure α}
@@ -75,7 +82,210 @@ lemma rnDeriv_pos [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) :
     ae_withDensity_iff (Measure.measurable_rnDeriv _ _), Measure.withDensity_rnDeriv_eq _ _  hμν]
   exact ae_of_all _ (fun x hx ↦ lt_of_le_of_ne (zero_le _) hx.symm)
 
-lemma inv_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) (hνμ : ν ≪ μ) :
+lemma rnDeriv_pos' [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
+    ∀ᵐ x ∂μ, 0 < ν.rnDeriv μ x := by
+  refine (absolutelyContinuous_withDensity_rnDeriv hμν).ae_le ?_
+  filter_upwards [Measure.rnDeriv_pos (withDensity_absolutelyContinuous μ (ν.rnDeriv μ)),
+    (withDensity_absolutelyContinuous μ (ν.rnDeriv μ)).ae_le
+    (Measure.rnDeriv_withDensity μ (Measure.measurable_rnDeriv ν μ))] with x hx hx2
+  rwa [← hx2]
+
+section rnDeriv_withDensity_leftRight
+
+variable {μ ν : Measure α} {f : α → ℝ≥0∞}
+
+/-- Auxiliary lemma for `rnDeriv_withDensity_left`. -/
+lemma rnDeriv_withDensity_withDensity_rnDeriv_left (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν]
+    (hf : AEMeasurable f μ) (hf_ne_top : ∀ᵐ x ∂μ, f x ≠ ∞) :
+    ((ν.withDensity (μ.rnDeriv ν)).withDensity f).rnDeriv ν =ᵐ[ν] (μ.withDensity f).rnDeriv ν := by
+  conv_rhs => rw [μ.haveLebesgueDecomposition_add ν, add_comm, withDensity_add_measure]
+  have : SigmaFinite ((μ.singularPart ν).withDensity f) :=
+    SigmaFinite.withDensity_of_ne_top (hf.singularPart ν)
+      ((Measure.absolutelyContinuous_of_le (Measure.singularPart_le _ _)).ae_le hf_ne_top)
+  have : SigmaFinite ((ν.withDensity (μ.rnDeriv ν)).withDensity f) :=
+    SigmaFinite.withDensity_of_ne_top (hf.withDensity_rnDeriv ν)
+      ((Measure.absolutelyContinuous_of_le (Measure.withDensity_rnDeriv_le _ _)).ae_le hf_ne_top)
+  exact (rnDeriv_add_of_mutuallySingular _ _ _ (mutuallySingular_singularPart μ ν).withDensity).symm
+
+/-- Auxiliary lemma for `rnDeriv_withDensity_right`. -/
+lemma rnDeriv_withDensity_withDensity_rnDeriv_right (μ ν : Measure α) [SigmaFinite μ]
+    [SigmaFinite ν] (hf : AEMeasurable f ν) (hf_ne_zero : ∀ᵐ x ∂ν, f x ≠ 0)
+    (hf_ne_top : ∀ᵐ x ∂ν, f x ≠ ∞) :
+    (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν.withDensity f) =ᵐ[ν] μ.rnDeriv (ν.withDensity f) := by
+  conv_rhs => rw [μ.haveLebesgueDecomposition_add ν, add_comm]
+  have hν_ac : ν ≪ ν.withDensity f := withDensity_absolutelyContinuous' hf hf_ne_zero hf_ne_top
+  refine hν_ac.ae_eq ?_
+  have : SigmaFinite (ν.withDensity f) := SigmaFinite.withDensity_of_ne_top hf hf_ne_top
+  refine (rnDeriv_add_of_mutuallySingular _ _ _ ?_).symm
+  exact ((mutuallySingular_singularPart μ ν).symm.withDensity).symm
+
+lemma rnDeriv_withDensity_left_of_absolutelyContinuous {ν : Measure α} [SigmaFinite μ]
+    [SigmaFinite ν] (hμν : μ ≪ ν) (hf : AEMeasurable f ν) :
+    (μ.withDensity f).rnDeriv ν =ᵐ[ν] fun x ↦ f x * μ.rnDeriv ν x := by
+  refine (Measure.eq_rnDeriv₀ ?_ Measure.MutuallySingular.zero_left ?_).symm
+  · exact hf.mul (Measure.measurable_rnDeriv _ _).aemeasurable
+  · ext1 s hs
+    rw [zero_add, withDensity_apply _ hs, withDensity_apply _ hs]
+    conv_lhs => rw [← Measure.withDensity_rnDeriv_eq _ _ hμν]
+    rw [set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable₀ _ _ _ hs]
+    · congr with x
+      rw [mul_comm]
+      simp only [Pi.mul_apply]
+    · refine ae_restrict_of_ae ?_
+      exact Measure.rnDeriv_lt_top _ _
+    · exact (Measure.measurable_rnDeriv _ _).aemeasurable
+
+lemma rnDeriv_withDensity_left {μ ν : Measure α} [SigmaFinite μ] [SigmaFinite ν]
+    (hfμ : AEMeasurable f μ) (hfν : AEMeasurable f ν)
+    (hf_ne_top : ∀ᵐ x ∂μ, f x ≠ ∞) :
+    (μ.withDensity f).rnDeriv ν =ᵐ[ν] fun x ↦ f x * μ.rnDeriv ν x := by
+  let μ' := ν.withDensity (μ.rnDeriv ν)
+  have hμ'ν : μ' ≪ ν := withDensity_absolutelyContinuous _ _
+  have h := rnDeriv_withDensity_left_of_absolutelyContinuous hμ'ν hfν
+  have h1 : μ'.rnDeriv ν =ᵐ[ν] μ.rnDeriv ν :=
+    Measure.rnDeriv_withDensity _ (Measure.measurable_rnDeriv _ _)
+  have h2 : (μ'.withDensity f).rnDeriv ν =ᵐ[ν] (μ.withDensity f).rnDeriv ν := by
+    exact rnDeriv_withDensity_withDensity_rnDeriv_left μ ν hfμ hf_ne_top
