probability.density ⟷ Mathlib.Probability.Density

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

@@ -100,7 +100,7 @@ theorem pdf_undef {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {
 theorem hasPDF_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E}
     (h : pdf X ℙ μ ≠ 0) : HasPDF X ℙ μ := by
   by_contra hpdf
-  rw [pdf, dif_neg hpdf] at h 
+  rw [pdf, dif_neg hpdf] at h
   exact hpdf (False.ndrec (has_pdf X ℙ μ) (h rfl))
 #align measure_theory.has_pdf_of_pdf_ne_zero MeasureTheory.hasPDF_of_pdf_ne_zero
 -/
@@ -241,7 +241,7 @@ theorem integral_pdf_smul [IsFiniteMeasure ℙ] {X : Ω → E} [HasPDF X ℙ μ]
       rw [lintegral_congr_ae this]
       exact hpdf.2
   · rw [integral_undef hpdf, integral_undef]
-    rwa [← integrable_iff_integrable_mul_pdf hf] at hpdf 
+    rwa [← integrable_iff_integrable_mul_pdf hf] at hpdf
     all_goals infer_instance
 #align measure_theory.pdf.integral_fun_mul_eq_integral MeasureTheory.pdf.integral_pdf_smul
 -/
@@ -311,7 +311,7 @@ theorem quasiMeasurePreserving_hasPDF {X : Ω → E} [HasPDF X ℙ μ] {g : E 
   refine' absolutely_continuous.mk fun s hsm hs => _
   rw [map_apply hg.measurable hsm, with_density_apply _ (hg.measurable hsm)]
   have := hg.absolutely_continuous hs
-  rw [map_apply hg.measurable hsm] at this 
+  rw [map_apply hg.measurable hsm] at this
   exact set_lintegral_measure_zero _ _ this
 #align measure_theory.pdf.quasi_measure_preserving_has_pdf MeasureTheory.pdf.quasiMeasurePreserving_hasPDF
 -/
@@ -403,7 +403,7 @@ theorem hasPDF {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ :
   hasPDF_of_pdf_ne_zero
     (by
       intro hpdf
-      rw [is_uniform, hpdf] at hu 
+      rw [is_uniform, hpdf] at hu
       suffices μ (s ∩ Function.support ((μ s)⁻¹ • 1)) = 0
         by
         have heq : Function.support ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) = Set.univ :=
@@ -411,7 +411,7 @@ theorem hasPDF {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ :
           ext x
           rw [Function.mem_support]
           simp [hnt]
-        rw [HEq, Set.inter_univ] at this 
+        rw [HEq, Set.inter_univ] at this
         exact hns this
       exact Set.indicator_ae_eq_zero.1 hu.symm)
 #align measure_theory.pdf.is_uniform.has_pdf MeasureTheory.pdf.IsUniform.hasPDF

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

@@ -74,13 +74,13 @@ class HasPDF {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
 #align measure_theory.has_pdf MeasureTheory.HasPDF
 -/
 
-#print MeasureTheory.HasPDF.measurable /-
+#print MeasureTheory.HasPDF.aemeasurable /-
 @[measurability]
-theorem HasPDF.measurable {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
+theorem HasPDF.aemeasurable {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
     (μ : Measure E := by exact MeasureTheory.MeasureSpace.volume) [hX : HasPDF X ℙ μ] :
     Measurable X :=
   hX.pdf'.1
-#align measure_theory.has_pdf.measurable MeasureTheory.HasPDF.measurable
+#align measure_theory.has_pdf.measurable MeasureTheory.HasPDF.aemeasurable
 -/
 
 #print MeasureTheory.pdf /-
@@ -92,11 +92,9 @@ def pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
 #align measure_theory.pdf MeasureTheory.pdf
 -/
 
-#print MeasureTheory.pdf_undef /-
 theorem pdf_undef {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E}
     (h : ¬HasPDF X ℙ μ) : pdf X ℙ μ = 0 := by simp only [pdf, dif_neg h]
 #align measure_theory.pdf_undef MeasureTheory.pdf_undef
--/
 
 #print MeasureTheory.hasPDF_of_pdf_ne_zero /-
 theorem hasPDF_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E}
@@ -107,18 +105,18 @@ theorem hasPDF_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ :
 #align measure_theory.has_pdf_of_pdf_ne_zero MeasureTheory.hasPDF_of_pdf_ne_zero
 -/
 
-#print MeasureTheory.pdf_eq_zero_of_not_measurable /-
-theorem pdf_eq_zero_of_not_measurable {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
-    {X : Ω → E} (hX : ¬Measurable X) : pdf X ℙ μ = 0 :=
+#print MeasureTheory.pdf_of_not_aemeasurable /-
+theorem pdf_of_not_aemeasurable {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E}
+    (hX : ¬Measurable X) : pdf X ℙ μ = 0 :=
   pdf_undef fun hpdf => hX hpdf.pdf'.1
-#align measure_theory.pdf_eq_zero_of_not_measurable MeasureTheory.pdf_eq_zero_of_not_measurable
+#align measure_theory.pdf_eq_zero_of_not_measurable MeasureTheory.pdf_of_not_aemeasurable
 -/
 
-#print MeasureTheory.measurable_of_pdf_ne_zero /-
-theorem measurable_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
+#print MeasureTheory.aemeasurable_of_pdf_ne_zero /-
+theorem aemeasurable_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
     (X : Ω → E) (h : pdf X ℙ μ ≠ 0) : Measurable X := by by_contra hX;
   exact h (pdf_eq_zero_of_not_measurable hX)
-#align measure_theory.measurable_of_pdf_ne_zero MeasureTheory.measurable_of_pdf_ne_zero
+#align measure_theory.measurable_of_pdf_ne_zero MeasureTheory.aemeasurable_of_pdf_ne_zero
 -/
 
 #print MeasureTheory.measurable_pdf /-
@@ -184,22 +182,22 @@ theorem ofReal_toReal_ae_eq [IsFiniteMeasure ℙ] {X : Ω → E} :
 #align measure_theory.pdf.of_real_to_real_ae_eq MeasureTheory.pdf.ofReal_toReal_ae_eq
 -/
 
-#print MeasureTheory.pdf.integrable_iff_integrable_mul_pdf /-
-theorem integrable_iff_integrable_mul_pdf [IsFiniteMeasure ℙ] {X : Ω → E} [HasPDF X ℙ μ] {f : E → ℝ}
+#print MeasureTheory.pdf.integrable_pdf_smul_iff /-
+theorem integrable_pdf_smul_iff [IsFiniteMeasure ℙ] {X : Ω → E} [HasPDF X ℙ μ] {f : E → ℝ}
     (hf : Measurable f) :
     Integrable (fun x => f (X x)) ℙ ↔ Integrable (fun x => f x * (pdf X ℙ μ x).toReal) μ :=
   by
   rw [← integrable_map_measure hf.ae_strongly_measurable (has_pdf.measurable X ℙ μ).AEMeasurable,
     map_eq_with_density_pdf X ℙ μ, integrable_with_density_iff (measurable_pdf _ _ _) ae_lt_top]
   infer_instance
-#align measure_theory.pdf.integrable_iff_integrable_mul_pdf MeasureTheory.pdf.integrable_iff_integrable_mul_pdf
+#align measure_theory.pdf.integrable_iff_integrable_mul_pdf MeasureTheory.pdf.integrable_pdf_smul_iff
 -/
 
-#print MeasureTheory.pdf.integral_fun_mul_eq_integral /-
+#print MeasureTheory.pdf.integral_pdf_smul /-
 /-- **The Law of the Unconscious Statistician**: Given a random variable `X` and a measurable
 function `f`, `f ∘ X` is a random variable with expectation `∫ x, f x * pdf X ∂μ`
 where `μ` is a measure on the codomain of `X`. -/
-theorem integral_fun_mul_eq_integral [IsFiniteMeasure ℙ] {X : Ω → E} [HasPDF X ℙ μ] {f : E → ℝ}
+theorem integral_pdf_smul [IsFiniteMeasure ℙ] {X : Ω → E} [HasPDF X ℙ μ] {f : E → ℝ}
     (hf : Measurable f) : ∫ x, f x * (pdf X ℙ μ x).toReal ∂μ = ∫ x, f (X x) ∂ℙ :=
   by
   by_cases hpdf : integrable (fun x => f x * (pdf X ℙ μ x).toReal) μ
@@ -245,34 +243,35 @@ theorem integral_fun_mul_eq_integral [IsFiniteMeasure ℙ] {X : Ω → E} [HasPD
   · rw [integral_undef hpdf, integral_undef]
     rwa [← integrable_iff_integrable_mul_pdf hf] at hpdf 
     all_goals infer_instance
-#align measure_theory.pdf.integral_fun_mul_eq_integral MeasureTheory.pdf.integral_fun_mul_eq_integral
+#align measure_theory.pdf.integral_fun_mul_eq_integral MeasureTheory.pdf.integral_pdf_smul
 -/
 
-#print MeasureTheory.pdf.map_absolutelyContinuous /-
-theorem map_absolutelyContinuous {X : Ω → E} [HasPDF X ℙ μ] : map X ℙ ≪ μ := by
+#print MeasureTheory.HasPDF.absolutelyContinuous /-
+theorem absolutelyContinuous {X : Ω → E} [HasPDF X ℙ μ] : map X ℙ ≪ μ := by
   rw [map_eq_with_density_pdf X ℙ μ]; exact with_density_absolutely_continuous _ _
-#align measure_theory.pdf.map_absolutely_continuous MeasureTheory.pdf.map_absolutelyContinuous
+#align measure_theory.pdf.map_absolutely_continuous MeasureTheory.HasPDF.absolutelyContinuous
 -/
 
-#print MeasureTheory.pdf.to_quasiMeasurePreserving /-
+#print MeasureTheory.HasPDF.quasiMeasurePreserving_of_measurable /-
 /-- A random variable that `has_pdf` is quasi-measure preserving. -/
-theorem to_quasiMeasurePreserving {X : Ω → E} [HasPDF X ℙ μ] : QuasiMeasurePreserving X ℙ μ :=
-  { Measurable := HasPDF.measurable X ℙ μ
-    AbsolutelyContinuous := map_absolutelyContinuous }
-#align measure_theory.pdf.to_quasi_measure_preserving MeasureTheory.pdf.to_quasiMeasurePreserving
+theorem quasiMeasurePreserving_of_measurable {X : Ω → E} [HasPDF X ℙ μ] :
+    QuasiMeasurePreserving X ℙ μ :=
+  { Measurable := HasPDF.aemeasurable X ℙ μ
+    AbsolutelyContinuous := absolutelyContinuous }
+#align measure_theory.pdf.to_quasi_measure_preserving MeasureTheory.HasPDF.quasiMeasurePreserving_of_measurable
 -/
 
-#print MeasureTheory.pdf.haveLebesgueDecomposition_of_hasPDF /-
-theorem haveLebesgueDecomposition_of_hasPDF {X : Ω → E} [hX' : HasPDF X ℙ μ] :
+#print MeasureTheory.HasPDF.haveLebesgueDecomposition /-
+theorem haveLebesgueDecomposition {X : Ω → E} [hX' : HasPDF X ℙ μ] :
     (map X ℙ).HaveLebesgueDecomposition μ :=
   ⟨⟨⟨0, pdf X ℙ μ⟩, by
       simp only [zero_add, measurable_pdf X ℙ μ, true_and_iff, mutually_singular.zero_left,
         map_eq_with_density_pdf X ℙ μ]⟩⟩
-#align measure_theory.pdf.have_lebesgue_decomposition_of_has_pdf MeasureTheory.pdf.haveLebesgueDecomposition_of_hasPDF
+#align measure_theory.pdf.have_lebesgue_decomposition_of_has_pdf MeasureTheory.HasPDF.haveLebesgueDecomposition
 -/
 
-#print MeasureTheory.pdf.hasPDF_iff /-
-theorem hasPDF_iff {X : Ω → E} :
+#print MeasureTheory.hasPDF_iff /-
+theorem MeasureTheory.hasPDF_iff {X : Ω → E} :
     HasPDF X ℙ μ ↔ Measurable X ∧ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ :=
   by
   constructor
@@ -282,14 +281,14 @@ theorem hasPDF_iff {X : Ω → E} :
     haveI := h_decomp
     refine' ⟨⟨hX, (measure.map X ℙ).rnDeriv μ, measurable_rn_deriv _ _, _⟩⟩
     rwa [with_density_rn_deriv_eq]
-#align measure_theory.pdf.has_pdf_iff MeasureTheory.pdf.hasPDF_iff
+#align measure_theory.pdf.has_pdf_iff MeasureTheory.hasPDF_iff
 -/
 
-#print MeasureTheory.pdf.hasPDF_iff_of_measurable /-
-theorem hasPDF_iff_of_measurable {X : Ω → E} (hX : Measurable X) :
+#print MeasureTheory.hasPDF_iff_of_aemeasurable /-
+theorem MeasureTheory.hasPDF_iff_of_aemeasurable {X : Ω → E} (hX : Measurable X) :
     HasPDF X ℙ μ ↔ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ := by rw [has_pdf_iff];
   simp only [hX, true_and_iff]
-#align measure_theory.pdf.has_pdf_iff_of_measurable MeasureTheory.pdf.hasPDF_iff_of_measurable
+#align measure_theory.pdf.has_pdf_iff_of_measurable MeasureTheory.hasPDF_iff_of_aemeasurable
 -/
 
 section
@@ -330,14 +329,14 @@ section Real
 
 variable [IsFiniteMeasure ℙ] {X : Ω → ℝ}
 
-#print Real.hasPDF_iff_of_measurable /-
+#print Real.hasPDF_iff_of_aemeasurable /-
 /-- A real-valued random variable `X` `has_pdf X ℙ λ` (where `λ` is the Lebesgue measure) if and
 only if the push-forward measure of `ℙ` along `X` is absolutely continuous with respect to `λ`. -/
-theorem Real.hasPDF_iff_of_measurable (hX : Measurable X) : HasPDF X ℙ ↔ map X ℙ ≪ volume :=
+theorem Real.hasPDF_iff_of_aemeasurable (hX : Measurable X) : HasPDF X ℙ ↔ map X ℙ ≪ volume :=
   by
   rw [has_pdf_iff_of_measurable hX, and_iff_right_iff_imp]
   exact fun h => inferInstance
-#align measure_theory.pdf.real.has_pdf_iff_of_measurable Real.hasPDF_iff_of_measurable
+#align measure_theory.pdf.real.has_pdf_iff_of_measurable Real.hasPDF_iff_of_aemeasurable
 -/
 
 #print Real.hasPDF_iff /-
@@ -355,7 +354,7 @@ theorem Real.hasPDF_iff : HasPDF X ℙ ↔ Measurable X ∧ map X ℙ ≪ volume
 /-- If `X` is a real-valued random variable that has pdf `f`, then the expectation of `X` equals
 `∫ x, x * f x ∂λ` where `λ` is the Lebesgue measure. -/
 theorem integral_mul_eq_integral [HasPDF X ℙ] : ∫ x, x * (pdf X ℙ volume x).toReal = ∫ x, X x ∂ℙ :=
-  integral_fun_mul_eq_integral measurable_id
+  integral_pdf_smul measurable_id
 #align measure_theory.pdf.integral_mul_eq_integral MeasureTheory.pdf.integral_mul_eq_integral
 -/
 

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

@@ -3,8 +3,8 @@ 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.RadonNikodym
-import Mathbin.MeasureTheory.Measure.Haar.OfBasis
+import MeasureTheory.Decomposition.RadonNikodym
+import MeasureTheory.Measure.Haar.OfBasis
 
 #align_import probability.density from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
 

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

@@ -2,15 +2,12 @@
 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 probability.density
-! leanprover-community/mathlib commit c14c8fcde993801fca8946b0d80131a1a81d1520
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.MeasureTheory.Decomposition.RadonNikodym
 import Mathbin.MeasureTheory.Measure.Haar.OfBasis
 
+#align_import probability.density from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
+
 /-!
 # Probability density function
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/bf2428c9486c407ca38b5b3fb10b87dad0bc99fa

@@ -417,7 +417,7 @@ theorem hasPDF {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ :
           simp [hnt]
         rw [HEq, Set.inter_univ] at this 
         exact hns this
-      exact MeasureTheory.Set.indicator_ae_eq_zero.1 hu.symm)
+      exact Set.indicator_ae_eq_zero.1 hu.symm)
 #align measure_theory.pdf.is_uniform.has_pdf MeasureTheory.pdf.IsUniform.hasPDF
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/6285167a053ad0990fc88e56c48ccd9fae6550eb

@@ -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 probability.density
-! leanprover-community/mathlib commit 44e2ae8cffc713925494e4975ee31ec1d06929b3
+! leanprover-community/mathlib commit c14c8fcde993801fca8946b0d80131a1a81d1520
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -417,7 +417,7 @@ theorem hasPDF {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ :
           simp [hnt]
         rw [HEq, Set.inter_univ] at this 
         exact hns this
-      exact MeasureTheory.Set.indicator_ae_eq_zero hu.symm)
+      exact MeasureTheory.Set.indicator_ae_eq_zero.1 hu.symm)
 #align measure_theory.pdf.is_uniform.has_pdf MeasureTheory.pdf.IsUniform.hasPDF
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8efcf8022aac8e01df8d302dcebdbc25d6a886c8

@@ -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 probability.density
-! leanprover-community/mathlib commit fd5edc43dc4f10b85abfe544b88f82cf13c5f844
+! leanprover-community/mathlib commit 44e2ae8cffc713925494e4975ee31ec1d06929b3
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.MeasureTheory.Measure.Haar.OfBasis
 /-!
 # Probability density function
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file defines the probability density function of random variables, by which we mean
 measurable functions taking values in a Borel space. In particular, a measurable function `f`
 is said to the probability density function of a random variable `X` if for all measurable

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

@@ -74,12 +74,14 @@ class HasPDF {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
 #align measure_theory.has_pdf MeasureTheory.HasPDF
 -/
 