+  filter_upwards [h, h1, h2] with x hx hx1 hx2
+  rw [← hx2, hx, hx1]
+
+/-- Auxiliary lemma for `rnDeriv_withDensity_right`. -/
+lemma rnDeriv_withDensity_right_of_absolutelyContinuous {ν : Measure α} [SigmaFinite μ]
+    [SigmaFinite ν] (hμν : μ ≪ ν) (hf : AEMeasurable f ν)
+    (hf_ne_zero : ∀ᵐ x ∂ν, f x ≠ 0) (hf_ne_top : ∀ᵐ x ∂ν, f x ≠ ∞) :
+    μ.rnDeriv (ν.withDensity f) =ᵐ[ν] fun x ↦ (f x)⁻¹ * μ.rnDeriv ν x := by
+  have : SigmaFinite (ν.withDensity f) := SigmaFinite.withDensity_of_ne_top hf hf_ne_top
+  refine (withDensity_absolutelyContinuous' hf hf_ne_zero hf_ne_top).ae_eq ?_
+  refine (Measure.eq_rnDeriv₀ (ν := ν.withDensity f) ?_ Measure.MutuallySingular.zero_left ?_).symm
+  · exact (hf.inv.mono_ac (withDensity_absolutelyContinuous _ _)).mul
+      (Measure.measurable_rnDeriv _ _).aemeasurable
+  · ext1 s hs
+    conv_lhs => rw [← Measure.withDensity_rnDeriv_eq _ _ hμν]
+    rw [zero_add, withDensity_apply _ hs, withDensity_apply _ hs]
+    rw [set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable₀ _ _ _ hs]
+    · simp only [Pi.mul_apply]
+      have : (fun a ↦ f a * ((f a)⁻¹ * μ.rnDeriv ν a)) =ᵐ[ν] μ.rnDeriv ν := by
+        filter_upwards [hf_ne_zero, hf_ne_top] with x hx1 hx2
+        simp [← mul_assoc, ENNReal.mul_inv_cancel, hx1, hx2]
+      rw [lintegral_congr_ae (ae_restrict_of_ae this)]
+    · refine ae_restrict_of_ae ?_
+      filter_upwards [hf_ne_top] with x hx using hx.lt_top
+    · exact hf.restrict
+
+lemma rnDeriv_withDensity_right (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν]
+    (hf : AEMeasurable f ν) (hf_ne_zero : ∀ᵐ x ∂ν, f x ≠ 0) (hf_ne_top : ∀ᵐ x ∂ν, f x ≠ ∞) :
+    μ.rnDeriv (ν.withDensity f) =ᵐ[ν] fun x ↦ (f x)⁻¹ * μ.rnDeriv ν x := by
+  let μ' := ν.withDensity (μ.rnDeriv ν)
+  have h₁ : μ'.rnDeriv (ν.withDensity f) =ᵐ[ν] μ.rnDeriv (ν.withDensity f) :=
+    rnDeriv_withDensity_withDensity_rnDeriv_right μ ν hf hf_ne_zero hf_ne_top
+  have h₂ : μ.rnDeriv ν =ᵐ[ν] μ'.rnDeriv ν :=
+    (Measure.rnDeriv_withDensity _ (Measure.measurable_rnDeriv _ _)).symm
+  have : SigmaFinite μ' := SigmaFinite.withDensity_of_ne_top
+    (Measure.measurable_rnDeriv _ _).aemeasurable (Measure.rnDeriv_ne_top _ _)
+  have hμ' := rnDeriv_withDensity_right_of_absolutelyContinuous
+    (withDensity_absolutelyContinuous ν (μ.rnDeriv ν)) hf hf_ne_zero hf_ne_top
+  filter_upwards [h₁, h₂, hμ'] with x hx₁ hx₂ hx_eq
+  rw [← hx₁, hx₂, hx_eq]
+
+end rnDeriv_withDensity_leftRight
+
+theorem rnDeriv_restrict (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν]
+    {s : Set α} (hs : MeasurableSet s) :
+    (μ.restrict s).rnDeriv ν =ᵐ[ν] s.indicator (μ.rnDeriv ν) := by
+  rw [← withDensity_indicator_one hs]
+  refine (rnDeriv_withDensity_left ?_ ?_ ?_).trans (ae_of_all _ (fun x ↦ ?_))
+  · exact measurable_one.aemeasurable.indicator hs
+  · exact measurable_one.aemeasurable.indicator hs
+  · refine ae_of_all _ (fun x ↦ ?_)
+    simp only [Set.indicator_apply, Pi.one_apply, ne_eq]
+    split_ifs <;> simp [ENNReal.zero_ne_top]
+  · simp [Set.indicator_apply]
+
+lemma rnDeriv_eq_zero_of_mutuallySingular [SigmaFinite μ] {ν' : Measure α}
+    [SigmaFinite ν'] (h : μ ⟂ₘ ν) (hνν' : ν ≪ ν') :
+    μ.rnDeriv ν' =ᵐ[ν] 0 := by
+  let t := h.nullSet
+  have ht : MeasurableSet t := h.measurableSet_nullSet
+  refine ae_of_ae_restrict_of_ae_restrict_compl t ?_ (by simp)
+  change μ.rnDeriv ν' =ᵐ[ν.restrict t] 0
+  have : μ.rnDeriv ν' =ᵐ[ν.restrict t] (μ.restrict t).rnDeriv ν' := by
+    have h : (μ.restrict t).rnDeriv ν' =ᵐ[ν] t.indicator (μ.rnDeriv ν') :=
+      hνν'.ae_le (rnDeriv_restrict μ ν' ht)
+    rw [Filter.EventuallyEq, ae_restrict_iff' ht]
+    filter_upwards [h] with x hx hxt
+    rw [hx, Set.indicator_of_mem hxt]
+  refine this.trans ?_
+  simp only [MutuallySingular.restrict_nullSet]
+  suffices (0 : Measure α).rnDeriv ν' =ᵐ[ν'] 0 by
+    have h_ac' : ν.restrict t ≪ ν' := restrict_le_self.absolutelyContinuous.trans hνν'
+    exact h_ac'.ae_le this
+  exact rnDeriv_zero _
+
+/-- Auxiliary lemma for `rnDeriv_add_right_of_mutuallySingular`. -/
+lemma rnDeriv_add_right_of_absolutelyContinuous_of_mutuallySingular {ν' : Measure α}
+    [SigmaFinite μ] [SigmaFinite ν] [SigmaFinite ν']
+    (hμν : μ ≪ ν) (hνν' : ν ⟂ₘ ν') :
+    μ.rnDeriv (ν + ν') =ᵐ[ν] μ.rnDeriv ν := by
+  let t := hνν'.nullSet
+  have ht : MeasurableSet t := hνν'.measurableSet_nullSet
+  refine ae_of_ae_restrict_of_ae_restrict_compl t (by simp) ?_
+  change μ.rnDeriv (ν + ν') =ᵐ[ν.restrict tᶜ] μ.rnDeriv ν
+  rw [← withDensity_eq_iff_of_sigmaFinite (μ := ν.restrict tᶜ)
+    (Measure.measurable_rnDeriv _ _).aemeasurable (Measure.measurable_rnDeriv _ _).aemeasurable]
+  have : (ν.restrict tᶜ).withDensity (μ.rnDeriv (ν + ν'))
+      = ((ν + ν').restrict tᶜ).withDensity (μ.rnDeriv (ν + ν')) := by simp
+  rw [this, ← restrict_withDensity ht.compl, ← restrict_withDensity ht.compl,
+      Measure.withDensity_rnDeriv_eq _ _ (hμν.add_right ν'), Measure.withDensity_rnDeriv_eq _ _ hμν]
+
+/-- Auxiliary lemma for `rnDeriv_add_right_of_mutuallySingular`. -/
+lemma rnDeriv_add_right_of_mutuallySingular' {ν' : Measure α}
+    [SigmaFinite μ] [SigmaFinite ν] [SigmaFinite ν']
+    (hμν' : μ ⟂ₘ ν') (hνν' : ν ⟂ₘ ν') :
+    μ.rnDeriv (ν + ν') =ᵐ[ν] μ.rnDeriv ν := by
+  have h_ac : ν ≪ ν + ν' := Measure.AbsolutelyContinuous.rfl.add_right _
+  rw [haveLebesgueDecomposition_add μ ν]
+  have h₁ := rnDeriv_add' (μ.singularPart ν) (ν.withDensity (μ.rnDeriv ν)) (ν + ν')
+  have h₂ := rnDeriv_add' (μ.singularPart ν) (ν.withDensity (μ.rnDeriv ν)) ν
+  refine (Filter.EventuallyEq.trans (h_ac.ae_le h₁) ?_).trans h₂.symm
+  simp only [Pi.add_apply]
+  have h₃ := rnDeriv_add_right_of_absolutelyContinuous_of_mutuallySingular
+    (withDensity_absolutelyContinuous ν (μ.rnDeriv ν)) hνν'
+  have h₄ : (μ.singularPart ν).rnDeriv (ν + ν') =ᵐ[ν] 0 := by
+    refine h_ac.ae_eq ?_
+    simp only [rnDeriv_eq_zero, MutuallySingular.add_right_iff]
+    exact ⟨mutuallySingular_singularPart μ ν, hμν'.singularPart ν⟩
+  have h₅ : (μ.singularPart ν).rnDeriv ν =ᵐ[ν] 0 := rnDeriv_singularPart μ ν
+  filter_upwards [h₃, h₄, h₅] with x hx₃ hx₄ hx₅
+  rw [hx₃, hx₄, Pi.add_apply, hx₅]
+
+lemma rnDeriv_add_right_of_mutuallySingular {ν' : Measure α}
+    [SigmaFinite μ] [SigmaFinite ν] [SigmaFinite ν'] (hνν' : ν ⟂ₘ ν') :
+    μ.rnDeriv (ν + ν') =ᵐ[ν] μ.rnDeriv ν := by
+  have h_ac : ν ≪ ν + ν' := Measure.AbsolutelyContinuous.rfl.add_right _
+  rw [haveLebesgueDecomposition_add μ ν']
+  have h₁ := rnDeriv_add' (μ.singularPart ν') (ν'.withDensity (μ.rnDeriv ν')) (ν + ν')
+  have h₂ := rnDeriv_add' (μ.singularPart ν') (ν'.withDensity (μ.rnDeriv ν')) ν
+  refine (Filter.EventuallyEq.trans (h_ac.ae_le h₁) ?_).trans h₂.symm
+  have h₃ := rnDeriv_add_right_of_mutuallySingular' (?_ : μ.singularPart ν' ⟂ₘ ν') hνν'
+  have h₄ : (ν'.withDensity (rnDeriv μ ν')).rnDeriv (ν + ν') =ᵐ[ν] 0 := by
+    refine rnDeriv_eq_zero_of_mutuallySingular ?_ h_ac
+    exact hνν'.symm.withDensity
+  have h₅ : (ν'.withDensity (rnDeriv μ ν')).rnDeriv ν =ᵐ[ν] 0 := by
+    rw [rnDeriv_eq_zero]
+    exact hνν'.symm.withDensity
+  filter_upwards [h₃, h₄, h₅] with x hx₃ hx₄ hx₅
+  rw [Pi.add_apply, Pi.add_apply, hx₃, hx₄, hx₅]
+  exact mutuallySingular_singularPart μ ν'
+
+lemma rnDeriv_withDensity_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
+    μ.rnDeriv (μ.withDensity (ν.rnDeriv μ)) =ᵐ[μ] μ.rnDeriv ν := by
+  conv_rhs => rw [ν.haveLebesgueDecomposition_add μ, add_comm]
+  refine (absolutelyContinuous_withDensity_rnDeriv hμν).ae_eq ?_
+  exact (rnDeriv_add_right_of_mutuallySingular
+    (Measure.mutuallySingular_singularPart ν μ).symm.withDensity).symm
+
+/-- Auxiliary lemma for `inv_rnDeriv`. -/
+lemma inv_rnDeriv_aux [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) (hνμ : ν ≪ μ) :
     (μ.rnDeriv ν)⁻¹ =ᵐ[μ] ν.rnDeriv μ := by
   suffices μ.withDensity (μ.rnDeriv ν)⁻¹ = μ.withDensity (ν.rnDeriv μ) by
     calc (μ.rnDeriv ν)⁻¹ =ᵐ[μ] (μ.withDensity (μ.rnDeriv ν)⁻¹).rnDeriv μ :=
@@ -89,6 +299,23 @@ lemma inv_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) (hνμ :
     (by filter_upwards [hνμ.ae_le (rnDeriv_pos hμν)] with x hx using hx.ne')
     (rnDeriv_ne_top _ _)]
 