+#print MeasureTheory.HasPDF.measurable /-
 @[measurability]
 theorem HasPDF.measurable {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
     (μ : Measure E := by exact MeasureTheory.MeasureSpace.volume) [hX : HasPDF X ℙ μ] :
     Measurable X :=
   hX.pdf'.1
 #align measure_theory.has_pdf.measurable MeasureTheory.HasPDF.measurable
+-/
 
 #print MeasureTheory.pdf /-
 /-- If `X` is a random variable that `has_pdf X ℙ μ`, then `pdf X` is the measurable function `f`
@@ -90,27 +92,36 @@ def pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
 #align measure_theory.pdf MeasureTheory.pdf
 -/
 
+#print MeasureTheory.pdf_undef /-
 theorem pdf_undef {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E}
     (h : ¬HasPDF X ℙ μ) : pdf X ℙ μ = 0 := by simp only [pdf, dif_neg h]
 #align measure_theory.pdf_undef MeasureTheory.pdf_undef
+-/
 
+#print MeasureTheory.hasPDF_of_pdf_ne_zero /-
 theorem hasPDF_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E}
     (h : pdf X ℙ μ ≠ 0) : HasPDF X ℙ μ := by
   by_contra hpdf
   rw [pdf, dif_neg hpdf] at h 
   exact hpdf (False.ndrec (has_pdf X ℙ μ) (h rfl))
 #align measure_theory.has_pdf_of_pdf_ne_zero MeasureTheory.hasPDF_of_pdf_ne_zero
+-/
 
+#print MeasureTheory.pdf_eq_zero_of_not_measurable /-
 theorem pdf_eq_zero_of_not_measurable {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
     {X : Ω → E} (hX : ¬Measurable X) : pdf X ℙ μ = 0 :=
   pdf_undef fun hpdf => hX hpdf.pdf'.1
 #align measure_theory.pdf_eq_zero_of_not_measurable MeasureTheory.pdf_eq_zero_of_not_measurable
+-/
 
+#print MeasureTheory.measurable_of_pdf_ne_zero /-
 theorem measurable_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
     (X : Ω → E) (h : pdf X ℙ μ ≠ 0) : Measurable X := by by_contra hX;
   exact h (pdf_eq_zero_of_not_measurable hX)
 #align measure_theory.measurable_of_pdf_ne_zero MeasureTheory.measurable_of_pdf_ne_zero
+-/
 
+#print MeasureTheory.measurable_pdf /-
 @[measurability]
 theorem measurable_pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
     (μ : Measure E := by exact MeasureTheory.MeasureSpace.volume) : Measurable (pdf X ℙ μ) :=
@@ -121,7 +132,9 @@ theorem measurable_pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω
   · rw [pdf, dif_neg hX]
     exact measurable_zero
 #align measure_theory.measurable_pdf MeasureTheory.measurable_pdf
+-/
 
+#print MeasureTheory.map_eq_withDensity_pdf /-
 theorem map_eq_withDensity_pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
     (μ : Measure E := by exact MeasureTheory.MeasureSpace.volume) [hX : HasPDF X ℙ μ] :
     Measure.map X ℙ = μ.withDensity (pdf X ℙ μ) :=
@@ -129,23 +142,29 @@ theorem map_eq_withDensity_pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Me
   rw [pdf, dif_pos hX]
   exact (Classical.choose_spec hX.pdf'.2).2
 #align measure_theory.map_eq_with_density_pdf MeasureTheory.map_eq_withDensity_pdf
+-/
 
+#print MeasureTheory.map_eq_set_lintegral_pdf /-
 theorem map_eq_set_lintegral_pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
     (μ : Measure E := by exact MeasureTheory.MeasureSpace.volume) [hX : HasPDF X ℙ μ] {s : Set E}
     (hs : MeasurableSet s) : Measure.map X ℙ s = ∫⁻ x in s, pdf X ℙ μ x ∂μ := by
   rw [← with_density_apply _ hs, map_eq_with_density_pdf X ℙ μ]
 #align measure_theory.map_eq_set_lintegral_pdf MeasureTheory.map_eq_set_lintegral_pdf
+-/
 
 namespace Pdf
 
 variable {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
 
+#print MeasureTheory.pdf.lintegral_eq_measure_univ /-
 theorem lintegral_eq_measure_univ {X : Ω → E} [HasPDF X ℙ μ] : ∫⁻ x, pdf X ℙ μ x ∂μ = ℙ Set.univ :=
   by
   rw [← set_lintegral_univ, ← map_eq_set_lintegral_pdf X ℙ μ MeasurableSet.univ,
     measure.map_apply (has_pdf.measurable X ℙ μ) MeasurableSet.univ, Set.preimage_univ]
 #align measure_theory.pdf.lintegral_eq_measure_univ MeasureTheory.pdf.lintegral_eq_measure_univ
+-/
 
+#print MeasureTheory.pdf.ae_lt_top /-
 theorem ae_lt_top [IsFiniteMeasure ℙ] {μ : Measure E} {X : Ω → E} : ∀ᵐ x ∂μ, pdf X ℙ μ x < ∞ :=
   by
   by_cases hpdf : has_pdf X ℙ μ
@@ -156,12 +175,16 @@ theorem ae_lt_top [IsFiniteMeasure ℙ] {μ : Measure E} {X : Ω → E} : ∀ᵐ
   · rw [pdf, dif_neg hpdf]
     exact Filter.eventually_of_forall fun x => WithTop.zero_lt_top
 #align measure_theory.pdf.ae_lt_top MeasureTheory.pdf.ae_lt_top
+-/
 
+#print MeasureTheory.pdf.ofReal_toReal_ae_eq /-
 theorem ofReal_toReal_ae_eq [IsFiniteMeasure ℙ] {X : Ω → E} :
     (fun x => ENNReal.ofReal (pdf X ℙ μ x).toReal) =ᵐ[μ] pdf X ℙ μ :=
   ofReal_toReal_ae_eq ae_lt_top
 #align measure_theory.pdf.of_real_to_real_ae_eq MeasureTheory.pdf.ofReal_toReal_ae_eq
+-/
 
+#print MeasureTheory.pdf.integrable_iff_integrable_mul_pdf /-
 theorem integrable_iff_integrable_mul_pdf [IsFiniteMeasure ℙ] {X : Ω → E} [HasPDF X ℙ μ] {f : E → ℝ}
     (hf : Measurable f) :
     Integrable (fun x => f (X x)) ℙ ↔ Integrable (fun x => f x * (pdf X ℙ μ x).toReal) μ :=
@@ -170,7 +193,9 @@ theorem integrable_iff_integrable_mul_pdf [IsFiniteMeasure ℙ] {X : Ω → E} [
     map_eq_with_density_pdf X ℙ μ, integrable_with_density_iff (measurable_pdf _ _ _) ae_lt_top]
   infer_instance
 #align measure_theory.pdf.integrable_iff_integrable_mul_pdf MeasureTheory.pdf.integrable_iff_integrable_mul_pdf
+-/
 
+#print MeasureTheory.pdf.integral_fun_mul_eq_integral /-
 /-- **The Law of the Unconscious Statistician**: Given a random variable `X` and a measurable
 function `f`, `f ∘ X` is a random variable with expectation `∫ x, f x * pdf X ∂μ`
 where `μ` is a measure on the codomain of `X`. -/
@@ -221,24 +246,32 @@ theorem integral_fun_mul_eq_integral [IsFiniteMeasure ℙ] {X : Ω → E} [HasPD
     rwa [← integrable_iff_integrable_mul_pdf hf] at hpdf 
     all_goals infer_instance
 #align measure_theory.pdf.integral_fun_mul_eq_integral MeasureTheory.pdf.integral_fun_mul_eq_integral
+-/
 
+#print MeasureTheory.pdf.map_absolutelyContinuous /-
 theorem map_absolutelyContinuous {X : Ω → E} [HasPDF X ℙ μ] : map X ℙ ≪ μ := by
   rw [map_eq_with_density_pdf X ℙ μ]; exact with_density_absolutely_continuous _ _
 #align measure_theory.pdf.map_absolutely_continuous MeasureTheory.pdf.map_absolutelyContinuous
+-/
 
+#print MeasureTheory.pdf.to_quasiMeasurePreserving /-
 /-- A random variable that `has_pdf` is quasi-measure preserving. -/
 theorem to_quasiMeasurePreserving {X : Ω → E} [HasPDF X ℙ μ] : QuasiMeasurePreserving X ℙ μ :=
   { Measurable := HasPDF.measurable X ℙ μ
     AbsolutelyContinuous := map_absolutelyContinuous }
 #align measure_theory.pdf.to_quasi_measure_preserving MeasureTheory.pdf.to_quasiMeasurePreserving
+-/
 
+#print MeasureTheory.pdf.haveLebesgueDecomposition_of_hasPDF /-
 theorem haveLebesgueDecomposition_of_hasPDF {X : Ω → E} [hX' : HasPDF X ℙ μ] :
     (map X ℙ).HaveLebesgueDecomposition μ :=
   ⟨⟨⟨0, pdf X ℙ μ⟩, by
       simp only [zero_add, measurable_pdf X ℙ μ, true_and_iff, mutually_singular.zero_left,
         map_eq_with_density_pdf X ℙ μ]⟩⟩
 #align measure_theory.pdf.have_lebesgue_decomposition_of_has_pdf MeasureTheory.pdf.haveLebesgueDecomposition_of_hasPDF
+-/
 
+#print MeasureTheory.pdf.hasPDF_iff /-
 theorem hasPDF_iff {X : Ω → E} :
     HasPDF X ℙ μ ↔ Measurable X ∧ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ :=
   by
@@ -250,16 +283,20 @@ theorem hasPDF_iff {X : Ω → E} :
     refine' ⟨⟨hX, (measure.map X ℙ).rnDeriv μ, measurable_rn_deriv _ _, _⟩⟩
     rwa [with_density_rn_deriv_eq]
 #align measure_theory.pdf.has_pdf_iff MeasureTheory.pdf.hasPDF_iff
+-/
 
+#print MeasureTheory.pdf.hasPDF_iff_of_measurable /-
 theorem hasPDF_iff_of_measurable {X : Ω → E} (hX : Measurable X) :
     HasPDF X ℙ μ ↔ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ := by rw [has_pdf_iff];
   simp only [hX, true_and_iff]
 #align measure_theory.pdf.has_pdf_iff_of_measurable MeasureTheory.pdf.hasPDF_iff_of_measurable
+-/
 
 section
 
 variable {F : Type _} [MeasurableSpace F] {ν : Measure F}
 
+#print MeasureTheory.pdf.quasiMeasurePreserving_hasPDF /-
 /-- A random variable that `has_pdf` transformed under a `quasi_measure_preserving`
 map also `has_pdf` if `(map g (map X ℙ)).have_lebesgue_decomposition μ`.
 
@@ -278,11 +315,14 @@ theorem quasiMeasurePreserving_hasPDF {X : Ω → E} [HasPDF X ℙ μ] {g : E 
   rw [map_apply hg.measurable hsm] at this 
   exact set_lintegral_measure_zero _ _ this
 #align measure_theory.pdf.quasi_measure_preserving_has_pdf MeasureTheory.pdf.quasiMeasurePreserving_hasPDF
+-/
 
+#print MeasureTheory.pdf.quasiMeasurePreserving_hasPDF' /-
 theorem quasiMeasurePreserving_hasPDF' [IsFiniteMeasure ℙ] [SigmaFinite ν] {X : Ω → E}
     [HasPDF X ℙ μ] {g : E → F} (hg : QuasiMeasurePreserving g μ ν) : HasPDF (g ∘ X) ℙ ν :=
   quasiMeasurePreserving_hasPDF hg inferInstance
 #align measure_theory.pdf.quasi_measure_preserving_has_pdf' MeasureTheory.pdf.quasiMeasurePreserving_hasPDF'
+-/
 
 end
 
@@ -311,12 +351,15 @@ theorem Real.hasPDF_iff : HasPDF X ℙ ↔ Measurable X ∧ map X ℙ ≪ volume
 #align measure_theory.pdf.real.has_pdf_iff Real.hasPDF_iff
 -/
 
+#print MeasureTheory.pdf.integral_mul_eq_integral /-
 /-- If `X` is a real-valued random variable that has pdf `f`, then the expectation of `X` equals
 `∫ x, x * f x ∂λ` where `λ` is the Lebesgue measure. -/
 theorem integral_mul_eq_integral [HasPDF X ℙ] : ∫ x, x * (pdf X ℙ volume x).toReal = ∫ x, X x ∂ℙ :=
   integral_fun_mul_eq_integral measurable_id
 #align measure_theory.pdf.integral_mul_eq_integral MeasureTheory.pdf.integral_mul_eq_integral
+-/
 
+#print MeasureTheory.pdf.hasFiniteIntegral_mul /-
 theorem hasFiniteIntegral_mul {f : ℝ → ℝ} {g : ℝ → ℝ≥0∞} (hg : pdf X ℙ =ᵐ[volume] g)
     (hgi : ∫⁻ x, ‖f x‖₊ * g x ≠ ∞) : HasFiniteIntegral fun x => f x * (pdf X ℙ volume x).toReal :=
   by
@@ -335,6 +378,7 @@ theorem hasFiniteIntegral_mul {f : ℝ → ℝ} {g : ℝ → ℝ≥0∞} (hg : p
     exact Real.ennnorm_eq_ofReal ENNReal.toReal_nonneg
   rwa [lt_top_iff_ne_top, ← lintegral_congr_ae this]
 #align measure_theory.pdf.has_finite_integral_mul MeasureTheory.pdf.hasFiniteIntegral_mul
+-/
 
 end Real
 
@@ -354,6 +398,7 @@ def IsUniform {m : MeasurableSpace Ω} (X : Ω → E) (support : Set E) (ℙ : M
 
 namespace IsUniform
 
+#print MeasureTheory.pdf.IsUniform.hasPDF /-
 theorem hasPDF {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E} {s : Set E}
     (hns : μ s ≠ 0) (hnt : μ s ≠ ∞) (hu : IsUniform X s ℙ μ) : HasPDF X ℙ μ :=
   hasPDF_of_pdf_ne_zero
@@ -371,14 +416,18 @@ theorem hasPDF {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ :
         exact hns this
       exact MeasureTheory.Set.indicator_ae_eq_zero hu.symm)
 #align measure_theory.pdf.is_uniform.has_pdf MeasureTheory.pdf.IsUniform.hasPDF
+-/
 
+#print MeasureTheory.pdf.IsUniform.pdf_toReal_ae_eq /-
 theorem pdf_toReal_ae_eq {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E}
     {s : Set E} (hX : IsUniform X s ℙ μ) :
     (fun x => (pdf X ℙ μ x).toReal) =ᵐ[μ] fun x =>
       (s.indicator ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) x).toReal :=
   Filter.EventuallyEq.fun_comp hX ENNReal.toReal
 #align measure_theory.pdf.is_uniform.pdf_to_real_ae_eq MeasureTheory.pdf.IsUniform.pdf_toReal_ae_eq
+-/
 
+#print MeasureTheory.pdf.IsUniform.measure_preimage /-
 theorem measure_preimage {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E}
     {s : Set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞) (hms : MeasurableSet s) (hu : IsUniform X s ℙ μ)
     {A : Set E} (hA : MeasurableSet A) : ℙ (X ⁻¹' A) = μ (s ∩ A) / μ s :=
@@ -390,7 +439,9 @@ theorem measure_preimage {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure
     mul_one, lintegral_const, restrict_apply', Set.univ_inter]
   rw [ENNReal.div_eq_inv_mul]
 #align measure_theory.pdf.is_uniform.measure_preimage MeasureTheory.pdf.IsUniform.measure_preimage
+-/
 
+#print MeasureTheory.pdf.IsUniform.isProbabilityMeasure /-
 theorem isProbabilityMeasure {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E}
     {s : Set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞) (hms : MeasurableSet s) (hu : IsUniform X s ℙ μ) :
     IsProbabilityMeasure ℙ :=
@@ -399,11 +450,11 @@ theorem isProbabilityMeasure {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Meas
     rw [← this, hu.measure_preimage hns hnt hms MeasurableSet.univ, Set.inter_univ,
       ENNReal.div_self hns hnt]⟩
 #align measure_theory.pdf.is_uniform.is_probability_measure MeasureTheory.pdf.IsUniform.isProbabilityMeasure
+-/
 
 variable {X : Ω → ℝ} {s : Set ℝ} (hms : MeasurableSet s) (hns : volume s ≠ 0)
 
-include hms hns
-
+#print MeasureTheory.pdf.IsUniform.mul_pdf_integrable /-
 theorem mul_pdf_integrable [IsFiniteMeasure ℙ] (hcs : IsCompact s) (huX : IsUniform X s ℙ) :
     Integrable fun x : ℝ => x * (pdf X ℙ volume x).toReal :=
   by
@@ -432,7 +483,9 @@ theorem mul_pdf_integrable [IsFiniteMeasure ℙ] (hcs : IsCompact s) (huX : IsUn
           (ENNReal.inv_lt_top.2 (pos_iff_ne_zero.mpr hns)).Ne).Ne
   · infer_instance
 #align measure_theory.pdf.is_uniform.mul_pdf_integrable MeasureTheory.pdf.IsUniform.mul_pdf_integrable
+-/
 
+#print MeasureTheory.pdf.IsUniform.integral_eq /-
 /-- A real uniform random variable `X` with support `s` has expectation
 `(λ s)⁻¹ * ∫ x in s, x ∂λ` where `λ` is the Lebesgue measure. -/
 theorem integral_eq (hnt : volume s ≠ ∞) (huX : IsUniform X s ℙ) :
@@ -455,6 +508,7 @@ theorem integral_eq (hnt : volume s ≠ ∞) (huX : IsUniform X s ℙ) :
   change ∫ x in s, x * (volume s)⁻¹.toReal • 1 ∂volume = _
   rw [integral_mul_right, mul_comm, Algebra.id.smul_eq_mul, mul_one]
 #align measure_theory.pdf.is_uniform.integral_eq MeasureTheory.pdf.IsUniform.integral_eq
+-/
 
 end IsUniform
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f

@@ -64,38 +64,42 @@ open TopologicalSpace MeasureTheory.Measure
 
 variable {Ω E : Type _} [MeasurableSpace E]
 