+lemma inv_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
+    (μ.rnDeriv ν)⁻¹ =ᵐ[μ] ν.rnDeriv μ := by
+  suffices (μ.rnDeriv ν)⁻¹ =ᵐ[μ] (μ.rnDeriv (μ.withDensity (ν.rnDeriv μ)))⁻¹
+      ∧ ν.rnDeriv μ =ᵐ[μ] (μ.withDensity (ν.rnDeriv μ)).rnDeriv μ by
+    refine (this.1.trans (Filter.EventuallyEq.trans ?_ this.2.symm))
+    exact Measure.inv_rnDeriv_aux (absolutelyContinuous_withDensity_rnDeriv hμν)
+      (withDensity_absolutelyContinuous _ _)
+  constructor
+  · filter_upwards [rnDeriv_withDensity_rnDeriv hμν] with x hx
+    simp only [Pi.inv_apply, inv_inj]
+    exact hx.symm
+  · exact (Measure.rnDeriv_withDensity μ (Measure.measurable_rnDeriv ν μ)).symm
+
+lemma inv_rnDeriv' [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
+    (ν.rnDeriv μ)⁻¹ =ᵐ[μ] μ.rnDeriv ν := by
+  filter_upwards [inv_rnDeriv hμν] with x hx; simp only [Pi.inv_apply, ← hx, inv_inv]
+
 lemma set_lintegral_rnDeriv_le (s : Set α) :
     ∫⁻ x in s, μ.rnDeriv ν x ∂ν ≤ μ s := by
   let t := toMeasurable μ s

fc758559

feat(MeasureTheory): integrals of rnDeriv without absolute continuity (#8343)

Add Lemmas about Lebesgue and Bochner integrals of rnDeriv without absolute continuity assumptions.

Also remove a duplicate lemma I introduced in a previous PR: set_integral_toReal_rnDeriv was already there under the name withDensity_rnDeriv_toReal_eq. I kept the name set_integral_toReal_rnDeriv .

708c4de3

Diff

@@ -69,17 +69,6 @@ theorem absolutelyContinuous_iff_withDensity_rnDeriv_eq
   ⟨withDensity_rnDeriv_eq μ ν, fun h => h ▸ withDensity_absolutelyContinuous _ _⟩
 #align measure_theory.measure.absolutely_continuous_iff_with_density_rn_deriv_eq MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq
 
-theorem withDensity_rnDeriv_toReal_eq [IsFiniteMeasure μ] [HaveLebesgueDecomposition μ ν]
-    (h : μ ≪ ν) {i : Set α} (hi : MeasurableSet i) :
-    (∫ x in i, (μ.rnDeriv ν x).toReal ∂ν) = (μ i).toReal := by
-  rw [integral_toReal, ← withDensity_apply _ hi, withDensity_rnDeriv_eq μ ν h]
-  · measurability
-  · refine' ae_lt_top (μ.measurable_rnDeriv ν)
-      (lt_of_le_of_lt (lintegral_mono_set i.subset_univ) _).ne
-    rw [← withDensity_apply _ MeasurableSet.univ, withDensity_rnDeriv_eq μ ν h]
-    exact measure_lt_top _ _
-#align measure_theory.measure.with_density_rn_deriv_to_real_eq MeasureTheory.Measure.withDensity_rnDeriv_toReal_eq
-
 lemma rnDeriv_pos [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) :
     ∀ᵐ x ∂μ, 0 < μ.rnDeriv ν x := by
   rw [← Measure.withDensity_rnDeriv_eq _ _  hμν,
@@ -100,24 +89,61 @@ lemma inv_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) (hνμ :
     (by filter_upwards [hνμ.ae_le (rnDeriv_pos hμν)] with x hx using hx.ne')
     (rnDeriv_ne_top _ _)]
 
+lemma set_lintegral_rnDeriv_le (s : Set α) :
+    ∫⁻ x in s, μ.rnDeriv ν x ∂ν ≤ μ s := by
+  let t := toMeasurable μ s
+  calc ∫⁻ x in s, μ.rnDeriv ν x ∂ν
+    ≤ ∫⁻ x in t, μ.rnDeriv ν x ∂ν := lintegral_mono_set (subset_toMeasurable μ s)
+  _ ≤ μ t := by
+        rw [← withDensity_apply _ (measurableSet_toMeasurable μ s)]
+        exact withDensity_rnDeriv_le _ _ _ (measurableSet_toMeasurable μ s)
+  _ = μ s := by rw [← measure_toMeasurable s]
+
 lemma set_lintegral_rnDeriv [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) {s : Set α}
     (hs : MeasurableSet s) :
     ∫⁻ x in s, μ.rnDeriv ν x ∂ν = μ s := by
-  conv_rhs => rw [← Measure.withDensity_rnDeriv_eq _ _ hμν, withDensity_apply _ hs]
+  rw [← withDensity_apply _ hs, Measure.withDensity_rnDeriv_eq _ _ hμν]
 
 lemma lintegral_rnDeriv [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) :
     ∫⁻ x, μ.rnDeriv ν x ∂ν = μ Set.univ := by
   rw [← set_lintegral_univ, set_lintegral_rnDeriv hμν MeasurableSet.univ]
 