+#print MeasureTheory.HasPDF /-
 /-- A random variable `X : Ω → E` is said to `has_pdf` with respect to the measure `ℙ` on `Ω` and
 `μ` on `E` if there exists a measurable function `f` such that the push-forward measure of `ℙ`
 along `X` equals `μ.with_density f`. -/
-class HasPdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
+class HasPDF {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
     (μ : Measure E := by exact MeasureTheory.MeasureSpace.volume) : Prop where
   pdf' : Measurable X ∧ ∃ f : E → ℝ≥0∞, Measurable f ∧ map X ℙ = μ.withDensity f
-#align measure_theory.has_pdf MeasureTheory.HasPdf
+#align measure_theory.has_pdf MeasureTheory.HasPDF
+-/
 
 @[measurability]
-theorem HasPdf.measurable {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
-    (μ : Measure E := by exact MeasureTheory.MeasureSpace.volume) [hX : HasPdf X ℙ μ] :
+theorem HasPDF.measurable {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
+    (μ : Measure E := by exact MeasureTheory.MeasureSpace.volume) [hX : HasPDF X ℙ μ] :
     Measurable X :=
   hX.pdf'.1
-#align measure_theory.has_pdf.measurable MeasureTheory.HasPdf.measurable
+#align measure_theory.has_pdf.measurable MeasureTheory.HasPDF.measurable
 
+#print MeasureTheory.pdf /-
 /-- If `X` is a random variable that `has_pdf X ℙ μ`, then `pdf X` is the measurable function `f`
 such that the push-forward measure of `ℙ` along `X` equals `μ.with_density f`. -/
 def pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
     (μ : Measure E := by exact MeasureTheory.MeasureSpace.volume) :=
-  if hX : HasPdf X ℙ μ then Classical.choose hX.pdf'.2 else 0
+  if hX : HasPDF X ℙ μ then Classical.choose hX.pdf'.2 else 0
 #align measure_theory.pdf MeasureTheory.pdf
+-/
 
 theorem pdf_undef {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E}
-    (h : ¬HasPdf X ℙ μ) : pdf X ℙ μ = 0 := by simp only [pdf, dif_neg h]
+    (h : ¬HasPDF X ℙ μ) : pdf X ℙ μ = 0 := by simp only [pdf, dif_neg h]
 #align measure_theory.pdf_undef MeasureTheory.pdf_undef
 
-theorem hasPdf_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E}
-    (h : pdf X ℙ μ ≠ 0) : HasPdf X ℙ μ := by
+theorem hasPDF_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E}
+    (h : pdf X ℙ μ ≠ 0) : HasPDF X ℙ μ := by
   by_contra hpdf
   rw [pdf, dif_neg hpdf] at h 
   exact hpdf (False.ndrec (has_pdf X ℙ μ) (h rfl))
-#align measure_theory.has_pdf_of_pdf_ne_zero MeasureTheory.hasPdf_of_pdf_ne_zero
+#align measure_theory.has_pdf_of_pdf_ne_zero MeasureTheory.hasPDF_of_pdf_ne_zero
 
 theorem pdf_eq_zero_of_not_measurable {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
     {X : Ω → E} (hX : ¬Measurable X) : pdf X ℙ μ = 0 :=
@@ -119,7 +123,7 @@ theorem measurable_pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω
 #align measure_theory.measurable_pdf MeasureTheory.measurable_pdf
 
 theorem map_eq_withDensity_pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
-    (μ : Measure E := by exact MeasureTheory.MeasureSpace.volume) [hX : HasPdf X ℙ μ] :
+    (μ : Measure E := by exact MeasureTheory.MeasureSpace.volume) [hX : HasPDF X ℙ μ] :
     Measure.map X ℙ = μ.withDensity (pdf X ℙ μ) :=
   by
   rw [pdf, dif_pos hX]
@@ -127,7 +131,7 @@ theorem map_eq_withDensity_pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Me
 #align measure_theory.map_eq_with_density_pdf MeasureTheory.map_eq_withDensity_pdf
 
 theorem map_eq_set_lintegral_pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
-    (μ : Measure E := by exact MeasureTheory.MeasureSpace.volume) [hX : HasPdf X ℙ μ] {s : Set E}
+    (μ : Measure E := by exact MeasureTheory.MeasureSpace.volume) [hX : HasPDF X ℙ μ] {s : Set E}
     (hs : MeasurableSet s) : Measure.map X ℙ s = ∫⁻ x in s, pdf X ℙ μ x ∂μ := by
   rw [← with_density_apply _ hs, map_eq_with_density_pdf X ℙ μ]
 #align measure_theory.map_eq_set_lintegral_pdf MeasureTheory.map_eq_set_lintegral_pdf
@@ -136,7 +140,7 @@ namespace Pdf
 
 variable {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
 
-theorem lintegral_eq_measure_univ {X : Ω → E} [HasPdf X ℙ μ] : ∫⁻ x, pdf X ℙ μ x ∂μ = ℙ Set.univ :=
+theorem lintegral_eq_measure_univ {X : Ω → E} [HasPDF X ℙ μ] : ∫⁻ x, pdf X ℙ μ x ∂μ = ℙ Set.univ :=
   by
   rw [← set_lintegral_univ, ← map_eq_set_lintegral_pdf X ℙ μ MeasurableSet.univ,
     measure.map_apply (has_pdf.measurable X ℙ μ) MeasurableSet.univ, Set.preimage_univ]
@@ -158,7 +162,7 @@ theorem ofReal_toReal_ae_eq [IsFiniteMeasure ℙ] {X : Ω → E} :
   ofReal_toReal_ae_eq ae_lt_top
 #align measure_theory.pdf.of_real_to_real_ae_eq MeasureTheory.pdf.ofReal_toReal_ae_eq
 
-theorem integrable_iff_integrable_mul_pdf [IsFiniteMeasure ℙ] {X : Ω → E} [HasPdf X ℙ μ] {f : E → ℝ}
+theorem integrable_iff_integrable_mul_pdf [IsFiniteMeasure ℙ] {X : Ω → E} [HasPDF X ℙ μ] {f : E → ℝ}
     (hf : Measurable f) :
     Integrable (fun x => f (X x)) ℙ ↔ Integrable (fun x => f x * (pdf X ℙ μ x).toReal) μ :=
   by
@@ -170,7 +174,7 @@ theorem integrable_iff_integrable_mul_pdf [IsFiniteMeasure ℙ] {X : Ω → E} [
 /-- **The Law of the Unconscious Statistician**: Given a random variable `X` and a measurable
 function `f`, `f ∘ X` is a random variable with expectation `∫ x, f x * pdf X ∂μ`
 where `μ` is a measure on the codomain of `X`. -/
-theorem integral_fun_mul_eq_integral [IsFiniteMeasure ℙ] {X : Ω → E} [HasPdf X ℙ μ] {f : E → ℝ}
+theorem integral_fun_mul_eq_integral [IsFiniteMeasure ℙ] {X : Ω → E} [HasPDF X ℙ μ] {f : E → ℝ}
     (hf : Measurable f) : ∫ x, f x * (pdf X ℙ μ x).toReal ∂μ = ∫ x, f (X x) ∂ℙ :=
   by
   by_cases hpdf : integrable (fun x => f x * (pdf X ℙ μ x).toReal) μ
@@ -218,25 +222,25 @@ theorem integral_fun_mul_eq_integral [IsFiniteMeasure ℙ] {X : Ω → E} [HasPd
     all_goals infer_instance
 #align measure_theory.pdf.integral_fun_mul_eq_integral MeasureTheory.pdf.integral_fun_mul_eq_integral
 
-theorem map_absolutelyContinuous {X : Ω → E} [HasPdf X ℙ μ] : map X ℙ ≪ μ := by
+theorem map_absolutelyContinuous {X : Ω → E} [HasPDF X ℙ μ] : map X ℙ ≪ μ := by
   rw [map_eq_with_density_pdf X ℙ μ]; exact with_density_absolutely_continuous _ _
 #align measure_theory.pdf.map_absolutely_continuous MeasureTheory.pdf.map_absolutelyContinuous
 
 /-- A random variable that `has_pdf` is quasi-measure preserving. -/
-theorem to_quasiMeasurePreserving {X : Ω → E} [HasPdf X ℙ μ] : QuasiMeasurePreserving X ℙ μ :=
-  { Measurable := HasPdf.measurable X ℙ μ
+theorem to_quasiMeasurePreserving {X : Ω → E} [HasPDF X ℙ μ] : QuasiMeasurePreserving X ℙ μ :=
+  { Measurable := HasPDF.measurable X ℙ μ
     AbsolutelyContinuous := map_absolutelyContinuous }
 #align measure_theory.pdf.to_quasi_measure_preserving MeasureTheory.pdf.to_quasiMeasurePreserving
 
-theorem haveLebesgueDecomposition_of_hasPdf {X : Ω → E} [hX' : HasPdf X ℙ μ] :
+theorem haveLebesgueDecomposition_of_hasPDF {X : Ω → E} [hX' : HasPDF X ℙ μ] :
     (map X ℙ).HaveLebesgueDecomposition μ :=
   ⟨⟨⟨0, pdf X ℙ μ⟩, by
       simp only [zero_add, measurable_pdf X ℙ μ, true_and_iff, mutually_singular.zero_left,
         map_eq_with_density_pdf X ℙ μ]⟩⟩
-#align measure_theory.pdf.have_lebesgue_decomposition_of_has_pdf MeasureTheory.pdf.haveLebesgueDecomposition_of_hasPdf
+#align measure_theory.pdf.have_lebesgue_decomposition_of_has_pdf MeasureTheory.pdf.haveLebesgueDecomposition_of_hasPDF
 
-theorem hasPdf_iff {X : Ω → E} :
-    HasPdf X ℙ μ ↔ Measurable X ∧ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ :=
+theorem hasPDF_iff {X : Ω → E} :
+    HasPDF X ℙ μ ↔ Measurable X ∧ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ :=
   by
   constructor
   · intro hX'
@@ -245,12 +249,12 @@ theorem hasPdf_iff {X : Ω → E} :
     haveI := h_decomp
     refine' ⟨⟨hX, (measure.map X ℙ).rnDeriv μ, measurable_rn_deriv _ _, _⟩⟩
     rwa [with_density_rn_deriv_eq]
-#align measure_theory.pdf.has_pdf_iff MeasureTheory.pdf.hasPdf_iff
+#align measure_theory.pdf.has_pdf_iff MeasureTheory.pdf.hasPDF_iff
 
-theorem hasPdf_iff_of_measurable {X : Ω → E} (hX : Measurable X) :
-    HasPdf X ℙ μ ↔ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ := by rw [has_pdf_iff];
+theorem hasPDF_iff_of_measurable {X : Ω → E} (hX : Measurable X) :
+    HasPDF X ℙ μ ↔ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ := by rw [has_pdf_iff];
   simp only [hX, true_and_iff]
-#align measure_theory.pdf.has_pdf_iff_of_measurable MeasureTheory.pdf.hasPdf_iff_of_measurable
+#align measure_theory.pdf.has_pdf_iff_of_measurable MeasureTheory.pdf.hasPDF_iff_of_measurable
 
 section
 
@@ -261,9 +265,9 @@ map also `has_pdf` if `(map g (map X ℙ)).have_lebesgue_decomposition μ`.
 
 `quasi_measure_preserving_has_pdf'` is more useful in the case we are working with a
 probability measure and a real-valued random variable. -/
-theorem quasiMeasurePreserving_hasPdf {X : Ω → E} [HasPdf X ℙ μ] {g : E → F}
+theorem quasiMeasurePreserving_hasPDF {X : Ω → E} [HasPDF X ℙ μ] {g : E → F}
     (hg : QuasiMeasurePreserving g μ ν) (hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν) :
-    HasPdf (g ∘ X) ℙ ν :=
+    HasPDF (g ∘ X) ℙ ν :=
   by
   rw [has_pdf_iff, ← map_map hg.measurable (has_pdf.measurable X ℙ μ)]
   refine' ⟨hg.measurable.comp (has_pdf.measurable X ℙ μ), hmap, _⟩
@@ -273,12 +277,12 @@ theorem quasiMeasurePreserving_hasPdf {X : Ω → E} [HasPdf X ℙ μ] {g : E 
   have := hg.absolutely_continuous hs
   rw [map_apply hg.measurable hsm] at this 
   exact set_lintegral_measure_zero _ _ this
-#align measure_theory.pdf.quasi_measure_preserving_has_pdf MeasureTheory.pdf.quasiMeasurePreserving_hasPdf
+#align measure_theory.pdf.quasi_measure_preserving_has_pdf MeasureTheory.pdf.quasiMeasurePreserving_hasPDF
 
-theorem quasiMeasurePreserving_has_pdf' [IsFiniteMeasure ℙ] [SigmaFinite ν] {X : Ω → E}
-    [HasPdf X ℙ μ] {g : E → F} (hg : QuasiMeasurePreserving g μ ν) : HasPdf (g ∘ X) ℙ ν :=
-  quasiMeasurePreserving_hasPdf hg inferInstance
-#align measure_theory.pdf.quasi_measure_preserving_has_pdf' MeasureTheory.pdf.quasiMeasurePreserving_has_pdf'
+theorem quasiMeasurePreserving_hasPDF' [IsFiniteMeasure ℙ] [SigmaFinite ν] {X : Ω → E}
+    [HasPDF X ℙ μ] {g : E → F} (hg : QuasiMeasurePreserving g μ ν) : HasPDF (g ∘ X) ℙ ν :=
+  quasiMeasurePreserving_hasPDF hg inferInstance
+#align measure_theory.pdf.quasi_measure_preserving_has_pdf' MeasureTheory.pdf.quasiMeasurePreserving_hasPDF'
 
 end
 
@@ -286,26 +290,30 @@ section Real
 
 variable [IsFiniteMeasure ℙ] {X : Ω → ℝ}
 
+#print Real.hasPDF_iff_of_measurable /-
 /-- A real-valued random variable `X` `has_pdf X ℙ λ` (where `λ` is the Lebesgue measure) if and
 only if the push-forward measure of `ℙ` along `X` is absolutely continuous with respect to `λ`. -/
-theorem Real.hasPdf_iff_of_measurable (hX : Measurable X) : HasPdf X ℙ ↔ map X ℙ ≪ volume :=
+theorem Real.hasPDF_iff_of_measurable (hX : Measurable X) : HasPDF X ℙ ↔ map X ℙ ≪ volume :=
   by
   rw [has_pdf_iff_of_measurable hX, and_iff_right_iff_imp]
   exact fun h => inferInstance
-#align measure_theory.pdf.real.has_pdf_iff_of_measurable MeasureTheory.pdf.Real.hasPdf_iff_of_measurable
+#align measure_theory.pdf.real.has_pdf_iff_of_measurable Real.hasPDF_iff_of_measurable
+-/
 
-theorem Real.hasPdf_iff : HasPdf X ℙ ↔ Measurable X ∧ map X ℙ ≪ volume :=
+#print Real.hasPDF_iff /-
+theorem Real.hasPDF_iff : HasPDF X ℙ ↔ Measurable X ∧ map X ℙ ≪ volume :=
   by
   by_cases hX : Measurable X
   · rw [real.has_pdf_iff_of_measurable hX, iff_and_self]
     exact fun h => hX
     infer_instance
   · exact ⟨fun h => False.elim (hX h.pdf'.1), fun h => False.elim (hX h.1)⟩
-#align measure_theory.pdf.real.has_pdf_iff MeasureTheory.pdf.Real.hasPdf_iff
+#align measure_theory.pdf.real.has_pdf_iff Real.hasPDF_iff
+-/
 
 /-- If `X` is a real-valued random variable that has pdf `f`, then the expectation of `X` equals
 `∫ x, x * f x ∂λ` where `λ` is the Lebesgue measure. -/
-theorem integral_mul_eq_integral [HasPdf X ℙ] : ∫ x, x * (pdf X ℙ volume x).toReal = ∫ x, X x ∂ℙ :=
+theorem integral_mul_eq_integral [HasPDF X ℙ] : ∫ x, x * (pdf X ℙ volume x).toReal = ∫ x, X x ∂ℙ :=
   integral_fun_mul_eq_integral measurable_id
 #align measure_theory.pdf.integral_mul_eq_integral MeasureTheory.pdf.integral_mul_eq_integral
 
@@ -335,18 +343,20 @@ section
 /-! **Uniform Distribution** -/
 
 
+#print MeasureTheory.pdf.IsUniform /-
 /-- A random variable `X` has uniform distribution if it has a probability density function `f`
 with support `s` such that `f = (μ s)⁻¹ 1ₛ` a.e. where `1ₛ` is the indicator function for `s`. -/
 def IsUniform {m : MeasurableSpace Ω} (X : Ω → E) (support : Set E) (ℙ : Measure Ω)
     (μ : Measure E := by exact MeasureTheory.MeasureSpace.volume) :=
   pdf X ℙ μ =ᵐ[μ] support.indicator ((μ support)⁻¹ • 1)
 #align measure_theory.pdf.is_uniform MeasureTheory.pdf.IsUniform
+-/
 
 namespace IsUniform
 
-theorem hasPdf {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E} {s : Set E}
-    (hns : μ s ≠ 0) (hnt : μ s ≠ ∞) (hu : IsUniform X s ℙ μ) : HasPdf X ℙ μ :=
-  hasPdf_of_pdf_ne_zero
+theorem hasPDF {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E} {s : Set E}
+    (hns : μ s ≠ 0) (hnt : μ s ≠ ∞) (hu : IsUniform X s ℙ μ) : HasPDF X ℙ μ :=
+  hasPDF_of_pdf_ne_zero
     (by
       intro hpdf
       rw [is_uniform, hpdf] at hu 
@@ -360,7 +370,7 @@ theorem hasPdf {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ :
         rw [HEq, Set.inter_univ] at this 
         exact hns this
       exact MeasureTheory.Set.indicator_ae_eq_zero hu.symm)
-#align measure_theory.pdf.is_uniform.has_pdf MeasureTheory.pdf.IsUniform.hasPdf
+#align measure_theory.pdf.is_uniform.has_pdf MeasureTheory.pdf.IsUniform.hasPDF
 
 theorem pdf_toReal_ae_eq {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E}
     {s : Set E} (hX : IsUniform X s ℙ μ) :

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

@@ -136,8 +136,8 @@ namespace Pdf
 
 variable {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
 
-theorem lintegral_eq_measure_univ {X : Ω → E} [HasPdf X ℙ μ] :
-    (∫⁻ x, pdf X ℙ μ x ∂μ) = ℙ Set.univ := by
+theorem lintegral_eq_measure_univ {X : Ω → E} [HasPdf X ℙ μ] : ∫⁻ x, pdf X ℙ μ x ∂μ = ℙ Set.univ :=
+  by
   rw [← set_lintegral_univ, ← map_eq_set_lintegral_pdf X ℙ μ MeasurableSet.univ,
     measure.map_apply (has_pdf.measurable X ℙ μ) MeasurableSet.univ, Set.preimage_univ]
 #align measure_theory.pdf.lintegral_eq_measure_univ MeasureTheory.pdf.lintegral_eq_measure_univ
@@ -171,7 +171,7 @@ theorem integrable_iff_integrable_mul_pdf [IsFiniteMeasure ℙ] {X : Ω → E} [
 function `f`, `f ∘ X` is a random variable with expectation `∫ x, f x * pdf X ∂μ`
 where `μ` is a measure on the codomain of `X`. -/
 theorem integral_fun_mul_eq_integral [IsFiniteMeasure ℙ] {X : Ω → E} [HasPdf X ℙ μ] {f : E → ℝ}
-    (hf : Measurable f) : (∫ x, f x * (pdf X ℙ μ x).toReal ∂μ) = ∫ x, f (X x) ∂ℙ :=
+    (hf : Measurable f) : ∫ x, f x * (pdf X ℙ μ x).toReal ∂μ = ∫ x, f (X x) ∂ℙ :=
   by
   by_cases hpdf : integrable (fun x => f x * (pdf X ℙ μ x).toReal) μ
   · rw [← integral_map (has_pdf.measurable X ℙ μ).AEMeasurable hf.ae_strongly_measurable,
@@ -305,13 +305,12 @@ theorem Real.hasPdf_iff : HasPdf X ℙ ↔ Measurable X ∧ map X ℙ ≪ volume
 
 /-- If `X` is a real-valued random variable that has pdf `f`, then the expectation of `X` equals
 `∫ x, x * f x ∂λ` where `λ` is the Lebesgue measure. -/
-theorem integral_mul_eq_integral [HasPdf X ℙ] :
-    (∫ x, x * (pdf X ℙ volume x).toReal) = ∫ x, X x ∂ℙ :=
+theorem integral_mul_eq_integral [HasPdf X ℙ] : ∫ x, x * (pdf X ℙ volume x).toReal = ∫ x, X x ∂ℙ :=
   integral_fun_mul_eq_integral measurable_id
 #align measure_theory.pdf.integral_mul_eq_integral MeasureTheory.pdf.integral_mul_eq_integral
 
 theorem hasFiniteIntegral_mul {f : ℝ → ℝ} {g : ℝ → ℝ≥0∞} (hg : pdf X ℙ =ᵐ[volume] g)
-    (hgi : (∫⁻ x, ‖f x‖₊ * g x) ≠ ∞) : HasFiniteIntegral fun x => f x * (pdf X ℙ volume x).toReal :=
+    (hgi : ∫⁻ x, ‖f x‖₊ * g x ≠ ∞) : HasFiniteIntegral fun x => f x * (pdf X ℙ volume x).toReal :=
   by
   rw [has_finite_integral]
   have : (fun x => ↑‖f x‖₊ * g x) =ᵐ[volume] fun x => ‖f x * (pdf X ℙ volume x).toReal‖₊ :=
@@ -427,7 +426,7 @@ theorem mul_pdf_integrable [IsFiniteMeasure ℙ] (hcs : IsCompact s) (huX : IsUn
 /-- A real uniform random variable `X` with support `s` has expectation
 `(λ s)⁻¹ * ∫ x in s, x ∂λ` where `λ` is the Lebesgue measure. -/
 theorem integral_eq (hnt : volume s ≠ ∞) (huX : IsUniform X s ℙ) :
-    (∫ x, X x ∂ℙ) = (volume s)⁻¹.toReal * ∫ x in s, x :=
+    ∫ x, X x ∂ℙ = (volume s)⁻¹.toReal * ∫ x in s, x :=
   by
   haveI := has_pdf hns hnt huX
   haveI := huX.is_probability_measure hns hnt hms
@@ -443,7 +442,7 @@ theorem integral_eq (hnt : volume s ≠ ∞) (huX : IsUniform X s ℙ) :
     · simp [Set.indicator_of_mem hx]
     · simp [Set.indicator_of_not_mem hx]
   simp_rw [this, ← s.indicator_mul_right fun x => x, integral_indicator hms]
-  change (∫ x in s, x * (volume s)⁻¹.toReal • 1 ∂volume) = _
+  change ∫ x in s, x * (volume s)⁻¹.toReal • 1 ∂volume = _
   rw [integral_mul_right, mul_comm, Algebra.id.smul_eq_mul, mul_one]
 #align measure_theory.pdf.is_uniform.integral_eq MeasureTheory.pdf.IsUniform.integral_eq
 

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

@@ -142,7 +142,7 @@ theorem lintegral_eq_measure_univ {X : Ω → E} [HasPdf X ℙ μ] :
     measure.map_apply (has_pdf.measurable X ℙ μ) MeasurableSet.univ, Set.preimage_univ]
 #align measure_theory.pdf.lintegral_eq_measure_univ MeasureTheory.pdf.lintegral_eq_measure_univ
 
-theorem ae_lt_top [FiniteMeasure ℙ] {μ : Measure E} {X : Ω → E} : ∀ᵐ x ∂μ, pdf X ℙ μ x < ∞ :=
+theorem ae_lt_top [IsFiniteMeasure ℙ] {μ : Measure E} {X : Ω → E} : ∀ᵐ x ∂μ, pdf X ℙ μ x < ∞ :=
   by
   by_cases hpdf : has_pdf X ℙ μ
   · haveI := hpdf
@@ -153,12 +153,12 @@ theorem ae_lt_top [FiniteMeasure ℙ] {μ : Measure E} {X : Ω → E} : ∀ᵐ x
     exact Filter.eventually_of_forall fun x => WithTop.zero_lt_top
 #align measure_theory.pdf.ae_lt_top MeasureTheory.pdf.ae_lt_top
 
-theorem ofReal_toReal_ae_eq [FiniteMeasure ℙ] {X : Ω → E} :
+theorem ofReal_toReal_ae_eq [IsFiniteMeasure ℙ] {X : Ω → E} :
     (fun x => ENNReal.ofReal (pdf X ℙ μ x).toReal) =ᵐ[μ] pdf X ℙ μ :=
   ofReal_toReal_ae_eq ae_lt_top
 #align measure_theory.pdf.of_real_to_real_ae_eq MeasureTheory.pdf.ofReal_toReal_ae_eq
 
-theorem integrable_iff_integrable_mul_pdf [FiniteMeasure ℙ] {X : Ω → E} [HasPdf X ℙ μ] {f : E → ℝ}
+theorem integrable_iff_integrable_mul_pdf [IsFiniteMeasure ℙ] {X : Ω → E} [HasPdf X ℙ μ] {f : E → ℝ}
     (hf : Measurable f) :
     Integrable (fun x => f (X x)) ℙ ↔ Integrable (fun x => f x * (pdf X ℙ μ x).toReal) μ :=
   by
@@ -170,7 +170,7 @@ theorem integrable_iff_integrable_mul_pdf [FiniteMeasure ℙ] {X : Ω → E} [Ha
 /-- **The Law of the Unconscious Statistician**: Given a random variable `X` and a measurable
 function `f`, `f ∘ X` is a random variable with expectation `∫ x, f x * pdf X ∂μ`
 where `μ` is a measure on the codomain of `X`. -/
-theorem integral_fun_mul_eq_integral [FiniteMeasure ℙ] {X : Ω → E} [HasPdf X ℙ μ] {f : E → ℝ}
+theorem integral_fun_mul_eq_integral [IsFiniteMeasure ℙ] {X : Ω → E} [HasPdf X ℙ μ] {f : E → ℝ}
     (hf : Measurable f) : (∫ x, f x * (pdf X ℙ μ x).toReal ∂μ) = ∫ x, f (X x) ∂ℙ :=
   by
   by_cases hpdf : integrable (fun x => f x * (pdf X ℙ μ x).toReal) μ
@@ -275,8 +275,8 @@ theorem quasiMeasurePreserving_hasPdf {X : Ω → E} [HasPdf X ℙ μ] {g : E 
   exact set_lintegral_measure_zero _ _ this
 #align measure_theory.pdf.quasi_measure_preserving_has_pdf MeasureTheory.pdf.quasiMeasurePreserving_hasPdf
 
-theorem quasiMeasurePreserving_has_pdf' [FiniteMeasure ℙ] [SigmaFinite ν] {X : Ω → E} [HasPdf X ℙ μ]
-    {g : E → F} (hg : QuasiMeasurePreserving g μ ν) : HasPdf (g ∘ X) ℙ ν :=
+theorem quasiMeasurePreserving_has_pdf' [IsFiniteMeasure ℙ] [SigmaFinite ν] {X : Ω → E}
+    [HasPdf X ℙ μ] {g : E → F} (hg : QuasiMeasurePreserving g μ ν) : HasPdf (g ∘ X) ℙ ν :=
   quasiMeasurePreserving_hasPdf hg inferInstance
 #align measure_theory.pdf.quasi_measure_preserving_has_pdf' MeasureTheory.pdf.quasiMeasurePreserving_has_pdf'
 
@@ -284,7 +284,7 @@ end
 
 section Real
 
-variable [FiniteMeasure ℙ] {X : Ω → ℝ}
+variable [IsFiniteMeasure ℙ] {X : Ω → ℝ}
 
 /-- A real-valued random variable `X` `has_pdf X ℙ λ` (where `λ` is the Lebesgue measure) if and
 only if the push-forward measure of `ℙ` along `X` is absolutely continuous with respect to `λ`. -/
@@ -382,20 +382,20 @@ theorem measure_preimage {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure
   rw [ENNReal.div_eq_inv_mul]
 #align measure_theory.pdf.is_uniform.measure_preimage MeasureTheory.pdf.IsUniform.measure_preimage
 
-theorem probabilityMeasure {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E}
+theorem isProbabilityMeasure {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E}
     {s : Set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞) (hms : MeasurableSet s) (hu : IsUniform X s ℙ μ) :
-    ProbabilityMeasure ℙ :=
+    IsProbabilityMeasure ℙ :=
   ⟨by
     have : X ⁻¹' Set.univ = Set.univ := by simp only [Set.preimage_univ]
     rw [← this, hu.measure_preimage hns hnt hms MeasurableSet.univ, Set.inter_univ,
       ENNReal.div_self hns hnt]⟩
-#align measure_theory.pdf.is_uniform.is_probability_measure MeasureTheory.pdf.IsUniform.probabilityMeasure
+#align measure_theory.pdf.is_uniform.is_probability_measure MeasureTheory.pdf.IsUniform.isProbabilityMeasure
 
 variable {X : Ω → ℝ} {s : Set ℝ} (hms : MeasurableSet s) (hns : volume s ≠ 0)
 
 include hms hns
 
-theorem mul_pdf_integrable [FiniteMeasure ℙ] (hcs : IsCompact s) (huX : IsUniform X s ℙ) :
+theorem mul_pdf_integrable [IsFiniteMeasure ℙ] (hcs : IsCompact s) (huX : IsUniform X s ℙ) :
     Integrable fun x : ℝ => x * (pdf X ℙ volume x).toReal :=
   by
   by_cases hsupp : volume s = ∞

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

@@ -68,7 +68,7 @@ variable {Ω E : Type _} [MeasurableSpace E]
 `μ` on `E` if there exists a measurable function `f` such that the push-forward measure of `ℙ`
 along `X` equals `μ.with_density f`. -/
 class HasPdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
-  (μ : Measure E := by exact MeasureTheory.MeasureSpace.volume) : Prop where
+    (μ : Measure E := by exact MeasureTheory.MeasureSpace.volume) : Prop where
   pdf' : Measurable X ∧ ∃ f : E → ℝ≥0∞, Measurable f ∧ map X ℙ = μ.withDensity f
 #align measure_theory.has_pdf MeasureTheory.HasPdf
 
@@ -93,7 +93,7 @@ theorem pdf_undef {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {
 theorem hasPdf_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E}
     (h : pdf X ℙ μ ≠ 0) : HasPdf X ℙ μ := by
   by_contra hpdf
-  rw [pdf, dif_neg hpdf] at h
+  rw [pdf, dif_neg hpdf] at h 
   exact hpdf (False.ndrec (has_pdf X ℙ μ) (h rfl))
 #align measure_theory.has_pdf_of_pdf_ne_zero MeasureTheory.hasPdf_of_pdf_ne_zero
 
@@ -214,7 +214,7 @@ theorem integral_fun_mul_eq_integral [FiniteMeasure ℙ] {X : Ω → E} [HasPdf
       rw [lintegral_congr_ae this]
       exact hpdf.2
   · rw [integral_undef hpdf, integral_undef]
-    rwa [← integrable_iff_integrable_mul_pdf hf] at hpdf
+    rwa [← integrable_iff_integrable_mul_pdf hf] at hpdf 
     all_goals infer_instance
 #align measure_theory.pdf.integral_fun_mul_eq_integral MeasureTheory.pdf.integral_fun_mul_eq_integral
 
@@ -271,7 +271,7 @@ theorem quasiMeasurePreserving_hasPdf {X : Ω → E} [HasPdf X ℙ μ] {g : E 
   refine' absolutely_continuous.mk fun s hsm hs => _
   rw [map_apply hg.measurable hsm, with_density_apply _ (hg.measurable hsm)]
   have := hg.absolutely_continuous hs
-  rw [map_apply hg.measurable hsm] at this
+  rw [map_apply hg.measurable hsm] at this 
   exact set_lintegral_measure_zero _ _ this
 #align measure_theory.pdf.quasi_measure_preserving_has_pdf MeasureTheory.pdf.quasiMeasurePreserving_hasPdf
 
@@ -350,7 +350,7 @@ theorem hasPdf {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ :
   hasPdf_of_pdf_ne_zero
     (by
       intro hpdf
-      rw [is_uniform, hpdf] at hu
+      rw [is_uniform, hpdf] at hu 
       suffices μ (s ∩ Function.support ((μ s)⁻¹ • 1)) = 0
         by
         have heq : Function.support ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) = Set.univ :=
@@ -358,7 +358,7 @@ theorem hasPdf {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ :
           ext x
           rw [Function.mem_support]
           simp [hnt]
-        rw [HEq, Set.inter_univ] at this
+        rw [HEq, Set.inter_univ] at this 
         exact hns this
       exact MeasureTheory.Set.indicator_ae_eq_zero hu.symm)
 #align measure_theory.pdf.is_uniform.has_pdf MeasureTheory.pdf.IsUniform.hasPdf

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

@@ -56,7 +56,7 @@ which we currently do not have.
 
 noncomputable section
 
-open Classical MeasureTheory NNReal ENNReal
+open scoped Classical MeasureTheory NNReal ENNReal
 
 namespace MeasureTheory
 

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

@@ -103,9 +103,7 @@ theorem pdf_eq_zero_of_not_measurable {m : MeasurableSpace Ω} {ℙ : Measure Ω
 #align measure_theory.pdf_eq_zero_of_not_measurable MeasureTheory.pdf_eq_zero_of_not_measurable
 
 theorem measurable_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
-    (X : Ω → E) (h : pdf X ℙ μ ≠ 0) : Measurable X :=
-  by
-  by_contra hX
+    (X : Ω → E) (h : pdf X ℙ μ ≠ 0) : Measurable X := by by_contra hX;
   exact h (pdf_eq_zero_of_not_measurable hX)
 #align measure_theory.measurable_of_pdf_ne_zero MeasureTheory.measurable_of_pdf_ne_zero
 
@@ -220,10 +218,8 @@ theorem integral_fun_mul_eq_integral [FiniteMeasure ℙ] {X : Ω → E} [HasPdf
     all_goals infer_instance
 #align measure_theory.pdf.integral_fun_mul_eq_integral MeasureTheory.pdf.integral_fun_mul_eq_integral
 
-theorem map_absolutelyContinuous {X : Ω → E} [HasPdf X ℙ μ] : map X ℙ ≪ μ :=
-  by
-  rw [map_eq_with_density_pdf X ℙ μ]
-  exact with_density_absolutely_continuous _ _
+theorem map_absolutelyContinuous {X : Ω → E} [HasPdf X ℙ μ] : map X ℙ ≪ μ := by
+  rw [map_eq_with_density_pdf X ℙ μ]; exact with_density_absolutely_continuous _ _
 #align measure_theory.pdf.map_absolutely_continuous MeasureTheory.pdf.map_absolutelyContinuous
 
 /-- A random variable that `has_pdf` is quasi-measure preserving. -/
@@ -252,9 +248,7 @@ theorem hasPdf_iff {X : Ω → E} :
 #align measure_theory.pdf.has_pdf_iff MeasureTheory.pdf.hasPdf_iff
 
 theorem hasPdf_iff_of_measurable {X : Ω → E} (hX : Measurable X) :
-    HasPdf X ℙ μ ↔ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ :=
-  by
-  rw [has_pdf_iff]
+    HasPdf X ℙ μ ↔ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ := by rw [has_pdf_iff];
   simp only [hX, true_and_iff]
 #align measure_theory.pdf.has_pdf_iff_of_measurable MeasureTheory.pdf.hasPdf_iff_of_measurable
 

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

@@ -414,7 +414,7 @@ theorem mul_pdf_integrable [FiniteMeasure ℙ] (hcs : IsCompact s) (huX : IsUnif
       Filter.EventuallyEq.mul (ae_eq_refl _) (Filter.EventuallyEq.fun_comp this.symm ENNReal.toReal)
   refine'
     ⟨ae_strongly_measurable_id.mul
-        (measurable_pdf X ℙ).AEMeasurable.ennreal_toReal.AeStronglyMeasurable,
+        (measurable_pdf X ℙ).AEMeasurable.ennreal_toReal.AEStronglyMeasurable,
       _⟩
   refine' has_finite_integral_mul huX _
   set ind := (volume s)⁻¹ • (1 : ℝ → ℝ≥0∞) with hind

mathlib commit https://github.com/leanprover-community/mathlib/commit/33c67ae661dd8988516ff7f247b0be3018cdd952