-lemma set_integral_toReal_rnDeriv {μ ν : Measure α} [SigmaFinite μ] [SigmaFinite ν]
-    (hμν : μ ≪ ν) {s : Set α} (hs : MeasurableSet s) :
-    ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (μ s).toReal := by
+lemma integrableOn_toReal_rnDeriv {s : Set α} (hμs : μ s ≠ ∞) :
+    IntegrableOn (fun x ↦ (μ.rnDeriv ν x).toReal) s ν := by
+  refine integrable_toReal_of_lintegral_ne_top (Measure.measurable_rnDeriv _ _).aemeasurable ?_
+  exact ((set_lintegral_rnDeriv_le _).trans_lt hμs.lt_top).ne
+
+lemma set_integral_toReal_rnDeriv_eq_withDensity [SigmaFinite μ]
+    {s : Set α} (hs : MeasurableSet s) :
+    ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (ν.withDensity (μ.rnDeriv ν) s).toReal := by
   rw [integral_toReal (Measure.measurable_rnDeriv _ _).aemeasurable]
-  · rw [ENNReal.toReal_eq_toReal_iff, set_lintegral_rnDeriv hμν hs]
+  · rw [ENNReal.toReal_eq_toReal_iff, ← withDensity_apply _ hs]
     simp
   · exact ae_restrict_of_ae (Measure.rnDeriv_lt_top _ _)
 
-lemma integral_toReal_rnDeriv {μ ν : Measure α} [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
+lemma set_integral_toReal_rnDeriv_le [SigmaFinite μ] {s : Set α} (hμs : μ s ≠ ∞) :
+    ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν ≤ (μ s).toReal := by
+  set t := toMeasurable μ s with ht
+  have ht_m : MeasurableSet t := measurableSet_toMeasurable μ s
+  have hμt : μ t ≠ ∞ := by rwa [ht, measure_toMeasurable s]
+  calc ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν
+    ≤ ∫ x in t, (μ.rnDeriv ν x).toReal ∂ν := by
+        refine set_integral_mono_set ?_ ?_ (HasSubset.Subset.eventuallyLE (subset_toMeasurable _ _))
+        · exact integrableOn_toReal_rnDeriv hμt
+        · exact ae_of_all _ (by simp)
+  _ ≤ (μ t).toReal := by
+        rw [set_integral_toReal_rnDeriv_eq_withDensity ht_m, ENNReal.toReal_le_toReal _ hμt]
+        · exact withDensity_rnDeriv_le _ _ _ ht_m
+        · exact ((withDensity_rnDeriv_le _ _ _ ht_m).trans_lt hμt.lt_top).ne
+  _ = (μ s).toReal := by rw [← measure_toMeasurable s]
+
+lemma set_integral_toReal_rnDeriv [SigmaFinite μ] [HaveLebesgueDecomposition μ ν]
+    (hμν : μ ≪ ν) {s : Set α} (hs : MeasurableSet s) :
+    ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (μ s).toReal := by
+  rw [set_integral_toReal_rnDeriv_eq_withDensity hs, Measure.withDensity_rnDeriv_eq _ _ hμν]
+#align measure_theory.measure.with_density_rn_deriv_to_real_eq MeasureTheory.Measure.set_integral_toReal_rnDeriv
+
+lemma integral_toReal_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
     ∫ x, (μ.rnDeriv ν x).toReal ∂ν = (μ Set.univ).toReal := by
   rw [← integral_univ, set_integral_toReal_rnDeriv hμν MeasurableSet.univ]
 
@@ -133,7 +159,7 @@ theorem withDensityᵥ_rnDeriv_eq (s : SignedMeasure α) (μ : Measure α) [Sigm
     totalVariation_absolutelyContinuous_iff] at h
   · ext1 i hi
     rw [withDensityᵥ_apply (integrable_rnDeriv _ _) hi, rnDeriv, integral_sub,
-      withDensity_rnDeriv_toReal_eq h.1 hi, withDensity_rnDeriv_toReal_eq h.2 hi]
+      set_integral_toReal_rnDeriv h.1 hi, set_integral_toReal_rnDeriv h.2 hi]
     · conv_rhs => rw [← s.toSignedMeasure_toJordanDecomposition]
       erw [VectorMeasure.sub_apply]
       rw [toSignedMeasure_apply_measurable hi, toSignedMeasure_apply_measurable hi]

fc758559

chore: split MeasureTheory.Decomposition.Lebesgue (#8272)