@@ -414,7 +414,7 @@ theorem mul_pdf_integrable [FiniteMeasure ℙ] (hcs : IsCompact s) (huX : IsUnif
       Filter.EventuallyEq.mul (ae_eq_refl _) (Filter.EventuallyEq.fun_comp this.symm ENNReal.toReal)
   refine'
     ⟨ae_strongly_measurable_id.mul
-        (measurable_pdf X ℙ).AEMeasurable.eNNReal_toReal.AeStronglyMeasurable,
+        (measurable_pdf X ℙ).AEMeasurable.ennreal_toReal.AeStronglyMeasurable,
       _⟩
   refine' has_finite_integral_mul huX _
   set ind := (volume s)⁻¹ • (1 : ℝ → ℝ≥0∞) with hind

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

@@ -4,12 +4,12 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kexing Ying
 
 ! This file was ported from Lean 3 source module probability.density
-! leanprover-community/mathlib commit f0c8bf9245297a541f468be517f1bde6195105e9
+! leanprover-community/mathlib commit fd5edc43dc4f10b85abfe544b88f82cf13c5f844
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.MeasureTheory.Decomposition.RadonNikodym
-import Mathbin.MeasureTheory.Measure.HaarOfBasis
+import Mathbin.MeasureTheory.Measure.Haar.OfBasis
 
 /-!
 # Probability density function

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

@@ -144,7 +144,7 @@ theorem lintegral_eq_measure_univ {X : Ω → E} [HasPdf X ℙ μ] :
     measure.map_apply (has_pdf.measurable X ℙ μ) MeasurableSet.univ, Set.preimage_univ]
 #align measure_theory.pdf.lintegral_eq_measure_univ MeasureTheory.pdf.lintegral_eq_measure_univ
 
-theorem ae_lt_top [IsFiniteMeasure ℙ] {μ : Measure E} {X : Ω → E} : ∀ᵐ x ∂μ, pdf X ℙ μ x < ∞ :=
+theorem ae_lt_top [FiniteMeasure ℙ] {μ : Measure E} {X : Ω → E} : ∀ᵐ x ∂μ, pdf X ℙ μ x < ∞ :=
   by
   by_cases hpdf : has_pdf X ℙ μ
   · haveI := hpdf
@@ -155,12 +155,12 @@ theorem ae_lt_top [IsFiniteMeasure ℙ] {μ : Measure E} {X : Ω → E} : ∀ᵐ
     exact Filter.eventually_of_forall fun x => WithTop.zero_lt_top
 #align measure_theory.pdf.ae_lt_top MeasureTheory.pdf.ae_lt_top
 
-theorem ofReal_toReal_ae_eq [IsFiniteMeasure ℙ] {X : Ω → E} :
+theorem ofReal_toReal_ae_eq [FiniteMeasure ℙ] {X : Ω → E} :
     (fun x => ENNReal.ofReal (pdf X ℙ μ x).toReal) =ᵐ[μ] pdf X ℙ μ :=
   ofReal_toReal_ae_eq ae_lt_top
 #align measure_theory.pdf.of_real_to_real_ae_eq MeasureTheory.pdf.ofReal_toReal_ae_eq
 
-theorem integrable_iff_integrable_mul_pdf [IsFiniteMeasure ℙ] {X : Ω → E} [HasPdf X ℙ μ] {f : E → ℝ}
+theorem integrable_iff_integrable_mul_pdf [FiniteMeasure ℙ] {X : Ω → E} [HasPdf X ℙ μ] {f : E → ℝ}
     (hf : Measurable f) :
     Integrable (fun x => f (X x)) ℙ ↔ Integrable (fun x => f x * (pdf X ℙ μ x).toReal) μ :=
   by
@@ -172,7 +172,7 @@ theorem integrable_iff_integrable_mul_pdf [IsFiniteMeasure ℙ] {X : Ω → E} [
 /-- **The Law of the Unconscious Statistician**: Given a random variable `X` and a measurable
 function `f`, `f ∘ X` is a random variable with expectation `∫ x, f x * pdf X ∂μ`
 where `μ` is a measure on the codomain of `X`. -/
-theorem integral_fun_mul_eq_integral [IsFiniteMeasure ℙ] {X : Ω → E} [HasPdf X ℙ μ] {f : E → ℝ}
+theorem integral_fun_mul_eq_integral [FiniteMeasure ℙ] {X : Ω → E} [HasPdf X ℙ μ] {f : E → ℝ}
     (hf : Measurable f) : (∫ x, f x * (pdf X ℙ μ x).toReal ∂μ) = ∫ x, f (X x) ∂ℙ :=
   by
   by_cases hpdf : integrable (fun x => f x * (pdf X ℙ μ x).toReal) μ
@@ -281,8 +281,8 @@ theorem quasiMeasurePreserving_hasPdf {X : Ω → E} [HasPdf X ℙ μ] {g : E 
   exact set_lintegral_measure_zero _ _ this
 #align measure_theory.pdf.quasi_measure_preserving_has_pdf MeasureTheory.pdf.quasiMeasurePreserving_hasPdf
 
-theorem quasiMeasurePreserving_has_pdf' [IsFiniteMeasure ℙ] [SigmaFinite ν] {X : Ω → E}
-    [HasPdf X ℙ μ] {g : E → F} (hg : QuasiMeasurePreserving g μ ν) : HasPdf (g ∘ X) ℙ ν :=
+theorem quasiMeasurePreserving_has_pdf' [FiniteMeasure ℙ] [SigmaFinite ν] {X : Ω → E} [HasPdf X ℙ μ]
+    {g : E → F} (hg : QuasiMeasurePreserving g μ ν) : HasPdf (g ∘ X) ℙ ν :=
   quasiMeasurePreserving_hasPdf hg inferInstance
 #align measure_theory.pdf.quasi_measure_preserving_has_pdf' MeasureTheory.pdf.quasiMeasurePreserving_has_pdf'
 
@@ -290,7 +290,7 @@ end
 
 section Real
 
-variable [IsFiniteMeasure ℙ] {X : Ω → ℝ}
+variable [FiniteMeasure ℙ] {X : Ω → ℝ}
 
 /-- A real-valued random variable `X` `has_pdf X ℙ λ` (where `λ` is the Lebesgue measure) if and
 only if the push-forward measure of `ℙ` along `X` is absolutely continuous with respect to `λ`. -/
@@ -388,20 +388,20 @@ theorem measure_preimage {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure
   rw [ENNReal.div_eq_inv_mul]
 #align measure_theory.pdf.is_uniform.measure_preimage MeasureTheory.pdf.IsUniform.measure_preimage
 
-theorem isProbabilityMeasure {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E}
+theorem probabilityMeasure {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E}
     {s : Set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞) (hms : MeasurableSet s) (hu : IsUniform X s ℙ μ) :
-    IsProbabilityMeasure ℙ :=
+    ProbabilityMeasure ℙ :=
   ⟨by
     have : X ⁻¹' Set.univ = Set.univ := by simp only [Set.preimage_univ]
     rw [← this, hu.measure_preimage hns hnt hms MeasurableSet.univ, Set.inter_univ,
       ENNReal.div_self hns hnt]⟩
-#align measure_theory.pdf.is_uniform.is_probability_measure MeasureTheory.pdf.IsUniform.isProbabilityMeasure
+#align measure_theory.pdf.is_uniform.is_probability_measure MeasureTheory.pdf.IsUniform.probabilityMeasure
 
 variable {X : Ω → ℝ} {s : Set ℝ} (hms : MeasurableSet s) (hns : volume s ≠ 0)
 
 include hms hns
 
-theorem mul_pdf_integrable [IsFiniteMeasure ℙ] (hcs : IsCompact s) (huX : IsUniform X s ℙ) :
+theorem mul_pdf_integrable [FiniteMeasure ℙ] (hcs : IsCompact s) (huX : IsUniform X s ℙ) :
     Integrable fun x : ℝ => x * (pdf X ℙ volume x).toReal :=
   by
   by_cases hsupp : volume s = ∞

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

@@ -4,12 +4,12 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kexing Ying
 
 ! This file was ported from Lean 3 source module probability.density
-! leanprover-community/mathlib commit 17ef379e997badd73e5eabb4d38f11919ab3c4b3
+! 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.MeasureTheory.Decomposition.RadonNikodym
-import Mathbin.MeasureTheory.Measure.Lebesgue
+import Mathbin.MeasureTheory.Measure.HaarOfBasis
 
 /-!
 # Probability density function

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

@@ -90,12 +90,12 @@ theorem pdf_undef {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {
     (h : ¬HasPdf X ℙ μ) : pdf X ℙ μ = 0 := by simp only [pdf, dif_neg h]
 #align measure_theory.pdf_undef MeasureTheory.pdf_undef
 
-theorem hasPdfOfPdfNeZero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E}
+theorem hasPdf_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E}
     (h : pdf X ℙ μ ≠ 0) : HasPdf X ℙ μ := by
   by_contra hpdf
   rw [pdf, dif_neg hpdf] at h
   exact hpdf (False.ndrec (has_pdf X ℙ μ) (h rfl))
-#align measure_theory.has_pdf_of_pdf_ne_zero MeasureTheory.hasPdfOfPdfNeZero
+#align measure_theory.has_pdf_of_pdf_ne_zero MeasureTheory.hasPdf_of_pdf_ne_zero
 
 theorem pdf_eq_zero_of_not_measurable {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
     {X : Ω → E} (hX : ¬Measurable X) : pdf X ℙ μ = 0 :=
@@ -164,7 +164,7 @@ theorem integrable_iff_integrable_mul_pdf [IsFiniteMeasure ℙ] {X : Ω → E} [
     (hf : Measurable f) :
     Integrable (fun x => f (X x)) ℙ ↔ Integrable (fun x => f x * (pdf X ℙ μ x).toReal) μ :=
   by
-  rw [← integrable_map_measure hf.ae_strongly_measurable (has_pdf.measurable X ℙ μ).AeMeasurable,
+  rw [← integrable_map_measure hf.ae_strongly_measurable (has_pdf.measurable X ℙ μ).AEMeasurable,
     map_eq_with_density_pdf X ℙ μ, integrable_with_density_iff (measurable_pdf _ _ _) ae_lt_top]
   infer_instance
 #align measure_theory.pdf.integrable_iff_integrable_mul_pdf MeasureTheory.pdf.integrable_iff_integrable_mul_pdf
@@ -176,7 +176,7 @@ theorem integral_fun_mul_eq_integral [IsFiniteMeasure ℙ] {X : Ω → E} [HasPd
     (hf : Measurable f) : (∫ x, f x * (pdf X ℙ μ x).toReal ∂μ) = ∫ x, f (X x) ∂ℙ :=
   by
   by_cases hpdf : integrable (fun x => f x * (pdf X ℙ μ x).toReal) μ
-  · rw [← integral_map (has_pdf.measurable X ℙ μ).AeMeasurable hf.ae_strongly_measurable,
+  · rw [← integral_map (has_pdf.measurable X ℙ μ).AEMeasurable hf.ae_strongly_measurable,
       map_eq_with_density_pdf X ℙ μ, integral_eq_lintegral_pos_part_sub_lintegral_neg_part hpdf,
       integral_eq_lintegral_pos_part_sub_lintegral_neg_part,
       lintegral_with_density_eq_lintegral_mul _ (measurable_pdf X ℙ μ) hf.neg.ennreal_of_real,
@@ -220,24 +220,24 @@ theorem integral_fun_mul_eq_integral [IsFiniteMeasure ℙ] {X : Ω → E} [HasPd
     all_goals infer_instance
 #align measure_theory.pdf.integral_fun_mul_eq_integral MeasureTheory.pdf.integral_fun_mul_eq_integral
 
-theorem mapAbsolutelyContinuous {X : Ω → E} [HasPdf X ℙ μ] : map X ℙ ≪ μ :=
+theorem map_absolutelyContinuous {X : Ω → E} [HasPdf X ℙ μ] : map X ℙ ≪ μ :=
   by
   rw [map_eq_with_density_pdf X ℙ μ]
   exact with_density_absolutely_continuous _ _
-#align measure_theory.pdf.map_absolutely_continuous MeasureTheory.pdf.mapAbsolutelyContinuous
+#align measure_theory.pdf.map_absolutely_continuous MeasureTheory.pdf.map_absolutelyContinuous
 
 /-- A random variable that `has_pdf` is quasi-measure preserving. -/
-theorem toQuasiMeasurePreserving {X : Ω → E} [HasPdf X ℙ μ] : QuasiMeasurePreserving X ℙ μ :=
+theorem to_quasiMeasurePreserving {X : Ω → E} [HasPdf X ℙ μ] : QuasiMeasurePreserving X ℙ μ :=
   { Measurable := HasPdf.measurable X ℙ μ
-    AbsolutelyContinuous := mapAbsolutelyContinuous }
-#align measure_theory.pdf.to_quasi_measure_preserving MeasureTheory.pdf.toQuasiMeasurePreserving
+    AbsolutelyContinuous := map_absolutelyContinuous }
+#align measure_theory.pdf.to_quasi_measure_preserving MeasureTheory.pdf.to_quasiMeasurePreserving
 
-theorem haveLebesgueDecompositionOfHasPdf {X : Ω → E} [hX' : HasPdf X ℙ μ] :
+theorem haveLebesgueDecomposition_of_hasPdf {X : Ω → E} [hX' : HasPdf X ℙ μ] :
     (map X ℙ).HaveLebesgueDecomposition μ :=
   ⟨⟨⟨0, pdf X ℙ μ⟩, by
       simp only [zero_add, measurable_pdf X ℙ μ, true_and_iff, mutually_singular.zero_left,
         map_eq_with_density_pdf X ℙ μ]⟩⟩
-#align measure_theory.pdf.have_lebesgue_decomposition_of_has_pdf MeasureTheory.pdf.haveLebesgueDecompositionOfHasPdf
+#align measure_theory.pdf.have_lebesgue_decomposition_of_has_pdf MeasureTheory.pdf.haveLebesgueDecomposition_of_hasPdf
 
 theorem hasPdf_iff {X : Ω → E} :
     HasPdf X ℙ μ ↔ Measurable X ∧ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ :=
@@ -267,7 +267,7 @@ map also `has_pdf` if `(map g (map X ℙ)).have_lebesgue_decomposition μ`.
 
 `quasi_measure_preserving_has_pdf'` is more useful in the case we are working with a
 probability measure and a real-valued random variable. -/
-theorem quasiMeasurePreservingHasPdf {X : Ω → E} [HasPdf X ℙ μ] {g : E → F}
+theorem quasiMeasurePreserving_hasPdf {X : Ω → E} [HasPdf X ℙ μ] {g : E → F}
     (hg : QuasiMeasurePreserving g μ ν) (hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν) :
     HasPdf (g ∘ X) ℙ ν :=
   by
@@ -279,12 +279,12 @@ theorem quasiMeasurePreservingHasPdf {X : Ω → E} [HasPdf X ℙ μ] {g : E →
   have := hg.absolutely_continuous hs
   rw [map_apply hg.measurable hsm] at this
   exact set_lintegral_measure_zero _ _ this
-#align measure_theory.pdf.quasi_measure_preserving_has_pdf MeasureTheory.pdf.quasiMeasurePreservingHasPdf
+#align measure_theory.pdf.quasi_measure_preserving_has_pdf MeasureTheory.pdf.quasiMeasurePreserving_hasPdf
 
-theorem quasiMeasurePreservingHasPdf' [IsFiniteMeasure ℙ] [SigmaFinite ν] {X : Ω → E} [HasPdf X ℙ μ]
-    {g : E → F} (hg : QuasiMeasurePreserving g μ ν) : HasPdf (g ∘ X) ℙ ν :=
-  quasiMeasurePreservingHasPdf hg inferInstance
-#align measure_theory.pdf.quasi_measure_preserving_has_pdf' MeasureTheory.pdf.quasiMeasurePreservingHasPdf'
+theorem quasiMeasurePreserving_has_pdf' [IsFiniteMeasure ℙ] [SigmaFinite ν] {X : Ω → E}
+    [HasPdf X ℙ μ] {g : E → F} (hg : QuasiMeasurePreserving g μ ν) : HasPdf (g ∘ X) ℙ ν :=
+  quasiMeasurePreserving_hasPdf hg inferInstance
+#align measure_theory.pdf.quasi_measure_preserving_has_pdf' MeasureTheory.pdf.quasiMeasurePreserving_has_pdf'
 
 end
 
@@ -316,7 +316,7 @@ theorem integral_mul_eq_integral [HasPdf X ℙ] :
   integral_fun_mul_eq_integral measurable_id
 #align measure_theory.pdf.integral_mul_eq_integral MeasureTheory.pdf.integral_mul_eq_integral
 
-theorem hasFiniteIntegralMul {f : ℝ → ℝ} {g : ℝ → ℝ≥0∞} (hg : pdf X ℙ =ᵐ[volume] g)
+theorem hasFiniteIntegral_mul {f : ℝ → ℝ} {g : ℝ → ℝ≥0∞} (hg : pdf X ℙ =ᵐ[volume] g)
     (hgi : (∫⁻ x, ‖f x‖₊ * g x) ≠ ∞) : HasFiniteIntegral fun x => f x * (pdf X ℙ volume x).toReal :=
   by
   rw [has_finite_integral]
@@ -333,7 +333,7 @@ theorem hasFiniteIntegralMul {f : ℝ → ℝ} {g : ℝ → ℝ≥0∞} (hg : pd
     ext1 x
     exact Real.ennnorm_eq_ofReal ENNReal.toReal_nonneg
   rwa [lt_top_iff_ne_top, ← lintegral_congr_ae this]
-#align measure_theory.pdf.has_finite_integral_mul MeasureTheory.pdf.hasFiniteIntegralMul
+#align measure_theory.pdf.has_finite_integral_mul MeasureTheory.pdf.hasFiniteIntegral_mul
 
 end Real
 
@@ -353,7 +353,7 @@ namespace IsUniform
 
 theorem hasPdf {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E} {s : Set E}
     (hns : μ s ≠ 0) (hnt : μ s ≠ ∞) (hu : IsUniform X s ℙ μ) : HasPdf X ℙ μ :=
-  hasPdfOfPdfNeZero
+  hasPdf_of_pdf_ne_zero
     (by
       intro hpdf
       rw [is_uniform, hpdf] at hu
@@ -366,7 +366,7 @@ theorem hasPdf {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ :
           simp [hnt]
         rw [HEq, Set.inter_univ] at this
         exact hns this
-      exact Set.indicator_ae_eq_zero hu.symm)
+      exact MeasureTheory.Set.indicator_ae_eq_zero hu.symm)
 #align measure_theory.pdf.is_uniform.has_pdf MeasureTheory.pdf.IsUniform.hasPdf
 
 theorem pdf_toReal_ae_eq {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E}
@@ -401,7 +401,7 @@ variable {X : Ω → ℝ} {s : Set ℝ} (hms : MeasurableSet s) (hns : volume s
 
 include hms hns
 
-theorem mulPdfIntegrable [IsFiniteMeasure ℙ] (hcs : IsCompact s) (huX : IsUniform X s ℙ) :
+theorem mul_pdf_integrable [IsFiniteMeasure ℙ] (hcs : IsCompact s) (huX : IsUniform X s ℙ) :
     Integrable fun x : ℝ => x * (pdf X ℙ volume x).toReal :=
   by
   by_cases hsupp : volume s = ∞
@@ -414,7 +414,7 @@ theorem mulPdfIntegrable [IsFiniteMeasure ℙ] (hcs : IsCompact s) (huX : IsUnif
       Filter.EventuallyEq.mul (ae_eq_refl _) (Filter.EventuallyEq.fun_comp this.symm ENNReal.toReal)
   refine'
     ⟨ae_strongly_measurable_id.mul
-        (measurable_pdf X ℙ).AeMeasurable.eNNReal_toReal.AeStronglyMeasurable,
+        (measurable_pdf X ℙ).AEMeasurable.eNNReal_toReal.AeStronglyMeasurable,
       _⟩
   refine' has_finite_integral_mul huX _
   set ind := (volume s)⁻¹ • (1 : ℝ → ℝ≥0∞) with hind
@@ -428,7 +428,7 @@ theorem mulPdfIntegrable [IsFiniteMeasure ℙ] (hcs : IsCompact s) (huX : IsUnif
       (ENNReal.mul_lt_top (set_lintegral_lt_top_of_is_compact hsupp hcs continuous_nnnorm).Ne
           (ENNReal.inv_lt_top.2 (pos_iff_ne_zero.mpr hns)).Ne).Ne
   · infer_instance
-#align measure_theory.pdf.is_uniform.mul_pdf_integrable MeasureTheory.pdf.IsUniform.mulPdfIntegrable
+#align measure_theory.pdf.is_uniform.mul_pdf_integrable MeasureTheory.pdf.IsUniform.mul_pdf_integrable
 
 /-- A real uniform random variable `X` with support `s` has expectation
 `(λ s)⁻¹ * ∫ x in s, x ∂λ` where `λ` is the Lebesgue measure. -/

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

@@ -56,7 +56,7 @@ which we currently do not have.
 
 noncomputable section
 
-open Classical MeasureTheory NNReal Ennreal
+open Classical MeasureTheory NNReal ENNReal
 
 namespace MeasureTheory
 
@@ -156,7 +156,7 @@ theorem ae_lt_top [IsFiniteMeasure ℙ] {μ : Measure E} {X : Ω → E} : ∀ᵐ
 #align measure_theory.pdf.ae_lt_top MeasureTheory.pdf.ae_lt_top
 
 theorem ofReal_toReal_ae_eq [IsFiniteMeasure ℙ] {X : Ω → E} :
-    (fun x => Ennreal.ofReal (pdf X ℙ μ x).toReal) =ᵐ[μ] pdf X ℙ μ :=
+    (fun x => ENNReal.ofReal (pdf X ℙ μ x).toReal) =ᵐ[μ] pdf X ℙ μ :=
   ofReal_toReal_ae_eq ae_lt_top
 #align measure_theory.pdf.of_real_to_real_ae_eq MeasureTheory.pdf.ofReal_toReal_ae_eq
 
@@ -184,20 +184,20 @@ theorem integral_fun_mul_eq_integral [IsFiniteMeasure ℙ] {X : Ω → E} [HasPd
     · congr 2
       · have :
           ∀ x,
-            Ennreal.ofReal (f x * (pdf X ℙ μ x).toReal) =
-              Ennreal.ofReal (pdf X ℙ μ x).toReal * Ennreal.ofReal (f x) :=
+            ENNReal.ofReal (f x * (pdf X ℙ μ x).toReal) =
+              ENNReal.ofReal (pdf X ℙ μ x).toReal * ENNReal.ofReal (f x) :=
           by
           intro x
-          rw [mul_comm, Ennreal.ofReal_mul Ennreal.toReal_nonneg]
+          rw [mul_comm, ENNReal.ofReal_mul ENNReal.toReal_nonneg]
         simp_rw [this]
         exact lintegral_congr_ae (Filter.EventuallyEq.mul of_real_to_real_ae_eq (ae_eq_refl _))
       · have :
           ∀ x,
-            Ennreal.ofReal (-(f x * (pdf X ℙ μ x).toReal)) =
-              Ennreal.ofReal (pdf X ℙ μ x).toReal * Ennreal.ofReal (-f x) :=
+            ENNReal.ofReal (-(f x * (pdf X ℙ μ x).toReal)) =
+              ENNReal.ofReal (pdf X ℙ μ x).toReal * ENNReal.ofReal (-f x) :=
           by
           intro x
-          rw [neg_mul_eq_neg_mul, mul_comm, Ennreal.ofReal_mul Ennreal.toReal_nonneg]
+          rw [neg_mul_eq_neg_mul, mul_comm, ENNReal.ofReal_mul ENNReal.toReal_nonneg]
         simp_rw [this]
         exact lintegral_congr_ae (Filter.EventuallyEq.mul of_real_to_real_ae_eq (ae_eq_refl _))
     · refine' ⟨hf.ae_strongly_measurable, _⟩
@@ -207,12 +207,12 @@ theorem integral_fun_mul_eq_integral [IsFiniteMeasure ℙ] {X : Ω → E} [HasPd
       have :
         (fun x => (pdf X ℙ μ * fun x => ↑‖f x‖₊) x) =ᵐ[μ] fun x => ‖f x * (pdf X ℙ μ x).toReal‖₊ :=
         by
-        simp_rw [← smul_eq_mul, nnnorm_smul, Ennreal.coe_mul]
+        simp_rw [← smul_eq_mul, nnnorm_smul, ENNReal.coe_mul]
         rw [smul_eq_mul, mul_comm]
         refine' Filter.EventuallyEq.mul (ae_eq_refl _) (ae_eq_trans of_real_to_real_ae_eq.symm _)
         convert ae_eq_refl _
         ext1 x
-        exact Real.ennnorm_eq_ofReal Ennreal.toReal_nonneg
+        exact Real.ennnorm_eq_ofReal ENNReal.toReal_nonneg
       rw [lintegral_congr_ae this]
       exact hpdf.2
   · rw [integral_undef hpdf, integral_undef]
@@ -327,11 +327,11 @@ theorem hasFiniteIntegralMul {f : ℝ → ℝ} {g : ℝ → ℝ≥0∞} (hg : pd
         (Filter.EventuallyEq.mul (ae_eq_refl fun x => ‖f x‖₊)
           (ae_eq_trans hg.symm of_real_to_real_ae_eq.symm))
         _
-    simp_rw [← smul_eq_mul, nnnorm_smul, Ennreal.coe_mul, smul_eq_mul]
+    simp_rw [← smul_eq_mul, nnnorm_smul, ENNReal.coe_mul, smul_eq_mul]
     refine' Filter.EventuallyEq.mul (ae_eq_refl _) _
     convert ae_eq_refl _
     ext1 x
-    exact Real.ennnorm_eq_ofReal Ennreal.toReal_nonneg
+    exact Real.ennnorm_eq_ofReal ENNReal.toReal_nonneg
   rwa [lt_top_iff_ne_top, ← lintegral_congr_ae this]
 #align measure_theory.pdf.has_finite_integral_mul MeasureTheory.pdf.hasFiniteIntegralMul
 
@@ -373,7 +373,7 @@ theorem pdf_toReal_ae_eq {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure
     {s : Set E} (hX : IsUniform X s ℙ μ) :
     (fun x => (pdf X ℙ μ x).toReal) =ᵐ[μ] fun x =>
       (s.indicator ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) x).toReal :=
-  Filter.EventuallyEq.fun_comp hX Ennreal.toReal
+  Filter.EventuallyEq.fun_comp hX ENNReal.toReal
 #align measure_theory.pdf.is_uniform.pdf_to_real_ae_eq MeasureTheory.pdf.IsUniform.pdf_toReal_ae_eq
 
 theorem measure_preimage {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E}
@@ -385,7 +385,7 @@ theorem measure_preimage {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure
     lintegral_congr_ae hu.restrict]
   simp only [hms, hA, lintegral_indicator, Pi.smul_apply, Pi.one_apply, Algebra.id.smul_eq_mul,
     mul_one, lintegral_const, restrict_apply', Set.univ_inter]
-  rw [Ennreal.div_eq_inv_mul]
+  rw [ENNReal.div_eq_inv_mul]
 #align measure_theory.pdf.is_uniform.measure_preimage MeasureTheory.pdf.IsUniform.measure_preimage
 
 theorem isProbabilityMeasure {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E}
@@ -394,7 +394,7 @@ theorem isProbabilityMeasure {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Meas
   ⟨by
     have : X ⁻¹' Set.univ = Set.univ := by simp only [Set.preimage_univ]
     rw [← this, hu.measure_preimage hns hnt hms MeasurableSet.univ, Set.inter_univ,
-      Ennreal.div_self hns hnt]⟩
+      ENNReal.div_self hns hnt]⟩
 #align measure_theory.pdf.is_uniform.is_probability_measure MeasureTheory.pdf.IsUniform.isProbabilityMeasure
 
 variable {X : Ω → ℝ} {s : Set ℝ} (hms : MeasurableSet s) (hns : volume s ≠ 0)
@@ -411,10 +411,10 @@ theorem mulPdfIntegrable [IsFiniteMeasure ℙ] (hcs : IsCompact s) (huX : IsUnif
     refine' integrable.congr (integrable_zero _ _ _) _
     rw [(by simp : (fun x => 0 : ℝ → ℝ) = fun x => x * (0 : ℝ≥0∞).toReal)]
     refine'
-      Filter.EventuallyEq.mul (ae_eq_refl _) (Filter.EventuallyEq.fun_comp this.symm Ennreal.toReal)
+      Filter.EventuallyEq.mul (ae_eq_refl _) (Filter.EventuallyEq.fun_comp this.symm ENNReal.toReal)
   refine'
     ⟨ae_strongly_measurable_id.mul
-        (measurable_pdf X ℙ).AeMeasurable.ennreal_toReal.AeStronglyMeasurable,
+        (measurable_pdf X ℙ).AeMeasurable.eNNReal_toReal.AeStronglyMeasurable,
       _⟩
   refine' has_finite_integral_mul huX _
   set ind := (volume s)⁻¹ • (1 : ℝ → ℝ≥0∞) with hind
@@ -425,8 +425,8 @@ theorem mulPdfIntegrable [IsFiniteMeasure ℙ] (hcs : IsCompact s) (huX : IsUnif
   rw [lintegral_mul_const _ measurable_nnnorm.coe_nnreal_ennreal]
   ·
     refine'
-      (Ennreal.mul_lt_top (set_lintegral_lt_top_of_is_compact hsupp hcs continuous_nnnorm).Ne
-          (Ennreal.inv_lt_top.2 (pos_iff_ne_zero.mpr hns)).Ne).Ne
+      (ENNReal.mul_lt_top (set_lintegral_lt_top_of_is_compact hsupp hcs continuous_nnnorm).Ne
+          (ENNReal.inv_lt_top.2 (pos_iff_ne_zero.mpr hns)).Ne).Ne
   · infer_instance
 #align measure_theory.pdf.is_uniform.mul_pdf_integrable MeasureTheory.pdf.IsUniform.mulPdfIntegrable
 

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

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

• converts "--porting note" into "-- Porting note";
• converts "porting note" into "Porting note".

@@ -241,7 +241,7 @@ variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F
 theorem integrable_pdf_smul_iff [IsFiniteMeasure ℙ] {X : Ω → E} [HasPDF X ℙ μ] {f : E → F}
     (hf : AEStronglyMeasurable f μ) :
     Integrable (fun x => (pdf X ℙ μ x).toReal • f x) μ ↔ Integrable (fun x => f (X x)) ℙ := by
-  -- porting note: using `erw` because `rw` doesn't recognize `(f <| X ·)` as `f ∘ X`
+  -- Porting note: using `erw` because `rw` doesn't recognize `(f <| X ·)` as `f ∘ X`
   -- https://github.com/leanprover-community/mathlib4/issues/5164
   erw [← integrable_map_measure (hf.mono' HasPDF.absolutelyContinuous) (HasPDF.aemeasurable X ℙ μ),
     map_eq_withDensity_pdf X ℙ μ, pdf_def, integrable_rnDeriv_smul_iff HasPDF.absolutelyContinuous]

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

@@ -175,10 +175,10 @@ theorem withDensity_pdf_le_map {_ : MeasurableSpace Ω} (X : Ω → E) (ℙ : Me
   withDensity_rnDeriv_le _ _
 
 theorem set_lintegral_pdf_le_map {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
-    (μ : Measure E := by volume_tac) {s : Set E} (hs : MeasurableSet s) :
+    (μ : Measure E := by volume_tac) (s : Set E) :
     ∫⁻ x in s, pdf X ℙ μ x ∂μ ≤ map X ℙ s := by
   apply (withDensity_apply_le _ s).trans
-  exact withDensity_pdf_le_map _ _ _ s hs
+  exact withDensity_pdf_le_map _ _ _ s
 
 theorem map_eq_withDensity_pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
     (μ : Measure E := by volume_tac) [hX : HasPDF X ℙ μ] :

feat : collect uniform distribution for measures and uniform PMF in one file.

This collects the various notions of uniform distribution in one file, to unify them with a forthcoming PR.

@@ -338,138 +338,6 @@ theorem hasFiniteIntegral_mul {f : ℝ → ℝ} {g : ℝ → ℝ≥0∞} (hg : p
 