Put all results about decomposition of signed measures into a new file. This does not significantly change the imports of the original file, because signed measures are used in the proof of the Lebesgue decomposition theorem for Measure.

fe7e9d03

Diff

@@ -3,7 +3,7 @@ Copyright (c) 2021 Kexing Ying. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kexing Ying
 -/
-import Mathlib.MeasureTheory.Decomposition.Lebesgue
+import Mathlib.MeasureTheory.Decomposition.SignedLebesgue
 
 #align_import measure_theory.decomposition.radon_nikodym from "leanprover-community/mathlib"@"fc75855907eaa8ff39791039710f567f37d4556f"

fc758559

feat: properties of rnDeriv (#7675)

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

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

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

9fa322ac

Diff

@@ -59,16 +59,18 @@ theorem withDensity_rnDeriv_eq (μ ν : Measure α) [HaveLebesgueDecomposition 
   exact hadd.symm
 #align measure_theory.measure.with_density_rn_deriv_eq MeasureTheory.Measure.withDensity_rnDeriv_eq
 
+variable {μ ν : Measure α}
+
 /-- **The Radon-Nikodym theorem**: Given two measures `μ` and `ν`, if
 `HaveLebesgueDecomposition μ ν`, then `μ` is absolutely continuous to `ν` if and only if
 `ν.withDensity (rnDeriv μ ν) = μ`. -/
-theorem absolutelyContinuous_iff_withDensity_rnDeriv_eq {μ ν : Measure α}
+theorem absolutelyContinuous_iff_withDensity_rnDeriv_eq
     [HaveLebesgueDecomposition μ ν] : μ ≪ ν ↔ ν.withDensity (rnDeriv μ ν) = μ :=
   ⟨withDensity_rnDeriv_eq μ ν, fun h => h ▸ withDensity_absolutelyContinuous _ _⟩
 #align measure_theory.measure.absolutely_continuous_iff_with_density_rn_deriv_eq MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq
 
-theorem withDensity_rnDeriv_toReal_eq {μ ν : Measure α} [IsFiniteMeasure μ]
-    [HaveLebesgueDecomposition μ ν] (h : μ ≪ ν) {i : Set α} (hi : MeasurableSet i) :
+theorem withDensity_rnDeriv_toReal_eq [IsFiniteMeasure μ] [HaveLebesgueDecomposition μ ν]
+    (h : μ ≪ ν) {i : Set α} (hi : MeasurableSet i) :
     (∫ x in i, (μ.rnDeriv ν x).toReal ∂ν) = (μ i).toReal := by
   rw [integral_toReal, ← withDensity_apply _ hi, withDensity_rnDeriv_eq μ ν h]
   · measurability
@@ -78,6 +80,47 @@ theorem withDensity_rnDeriv_toReal_eq {μ ν : Measure α} [IsFiniteMeasure μ]
     exact measure_lt_top _ _
 #align measure_theory.measure.with_density_rn_deriv_to_real_eq MeasureTheory.Measure.withDensity_rnDeriv_toReal_eq
 