 end Real
 
-section
-
-/-! **Uniform Distribution** -/
-
-/-- A random variable `X` has uniform distribution on `s` if its push-forward measure is
-`(μ s)⁻¹ • μ.restrict s`. -/
-def IsUniform (X : Ω → E) (support : Set E) (ℙ : Measure Ω) (μ : Measure E := by volume_tac) :=
-  map X ℙ = (μ support)⁻¹ • μ.restrict support
-#align measure_theory.pdf.is_uniform MeasureTheory.pdf.IsUniform
-
-namespace IsUniform
-
-theorem aemeasurable {X : Ω → E} {s : Set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞)
-    (hu : IsUniform X s ℙ μ) : AEMeasurable X ℙ := by
-  dsimp [IsUniform] at hu
-  by_contra h
-  rw [map_of_not_aemeasurable h] at hu
-  apply zero_ne_one' ℝ≥0∞
-  calc
-    0 = (0 : Measure E) Set.univ := rfl
-    _ = _ := by rw [hu, smul_apply, restrict_apply MeasurableSet.univ,
-      Set.univ_inter, smul_eq_mul, ENNReal.inv_mul_cancel hns hnt]
-
-theorem absolutelyContinuous {X : Ω → E} {s : Set E} (hu : IsUniform X s ℙ μ) : map X ℙ ≪ μ := by
-  intro t ht
-  rw [hu, smul_apply, smul_eq_mul]
-  apply mul_eq_zero.mpr
-  refine Or.inr (le_antisymm ?_ (zero_le _))
-  exact ht ▸ restrict_apply_le s t
-
-theorem measure_preimage {X : Ω → E} {s : Set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞)
-    (hu : IsUniform X s ℙ μ) {A : Set E} (hA : MeasurableSet A) :
-    ℙ (X ⁻¹' A) = μ (s ∩ A) / μ s := by
-  rw [← map_apply_of_aemeasurable (hu.aemeasurable hns hnt) hA, hu, smul_apply, restrict_apply hA,
-    ENNReal.div_eq_inv_mul, smul_eq_mul, Set.inter_comm]
-#align measure_theory.pdf.is_uniform.measure_preimage MeasureTheory.pdf.IsUniform.measure_preimage
-
-theorem isProbabilityMeasure {X : Ω → E} {s : Set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞)
-    (hu : IsUniform X s ℙ μ) : IsProbabilityMeasure ℙ :=
-  ⟨by
-    have : X ⁻¹' Set.univ = Set.univ := Set.preimage_univ
-    rw [← this, hu.measure_preimage hns hnt MeasurableSet.univ, Set.inter_univ,
-      ENNReal.div_self hns hnt]⟩
-#align measure_theory.pdf.is_uniform.is_probability_measure MeasureTheory.pdf.IsUniform.isProbabilityMeasure
-
-theorem toMeasurable_iff {X : Ω → E} {s : Set E} :
-    IsUniform X (toMeasurable μ s) ℙ μ ↔ IsUniform X s ℙ μ := by
-  by_cases hnt : μ s = ∞
-  · simp [IsUniform, hnt]
-  · simp [IsUniform, restrict_toMeasurable hnt]
-
-protected theorem toMeasurable {X : Ω → E} {s : Set E} (hu : IsUniform X s ℙ μ) :
-    IsUniform X (toMeasurable μ s) ℙ μ := toMeasurable_iff.2 hu
-
-theorem hasPDF {X : Ω → E} {s : Set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞)
-    (hu : IsUniform X s ℙ μ) : HasPDF X ℙ μ := by
-  let t := toMeasurable μ s
-  apply hasPDF_of_map_eq_withDensity (hu.aemeasurable hns hnt) (t.indicator ((μ t)⁻¹ • 1)) <|
-    (measurable_one.aemeasurable.const_smul (μ t)⁻¹).indicator (measurableSet_toMeasurable μ s)
-  rw [hu, withDensity_indicator (measurableSet_toMeasurable μ s), withDensity_smul _ measurable_one,
-    withDensity_one, restrict_toMeasurable hnt, measure_toMeasurable]
-#align measure_theory.pdf.is_uniform.has_pdf MeasureTheory.pdf.IsUniform.hasPDF
-
-theorem pdf_eq_zero_of_measure_eq_zero_or_top {X : Ω → E} {s : Set E}
-    (hu : IsUniform X s ℙ μ) (hμs : μ s = 0 ∨ μ s = ∞) : pdf X ℙ μ =ᵐ[μ] 0 := by
-  rcases hμs with H|H
-  · simp only [IsUniform, H, ENNReal.inv_zero, restrict_eq_zero.mpr H, smul_zero] at hu
-    simp [pdf, hu]
-  · simp only [IsUniform, H, ENNReal.inv_top, zero_smul] at hu
-    simp [pdf, hu]
-
-theorem pdf_eq {X : Ω → E} {s : Set E} (hms : MeasurableSet s)
-    (hu : IsUniform X s ℙ μ) : pdf X ℙ μ =ᵐ[μ] s.indicator ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) := by
-  by_cases hnt : μ s = ∞
-  · simp [pdf_eq_zero_of_measure_eq_zero_or_top hu (Or.inr hnt), hnt]
-  by_cases hns : μ s = 0
-  · filter_upwards [measure_zero_iff_ae_nmem.mp hns,
-      pdf_eq_zero_of_measure_eq_zero_or_top hu (Or.inl hns)] with x hx h'x
-    simp [hx, h'x, hns]
-  have : HasPDF X ℙ μ := hasPDF hns hnt hu
-  have : IsProbabilityMeasure ℙ := isProbabilityMeasure hns hnt hu
-  apply (eq_of_map_eq_withDensity _ _).mp
-  · rw [hu, withDensity_indicator hms, withDensity_smul _ measurable_one, withDensity_one]
-  · exact (measurable_one.aemeasurable.const_smul (μ s)⁻¹).indicator hms
-
-theorem pdf_toReal_ae_eq {X : Ω → E} {s : Set E} (hms : MeasurableSet s)
-    (hX : IsUniform X s ℙ μ) :
-    (fun x => (pdf X ℙ μ x).toReal) =ᵐ[μ] fun x =>
-      (s.indicator ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) x).toReal :=
-  Filter.EventuallyEq.fun_comp (pdf_eq hms hX) ENNReal.toReal
-#align measure_theory.pdf.is_uniform.pdf_to_real_ae_eq MeasureTheory.pdf.IsUniform.pdf_toReal_ae_eq
-
-variable {X : Ω → ℝ} {s : Set ℝ}
-
-theorem mul_pdf_integrable (hcs : IsCompact s) (huX : IsUniform X s ℙ) :
-    Integrable fun x : ℝ => x * (pdf X ℙ volume x).toReal := by
-  by_cases hnt : volume s = 0 ∨ volume s = ∞
-  · have I : Integrable (fun x ↦ x * ENNReal.toReal (0)) := by simp
-    apply I.congr
-    filter_upwards [pdf_eq_zero_of_measure_eq_zero_or_top huX hnt] with x hx
-    simp [hx]
-  simp only [not_or] at hnt
-  have : IsProbabilityMeasure ℙ := isProbabilityMeasure hnt.1 hnt.2 huX
-  constructor
-  · exact aestronglyMeasurable_id.mul
-      (measurable_pdf X ℙ).aemeasurable.ennreal_toReal.aestronglyMeasurable
-  refine' hasFiniteIntegral_mul (pdf_eq hcs.measurableSet huX) _
-  set ind := (volume s)⁻¹ • (1 : ℝ → ℝ≥0∞)
-  have : ∀ x, ↑‖x‖₊ * s.indicator ind x = s.indicator (fun x => ‖x‖₊ * ind x) x := fun x =>
-    (s.indicator_mul_right (fun x => ↑‖x‖₊) ind).symm
-  simp only [this, lintegral_indicator _ hcs.measurableSet, mul_one, Algebra.id.smul_eq_mul,
-    Pi.one_apply, Pi.smul_apply]
-  rw [lintegral_mul_const _ measurable_nnnorm.coe_nnreal_ennreal]
-  exact (ENNReal.mul_lt_top (set_lintegral_lt_top_of_isCompact hnt.2 hcs continuous_nnnorm).ne
-    (ENNReal.inv_lt_top.2 (pos_iff_ne_zero.mpr hnt.1)).ne).ne
-#align measure_theory.pdf.is_uniform.mul_pdf_integrable MeasureTheory.pdf.IsUniform.mul_pdf_integrable
-
-/-- A real uniform random variable `X` with support `s` has expectation
-`(λ s)⁻¹ * ∫ x in s, x ∂λ` where `λ` is the Lebesgue measure. -/
-theorem integral_eq (huX : IsUniform X s ℙ) :
-    ∫ x, X x ∂ℙ = (volume s)⁻¹.toReal * ∫ x in s, x := by
-  rw [← smul_eq_mul, ← integral_smul_measure, ← huX]
-  by_cases hX : AEMeasurable X ℙ
-  · exact (integral_map hX aestronglyMeasurable_id).symm
-  · rw [map_of_not_aemeasurable hX, integral_zero_measure, integral_non_aestronglyMeasurable]
-    rwa [aestronglyMeasurable_iff_aemeasurable]
-#align measure_theory.pdf.is_uniform.integral_eq MeasureTheory.pdf.IsUniform.integral_eq
-
-end IsUniform
-
-end
-
 section TwoVariables
 
 open ProbabilityTheory

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

@@ -378,79 +378,92 @@ theorem measure_preimage {X : Ω → E} {s : Set E} (hns : μ s ≠ 0) (hnt : μ
 theorem isProbabilityMeasure {X : Ω → E} {s : Set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞)
     (hu : IsUniform X s ℙ μ) : IsProbabilityMeasure ℙ :=
   ⟨by
-    have : X ⁻¹' Set.univ = Set.univ := by simp only [Set.preimage_univ]
+    have : X ⁻¹' Set.univ = Set.univ := Set.preimage_univ
     rw [← this, hu.measure_preimage hns hnt MeasurableSet.univ, Set.inter_univ,
       ENNReal.div_self hns hnt]⟩
 #align measure_theory.pdf.is_uniform.is_probability_measure MeasureTheory.pdf.IsUniform.isProbabilityMeasure
 
-theorem hasPDF {X : Ω → E} {s : Set E} (hms : MeasurableSet s) (hns : μ s ≠ 0) (hnt : μ s ≠ ∞)
+theorem toMeasurable_iff {X : Ω → E} {s : Set E} :
+    IsUniform X (toMeasurable μ s) ℙ μ ↔ IsUniform X s ℙ μ := by
+  by_cases hnt : μ s = ∞
+  · simp [IsUniform, hnt]
+  · simp [IsUniform, restrict_toMeasurable hnt]
+
+protected theorem toMeasurable {X : Ω → E} {s : Set E} (hu : IsUniform X s ℙ μ) :
+    IsUniform X (toMeasurable μ s) ℙ μ := toMeasurable_iff.2 hu
+
+theorem hasPDF {X : Ω → E} {s : Set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞)
     (hu : IsUniform X s ℙ μ) : HasPDF X ℙ μ := by
-  apply hasPDF_of_map_eq_withDensity (hu.aemeasurable hns hnt) (s.indicator ((μ s)⁻¹ • 1)) <|
-    (measurable_one.aemeasurable.const_smul (μ s)⁻¹).indicator hms
-  rw [hu, withDensity_indicator hms, withDensity_smul _ measurable_one, withDensity_one]
+  let t := toMeasurable μ s
+  apply hasPDF_of_map_eq_withDensity (hu.aemeasurable hns hnt) (t.indicator ((μ t)⁻¹ • 1)) <|
+    (measurable_one.aemeasurable.const_smul (μ t)⁻¹).indicator (measurableSet_toMeasurable μ s)
+  rw [hu, withDensity_indicator (measurableSet_toMeasurable μ s), withDensity_smul _ measurable_one,
+    withDensity_one, restrict_toMeasurable hnt, measure_toMeasurable]
 #align measure_theory.pdf.is_uniform.has_pdf MeasureTheory.pdf.IsUniform.hasPDF
 
-theorem hasPDF₀ {X : Ω → E} {s : Set E} (hms : NullMeasurableSet s μ) (hns : μ s ≠ 0)
-    (hnt : μ s ≠ ∞) (hu : IsUniform X s ℙ μ) : HasPDF X ℙ μ := by
-  have hms' := measurableSet_toMeasurable μ s
-  apply hasPDF (m := m) (ℙ := ℙ) (μ := μ) hms'
-    (measure_toMeasurable s ▸ hns) (measure_toMeasurable s ▸ hnt) _
-  unfold IsUniform
-  rw [measure_toMeasurable, restrict_congr_set hms.toMeasurable_ae_eq, hu]
+theorem pdf_eq_zero_of_measure_eq_zero_or_top {X : Ω → E} {s : Set E}
+    (hu : IsUniform X s ℙ μ) (hμs : μ s = 0 ∨ μ s = ∞) : pdf X ℙ μ =ᵐ[μ] 0 := by
+  rcases hμs with H|H
+  · simp only [IsUniform, H, ENNReal.inv_zero, restrict_eq_zero.mpr H, smul_zero] at hu
+    simp [pdf, hu]
+  · simp only [IsUniform, H, ENNReal.inv_top, zero_smul] at hu
+    simp [pdf, hu]
 
-theorem pdf_eq {X : Ω → E} {s : Set E} (hms : MeasurableSet s) (hns : μ s ≠ 0) (hnt : μ s ≠ ∞)
+theorem pdf_eq {X : Ω → E} {s : Set E} (hms : MeasurableSet s)
     (hu : IsUniform X s ℙ μ) : pdf X ℙ μ =ᵐ[μ] s.indicator ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) := by
-  have : HasPDF X ℙ μ := hasPDF hms hns hnt hu
+  by_cases hnt : μ s = ∞
+  · simp [pdf_eq_zero_of_measure_eq_zero_or_top hu (Or.inr hnt), hnt]
+  by_cases hns : μ s = 0
+  · filter_upwards [measure_zero_iff_ae_nmem.mp hns,
+      pdf_eq_zero_of_measure_eq_zero_or_top hu (Or.inl hns)] with x hx h'x
+    simp [hx, h'x, hns]
+  have : HasPDF X ℙ μ := hasPDF hns hnt hu
   have : IsProbabilityMeasure ℙ := isProbabilityMeasure hns hnt hu
   apply (eq_of_map_eq_withDensity _ _).mp
   · rw [hu, withDensity_indicator hms, withDensity_smul _ measurable_one, withDensity_one]
   · exact (measurable_one.aemeasurable.const_smul (μ s)⁻¹).indicator hms
 
-theorem pdf_toReal_ae_eq {X : Ω → E}
-    {s : Set E} (hms : MeasurableSet s) (hns : μ s ≠ 0) (hnt : μ s ≠ ∞)
+theorem pdf_toReal_ae_eq {X : Ω → E} {s : Set E} (hms : MeasurableSet s)
     (hX : IsUniform X s ℙ μ) :
     (fun x => (pdf X ℙ μ x).toReal) =ᵐ[μ] fun x =>
       (s.indicator ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) x).toReal :=
-  Filter.EventuallyEq.fun_comp (pdf_eq hms hns hnt hX) ENNReal.toReal
+  Filter.EventuallyEq.fun_comp (pdf_eq hms hX) ENNReal.toReal
 #align measure_theory.pdf.is_uniform.pdf_to_real_ae_eq MeasureTheory.pdf.IsUniform.pdf_toReal_ae_eq
 
-variable {X : Ω → ℝ} {s : Set ℝ} (hms : MeasurableSet s) (hns : volume s ≠ 0) (hnt : volume s ≠ ∞)
+variable {X : Ω → ℝ} {s : Set ℝ}
 
-theorem mul_pdf_integrable [IsFiniteMeasure ℙ] (hcs : IsCompact s) (huX : IsUniform X s ℙ) :
+theorem mul_pdf_integrable (hcs : IsCompact s) (huX : IsUniform X s ℙ) :
     Integrable fun x : ℝ => x * (pdf X ℙ volume x).toReal := by
-  constructor -- porting note: `refine` was failing, don't know why
+  by_cases hnt : volume s = 0 ∨ volume s = ∞
+  · have I : Integrable (fun x ↦ x * ENNReal.toReal (0)) := by simp
+    apply I.congr
+    filter_upwards [pdf_eq_zero_of_measure_eq_zero_or_top huX hnt] with x hx
+    simp [hx]
+  simp only [not_or] at hnt
+  have : IsProbabilityMeasure ℙ := isProbabilityMeasure hnt.1 hnt.2 huX
+  constructor
   · exact aestronglyMeasurable_id.mul
       (measurable_pdf X ℙ).aemeasurable.ennreal_toReal.aestronglyMeasurable
-  refine' hasFiniteIntegral_mul (pdf_eq hms hns hnt huX) _
+  refine' hasFiniteIntegral_mul (pdf_eq hcs.measurableSet huX) _
   set ind := (volume s)⁻¹ • (1 : ℝ → ℝ≥0∞)
   have : ∀ x, ↑‖x‖₊ * s.indicator ind x = s.indicator (fun x => ‖x‖₊ * ind x) x := fun x =>
     (s.indicator_mul_right (fun x => ↑‖x‖₊) ind).symm
-  simp only [this, lintegral_indicator _ hms, mul_one, Algebra.id.smul_eq_mul, Pi.one_apply,
-    Pi.smul_apply]
+  simp only [this, lintegral_indicator _ hcs.measurableSet, mul_one, Algebra.id.smul_eq_mul,
+    Pi.one_apply, Pi.smul_apply]
   rw [lintegral_mul_const _ measurable_nnnorm.coe_nnreal_ennreal]
-  refine' (ENNReal.mul_lt_top (set_lintegral_lt_top_of_isCompact hnt hcs continuous_nnnorm).ne
-    (ENNReal.inv_lt_top.2 (pos_iff_ne_zero.mpr hns)).ne).ne
+  exact (ENNReal.mul_lt_top (set_lintegral_lt_top_of_isCompact hnt.2 hcs continuous_nnnorm).ne
+    (ENNReal.inv_lt_top.2 (pos_iff_ne_zero.mpr hnt.1)).ne).ne
 #align measure_theory.pdf.is_uniform.mul_pdf_integrable MeasureTheory.pdf.IsUniform.mul_pdf_integrable
 
 /-- A real uniform random variable `X` with support `s` has expectation
 `(λ s)⁻¹ * ∫ x in s, x ∂λ` where `λ` is the Lebesgue measure. -/
 theorem integral_eq (huX : IsUniform X s ℙ) :
     ∫ x, X x ∂ℙ = (volume s)⁻¹.toReal * ∫ x in s, x := by
-  haveI := hasPDF hms hns hnt huX
-  haveI := huX.isProbabilityMeasure hns hnt
-  rw [← integral_mul_eq_integral]
-  rw [integral_congr_ae (Filter.EventuallyEq.mul (ae_eq_refl _) (pdf_toReal_ae_eq hms hns hnt huX))]
-  have :
-    ∀ x,
-      x * (s.indicator ((volume s)⁻¹ • (1 : ℝ → ℝ≥0∞)) x).toReal =
-        x * s.indicator ((volume s)⁻¹.toReal • (1 : ℝ → ℝ)) x := by
-    refine' fun x => congr_arg (x * ·) _
-    by_cases hx : x ∈ s
-    · simp [Set.indicator_of_mem hx]
-    · simp [Set.indicator_of_not_mem hx]
-  simp_rw [this, ← s.indicator_mul_right fun x => x, integral_indicator hms]
-  change ∫ x in s, x * (volume s)⁻¹.toReal • (1 : ℝ) = _
-  rw [integral_mul_right, mul_comm, smul_eq_mul, mul_one]
+  rw [← smul_eq_mul, ← integral_smul_measure, ← huX]
+  by_cases hX : AEMeasurable X ℙ
+  · exact (integral_map hX aestronglyMeasurable_id).symm
+  · rw [map_of_not_aemeasurable hX, integral_zero_measure, integral_non_aestronglyMeasurable]
+    rwa [aestronglyMeasurable_iff_aemeasurable]
 #align measure_theory.pdf.is_uniform.integral_eq MeasureTheory.pdf.IsUniform.integral_eq
 
 end IsUniform

I used the regex \(\(· (.) ·\) (.)\), replacing with ($2 $1 ·).

@@ -444,7 +444,7 @@ theorem integral_eq (huX : IsUniform X s ℙ) :
     ∀ x,
       x * (s.indicator ((volume s)⁻¹ • (1 : ℝ → ℝ≥0∞)) x).toReal =
         x * s.indicator ((volume s)⁻¹.toReal • (1 : ℝ → ℝ)) x := by
-    refine' fun x => congr_arg ((· * ·) x) _
+    refine' fun x => congr_arg (x * ·) _
     by_cases hx : x ∈ s
     · simp [Set.indicator_of_mem hx]
     · simp [Set.indicator_of_not_mem hx]

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

@@ -13,7 +13,8 @@ import Mathlib.Probability.Independence.Basic
 # Probability density function
 
 This file defines the probability density function of random variables, by which we mean
-measurable functions taking values in a Borel space. In particular, a measurable function `f`
+measurable functions taking values in a Borel space. The probability density function is defined
+as the Radon–Nikodym derivative of the law of `X`. In particular, a measurable function `f`
 is said to the probability density function of a random variable `X` if for all measurable
 sets `S`, `ℙ(X ∈ S) = ∫ x in S, f x dx`. Probability density functions are one way of describing
 the distribution of a random variable, and are useful for calculating probabilities and
@@ -25,19 +26,18 @@ random variables with this distribution.
 ## Main definitions
 
 * `MeasureTheory.HasPDF` : A random variable `X : Ω → E` is said to `HasPDF` with
-  respect to the measure `ℙ` on `Ω` and `μ` on `E` if there exists a measurable function `f`
-  such that the push-forward measure of `ℙ` along `X` equals `μ.withDensity f`.
+  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`.
 * `MeasureTheory.pdf` : If `X` is a random variable that `HasPDF X ℙ μ`, then `pdf X`
-  is the measurable function `f` such that the push-forward measure of `ℙ` along `X` equals
-  `μ.withDensity f`.
+  is the Radon–Nikodym derivative of the push-forward measure of `ℙ` along `X` with respect to `μ`.
 * `MeasureTheory.pdf.IsUniform` : A random variable `X` is said to follow the uniform
   distribution if it has a constant probability density function with a compact, non-null support.
 