+lemma rnDeriv_pos [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) :
+    ∀ᵐ x ∂μ, 0 < μ.rnDeriv ν x := by
+  rw [← Measure.withDensity_rnDeriv_eq _ _  hμν,
+    ae_withDensity_iff (Measure.measurable_rnDeriv _ _), Measure.withDensity_rnDeriv_eq _ _  hμν]
+  exact ae_of_all _ (fun x hx ↦ lt_of_le_of_ne (zero_le _) hx.symm)
+
+lemma inv_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) (hνμ : ν ≪ μ) :
+    (μ.rnDeriv ν)⁻¹ =ᵐ[μ] ν.rnDeriv μ := by
+  suffices μ.withDensity (μ.rnDeriv ν)⁻¹ = μ.withDensity (ν.rnDeriv μ) by
+    calc (μ.rnDeriv ν)⁻¹ =ᵐ[μ] (μ.withDensity (μ.rnDeriv ν)⁻¹).rnDeriv μ :=
+          (rnDeriv_withDensity _ (measurable_rnDeriv _ _).inv).symm
+    _ = (μ.withDensity (ν.rnDeriv μ)).rnDeriv μ := by rw [this]
+    _ =ᵐ[μ] ν.rnDeriv μ := rnDeriv_withDensity _ (measurable_rnDeriv _ _)
+  rw [withDensity_rnDeriv_eq _ _ hνμ, ← withDensity_rnDeriv_eq _ _ hμν]
+  conv in ((ν.withDensity (μ.rnDeriv ν)).rnDeriv ν)⁻¹ => rw [withDensity_rnDeriv_eq _ _ hμν]
+  change (ν.withDensity (μ.rnDeriv ν)).withDensity (fun x ↦ (μ.rnDeriv ν x)⁻¹) = ν
+  rw [withDensity_inv_same (measurable_rnDeriv _ _)
+    (by filter_upwards [hνμ.ae_le (rnDeriv_pos hμν)] with x hx using hx.ne')
+    (rnDeriv_ne_top _ _)]
+
+lemma set_lintegral_rnDeriv [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) {s : Set α}
+    (hs : MeasurableSet s) :
+    ∫⁻ x in s, μ.rnDeriv ν x ∂ν = μ s := by
+  conv_rhs => rw [← Measure.withDensity_rnDeriv_eq _ _ hμν, withDensity_apply _ hs]
+
+lemma lintegral_rnDeriv [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) :
+    ∫⁻ x, μ.rnDeriv ν x ∂ν = μ Set.univ := by
+  rw [← set_lintegral_univ, set_lintegral_rnDeriv hμν MeasurableSet.univ]
+
+lemma set_integral_toReal_rnDeriv {μ ν : Measure α} [SigmaFinite μ] [SigmaFinite ν]
+    (hμν : μ ≪ ν) {s : Set α} (hs : MeasurableSet s) :
+    ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (μ s).toReal := by
+  rw [integral_toReal (Measure.measurable_rnDeriv _ _).aemeasurable]
+  · rw [ENNReal.toReal_eq_toReal_iff, set_lintegral_rnDeriv hμν hs]
+    simp
+  · exact ae_restrict_of_ae (Measure.rnDeriv_lt_top _ _)
+
+lemma integral_toReal_rnDeriv {μ ν : Measure α} [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
+    ∫ x, (μ.rnDeriv ν x).toReal ∂ν = (μ Set.univ).toReal := by
+  rw [← integral_univ, set_integral_toReal_rnDeriv hμν MeasurableSet.univ]
+
 end Measure
 
 namespace SignedMeasure

fc758559

chore: banish Type _ and Sort _ (#6499)

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

This has nice performance benefits.

32de3f67

Diff

@@ -37,7 +37,7 @@ noncomputable section
 
 open scoped Classical MeasureTheory NNReal ENNReal
 
-variable {α β : Type _} {m : MeasurableSpace α}
+variable {α β : Type*} {m : MeasurableSpace α}
 
 namespace MeasureTheory

fc758559

chore: script to replace headers with #align_import statements (#5979)

depends on: #5966

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

89aa3a3b

Diff

@@ -2,14 +2,11 @@
 Copyright (c) 2021 Kexing Ying. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kexing Ying
-
-! This file was ported from Lean 3 source module measure_theory.decomposition.radon_nikodym
-! leanprover-community/mathlib commit fc75855907eaa8ff39791039710f567f37d4556f
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.MeasureTheory.Decomposition.Lebesgue
 
+#align_import measure_theory.decomposition.radon_nikodym from "leanprover-community/mathlib"@"fc75855907eaa8ff39791039710f567f37d4556f"
+
 /-!
 # Radon-Nikodym theorem

fc758559

fix: change compl precedence (#5586)

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

4d4f0f25

Diff

@@ -55,7 +55,7 @@ theorem withDensity_rnDeriv_eq (μ ν : Measure α) [HaveLebesgueDecomposition 
       rw [Measure.coe_zero, Pi.zero_apply, ← this]
       exact measure_mono (Set.subset_univ _)
     rw [← measure_add_measure_compl hE₁, hE₂, zero_add]
-    have : (singularPart μ ν + ν.withDensity (rnDeriv μ ν)) (Eᶜ) = μ (Eᶜ) := by rw [← hadd]
+    have : (singularPart μ ν + ν.withDensity (rnDeriv μ ν)) Eᶜ = μ Eᶜ := by rw [← hadd]
     rw [Measure.coe_add, Pi.add_apply, h hE₃] at this
     exact (add_eq_zero_iff.1 this).1
   rw [this, zero_add] at hadd

fc758559

feat: port MeasureTheory.Decomposition.RadonNikodym (#4781)

21511d60

`measure_theory.decomposition.radon_nikodym` ⟷ `Mathlib.MeasureTheory.Decomposition.RadonNikodym`

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Reasons

Other changes

Dependencies 12 + 979

measure_theory.decomposition.radon_nikodym ⟷ Mathlib.MeasureTheory.Decomposition.RadonNikodym

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Reasons

Other changes

Dependencies 12 + 979

`measure_theory.decomposition.radon_nikodym` ⟷ `Mathlib.MeasureTheory.Decomposition.RadonNikodym`