 ## Main results
 
-* `MeasureTheory.pdf.integral_fun_mul_eq_integral` : Law of the unconscious statistician,
-  i.e. if a random variable `X : Ω → E` has pdf `f`, then `𝔼(g(X)) = ∫ x, g x * f x dx` for
-  all measurable `g : E → ℝ`.
+* `MeasureTheory.pdf.integral_pdf_smul` : Law of the unconscious statistician,
+  i.e. if a random variable `X : Ω → E` has pdf `f`, then `𝔼(g(X)) = ∫ x, f x • g x dx` for
+  all measurable `g : E → F`.
 * `MeasureTheory.pdf.integral_mul_eq_integral` : A real-valued random variable `X` with
   pdf `f` has expectation `∫ x, x * f x dx`.
 * `MeasureTheory.pdf.IsUniform.integral_eq` : If `X` follows the uniform distribution with
@@ -63,67 +63,132 @@ namespace MeasureTheory
 variable {Ω E : Type*} [MeasurableSpace E]
 
 /-- A random variable `X : Ω → E` is said to `HasPDF` with respect to the measure `ℙ` on `Ω` and
-`μ` on `E` if there exists a measurable function `f` such that the push-forward measure of `ℙ`
-along `X` equals `μ.withDensity f`. -/
+`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
-  -- porting note: TODO: split into fields `Measurable` and `exists_pdf`
-  pdf' : Measurable X ∧ ∃ f : E → ℝ≥0∞, Measurable f ∧ map X ℙ = μ.withDensity f
+  pdf' : AEMeasurable X ℙ ∧ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ
 #align measure_theory.has_pdf MeasureTheory.HasPDF
 
+section HasPDF
+
+variable {_ : MeasurableSpace Ω}
+
+theorem hasPDF_iff {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E} :
+    HasPDF X ℙ μ ↔ AEMeasurable X ℙ ∧ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ :=
+  ⟨@HasPDF.pdf' _ _ _ _ _ _ _, HasPDF.mk⟩
+#align measure_theory.pdf.has_pdf_iff MeasureTheory.hasPDF_iff
+
+theorem hasPDF_iff_of_aemeasurable {X : Ω → E} {ℙ : Measure Ω}
+    {μ : Measure E} (hX : AEMeasurable X ℙ) :
+    HasPDF X ℙ μ ↔ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ := by
+  rw [hasPDF_iff]
+  simp only [hX, true_and]
+#align measure_theory.pdf.has_pdf_iff_of_measurable MeasureTheory.hasPDF_iff_of_aemeasurable
+
 @[measurability]
-theorem HasPDF.measurable {_ : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
-    (μ : Measure E := by volume_tac) [hX : HasPDF X ℙ μ] :
-    Measurable X :=
+theorem HasPDF.aemeasurable (X : Ω → E) (ℙ : Measure Ω)
+    (μ : Measure E) [hX : HasPDF X ℙ μ] : AEMeasurable X ℙ :=
   hX.pdf'.1
-#align measure_theory.has_pdf.measurable MeasureTheory.HasPDF.measurable
+#align measure_theory.has_pdf.measurable MeasureTheory.HasPDF.aemeasurable
+
+instance HasPDF.haveLebesgueDecomposition {X : Ω → E} {ℙ : Measure Ω}
+    {μ : Measure E} [hX : HasPDF X ℙ μ] : (map X ℙ).HaveLebesgueDecomposition μ :=
+  hX.pdf'.2.1
+#align measure_theory.pdf.have_lebesgue_decomposition_of_has_pdf MeasureTheory.HasPDF.haveLebesgueDecomposition
+
+theorem HasPDF.absolutelyContinuous {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E}
+    [hX : HasPDF X ℙ μ] : map X ℙ ≪ μ :=
+  hX.pdf'.2.2
+#align measure_theory.pdf.map_absolutely_continuous MeasureTheory.HasPDF.absolutelyContinuous
 
-/-- If `X` is a random variable that `HasPDF X ℙ μ`, then `pdf X` is the measurable function `f`
-such that the push-forward measure of `ℙ` along `X` equals `μ.withDensity f`. -/
+/-- A random variable that `HasPDF` is quasi-measure preserving. -/
+theorem HasPDF.quasiMeasurePreserving_of_measurable (X : Ω → E) (ℙ : Measure Ω) (μ : Measure E)
+    [HasPDF X ℙ μ] (h : Measurable X) : QuasiMeasurePreserving X ℙ μ :=
+  { measurable := h
+    absolutelyContinuous := HasPDF.absolutelyContinuous }
+#align measure_theory.pdf.to_quasi_measure_preserving MeasureTheory.HasPDF.quasiMeasurePreserving_of_measurable
+
+theorem HasPDF.congr {X Y : Ω → E} {ℙ : Measure Ω} {μ : Measure E} (hXY : X =ᵐ[ℙ] Y)
+    [hX : HasPDF X ℙ μ] : HasPDF Y ℙ μ :=
+  ⟨(HasPDF.aemeasurable X ℙ μ).congr hXY, ℙ.map_congr hXY ▸ hX.haveLebesgueDecomposition,
+    ℙ.map_congr hXY ▸ hX.absolutelyContinuous⟩
+
+theorem HasPDF.congr' {X Y : Ω → E} {ℙ : Measure Ω} {μ : Measure E} (hXY : X =ᵐ[ℙ] Y) :
+    HasPDF X ℙ μ ↔ HasPDF Y ℙ μ :=
+  ⟨fun _ ↦ HasPDF.congr hXY, fun _ ↦ HasPDF.congr hXY.symm⟩
+
+/-- X `HasPDF` if there is a pdf `f` such that `map X ℙ = μ.withDensity f`. -/
+theorem hasPDF_of_map_eq_withDensity {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E}
+    (hX : AEMeasurable X ℙ) (f : E → ℝ≥0∞) (hf : AEMeasurable f μ) (h : map X ℙ = μ.withDensity f) :
+    HasPDF X ℙ μ := by
+  refine ⟨hX, ?_, ?_⟩ <;> rw [h]
+  · rw [withDensity_congr_ae hf.ae_eq_mk]
+    exact haveLebesgueDecomposition_withDensity μ hf.measurable_mk
+  · exact withDensity_absolutelyContinuous μ f
+
+end HasPDF
+
+/-- If `X` is a random variable, 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 hX : HasPDF X ℙ μ then Classical.choose hX.pdf'.2 else 0
+  (map X ℙ).rnDeriv μ
 #align measure_theory.pdf MeasureTheory.pdf
 
-theorem pdf_undef {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E}
-    (h : ¬HasPDF X ℙ μ) : pdf X ℙ μ = 0 := by simp only [pdf, dif_neg h]
-#align measure_theory.pdf_undef MeasureTheory.pdf_undef
+theorem pdf_def {_ : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E} :
+    pdf X ℙ μ = (map X ℙ).rnDeriv μ := rfl
 
-theorem hasPDF_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E}
-    (h : pdf X ℙ μ ≠ 0) : HasPDF X ℙ μ := by
-  by_contra hpdf
-  simp [pdf, hpdf] at h
-#align measure_theory.has_pdf_of_pdf_ne_zero MeasureTheory.hasPDF_of_pdf_ne_zero
+theorem pdf_of_not_aemeasurable {_ : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
+    {X : Ω → E} (hX : ¬AEMeasurable X ℙ) : pdf X ℙ μ =ᵐ[μ] 0 := by
+  rw [pdf_def, map_of_not_aemeasurable hX]
+  exact rnDeriv_zero μ
+#align measure_theory.pdf_eq_zero_of_not_measurable MeasureTheory.pdf_of_not_aemeasurable
 
-theorem pdf_eq_zero_of_not_measurable {_ : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
-    {X : Ω → E} (hX : ¬Measurable X) : pdf X ℙ μ = 0 :=
-  pdf_undef fun hpdf => hX hpdf.pdf'.1
-#align measure_theory.pdf_eq_zero_of_not_measurable MeasureTheory.pdf_eq_zero_of_not_measurable
+theorem pdf_of_not_haveLebesgueDecomposition {_ : MeasurableSpace Ω} {ℙ : Measure Ω}
+    {μ : Measure E} {X : Ω → E} (h : ¬(map X ℙ).HaveLebesgueDecomposition μ) : pdf X ℙ μ = 0 :=
+  rnDeriv_of_not_haveLebesgueDecomposition h
 
-theorem measurable_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
-    (X : Ω → E) (h : pdf X ℙ μ ≠ 0) : Measurable X := by
-  by_contra hX
-  exact h (pdf_eq_zero_of_not_measurable hX)
-#align measure_theory.measurable_of_pdf_ne_zero MeasureTheory.measurable_of_pdf_ne_zero
+theorem aemeasurable_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
+    (X : Ω → E) (h : ¬pdf X ℙ μ =ᵐ[μ] 0) : AEMeasurable X ℙ := by
+  contrapose! h
+  exact pdf_of_not_aemeasurable h
+#align measure_theory.measurable_of_pdf_ne_zero MeasureTheory.aemeasurable_of_pdf_ne_zero
+
+theorem hasPDF_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E}
+    (hac : map X ℙ ≪ μ) (hpdf : ¬pdf X ℙ μ =ᵐ[μ] 0) : HasPDF X ℙ μ := by
+  refine ⟨?_, ?_, hac⟩
+  · exact aemeasurable_of_pdf_ne_zero X hpdf
+  · contrapose! hpdf
+    have := pdf_of_not_haveLebesgueDecomposition hpdf
+    filter_upwards using congrFun this
+#align measure_theory.has_pdf_of_pdf_ne_zero MeasureTheory.hasPDF_of_pdf_ne_zero
 
 @[measurability]
 theorem measurable_pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
     (μ : Measure E := by volume_tac) : Measurable (pdf X ℙ μ) := by
-  unfold pdf
-  split_ifs with h
-  exacts [(Classical.choose_spec h.1.2).1, measurable_zero]
+  exact measurable_rnDeriv _ _
 #align measure_theory.measurable_pdf MeasureTheory.measurable_pdf
 
+theorem withDensity_pdf_le_map {_ : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
+    (μ : Measure E := by volume_tac) : μ.withDensity (pdf X ℙ μ) ≤ map X ℙ :=
+  withDensity_rnDeriv_le _ _
+
+theorem set_lintegral_pdf_le_map {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
+    (μ : Measure E := by volume_tac) {s : Set E} (hs : MeasurableSet s) :
+    ∫⁻ x in s, pdf X ℙ μ x ∂μ ≤ map X ℙ s := by
+  apply (withDensity_apply_le _ s).trans
+  exact withDensity_pdf_le_map _ _ _ s hs
+
 theorem map_eq_withDensity_pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
     (μ : Measure E := by volume_tac) [hX : HasPDF X ℙ μ] :
-    Measure.map X ℙ = μ.withDensity (pdf X ℙ μ) := by
-  simp only [pdf, dif_pos hX]
-  exact (Classical.choose_spec hX.pdf'.2).2
+    map X ℙ = μ.withDensity (pdf X ℙ μ) := by
+  rw [pdf_def, withDensity_rnDeriv_eq _ _ hX.absolutelyContinuous]
 #align measure_theory.map_eq_with_density_pdf MeasureTheory.map_eq_withDensity_pdf
 
 theorem map_eq_set_lintegral_pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
     (μ : Measure E := by volume_tac) [hX : HasPDF X ℙ μ] {s : Set E}
-    (hs : MeasurableSet s) : Measure.map X ℙ s = ∫⁻ x in s, pdf X ℙ μ x ∂μ := by
+    (hs : MeasurableSet s) : map X ℙ s = ∫⁻ x in s, pdf X ℙ μ x ∂μ := by
   rw [← withDensity_apply _ hs, map_eq_withDensity_pdf X ℙ μ]
 #align measure_theory.map_eq_set_lintegral_pdf MeasureTheory.map_eq_set_lintegral_pdf
 
@@ -131,32 +196,30 @@ namespace pdf
 
 variable {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
 
+protected theorem congr {X Y : Ω → E} (hXY : X =ᵐ[ℙ] Y) : pdf X ℙ μ = pdf Y ℙ μ :=
+  by rw [pdf_def, pdf_def, map_congr hXY]
+
 theorem lintegral_eq_measure_univ {X : Ω → E} [HasPDF X ℙ μ] :
     ∫⁻ x, pdf X ℙ μ x ∂μ = ℙ Set.univ := by
   rw [← set_lintegral_univ, ← map_eq_set_lintegral_pdf X ℙ μ MeasurableSet.univ,
-    Measure.map_apply (HasPDF.measurable X ℙ μ) MeasurableSet.univ, Set.preimage_univ]
+    map_apply_of_aemeasurable (HasPDF.aemeasurable X ℙ μ) MeasurableSet.univ, Set.preimage_univ]
 #align measure_theory.pdf.lintegral_eq_measure_univ MeasureTheory.pdf.lintegral_eq_measure_univ
 
-theorem unique [IsFiniteMeasure ℙ] {X : Ω → E} [HasPDF X ℙ μ] (f : E → ℝ≥0∞)
-    (hmf : AEMeasurable f μ) : ℙ.map X = μ.withDensity f ↔ pdf X ℙ μ =ᵐ[μ] f := by
+theorem eq_of_map_eq_withDensity [IsFiniteMeasure ℙ] {X : Ω → E} [HasPDF X ℙ μ] (f : E → ℝ≥0∞)
+    (hmf : AEMeasurable f μ) : map X ℙ = μ.withDensity f ↔ pdf X ℙ μ =ᵐ[μ] f := by
   rw [map_eq_withDensity_pdf X ℙ μ]
   apply withDensity_eq_iff (measurable_pdf X ℙ μ).aemeasurable hmf
   rw [lintegral_eq_measure_univ]
   exact measure_ne_top _ _
 
-theorem unique' [SigmaFinite μ] {X : Ω → E} [HasPDF X ℙ μ] (f : E → ℝ≥0∞) (hmf : AEMeasurable f μ) :
-    ℙ.map X = μ.withDensity f ↔ pdf X ℙ μ =ᵐ[μ] f :=
+theorem eq_of_map_eq_withDensity' [SigmaFinite μ] {X : Ω → E} [HasPDF X ℙ μ] (f : E → ℝ≥0∞)
+    (hmf : AEMeasurable f μ) : map X ℙ = μ.withDensity f ↔ pdf X ℙ μ =ᵐ[μ] f :=
   map_eq_withDensity_pdf X ℙ μ ▸
     withDensity_eq_iff_of_sigmaFinite (measurable_pdf X ℙ μ).aemeasurable hmf
 
 nonrec theorem ae_lt_top [IsFiniteMeasure ℙ] {μ : Measure E} {X : Ω → E} :
-    ∀ᵐ x ∂μ, pdf X ℙ μ x < ∞ := by
-  by_cases hpdf : HasPDF X ℙ μ
-  · haveI := hpdf
-    refine' ae_lt_top (measurable_pdf X ℙ μ) _
-    rw [lintegral_eq_measure_univ]
-    exact measure_ne_top _ _
-  · simp [pdf, hpdf]
+    ∀ᵐ x ∂μ, pdf X ℙ μ x < ∞ :=
+  rnDeriv_lt_top (map X ℙ) μ
 #align measure_theory.pdf.ae_lt_top MeasureTheory.pdf.ae_lt_top
 
 nonrec theorem ofReal_toReal_ae_eq [IsFiniteMeasure ℙ] {X : Ω → E} :
@@ -164,89 +227,39 @@ nonrec theorem ofReal_toReal_ae_eq [IsFiniteMeasure ℙ] {X : Ω → E} :
   ofReal_toReal_ae_eq ae_lt_top
 #align measure_theory.pdf.of_real_to_real_ae_eq MeasureTheory.pdf.ofReal_toReal_ae_eq
 
-theorem integrable_iff_integrable_mul_pdf [IsFiniteMeasure ℙ] {X : Ω → E} [HasPDF X ℙ μ] {f : E → ℝ}
-    (hf : Measurable f) :
-    Integrable (fun x => f (X x)) ℙ ↔ Integrable (fun x => f x * (pdf X ℙ μ x).toReal) μ := by
+section IntegralPDFMul
+
+/-- **The Law of the Unconscious Statistician** for nonnegative random variables. -/
+theorem lintegral_pdf_mul {X : Ω → E} [HasPDF X ℙ μ] {f : E → ℝ≥0∞}
+    (hf : AEMeasurable f μ) : ∫⁻ x, pdf X ℙ μ x * f x ∂μ = ∫⁻ x, f (X x) ∂ℙ := by
+  rw [pdf_def,
+    ← lintegral_map' (hf.mono_ac HasPDF.absolutelyContinuous) (HasPDF.aemeasurable X ℙ μ),
+    lintegral_rnDeriv_mul HasPDF.absolutelyContinuous hf]
+
+variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F]
+
+theorem integrable_pdf_smul_iff [IsFiniteMeasure ℙ] {X : Ω → E} [HasPDF X ℙ μ] {f : E → F}
+    (hf : AEStronglyMeasurable f μ) :
+    Integrable (fun x => (pdf X ℙ μ x).toReal • f x) μ ↔ Integrable (fun x => f (X x)) ℙ := by
   -- porting note: using `erw` because `rw` doesn't recognize `(f <| X ·)` as `f ∘ X`
   -- https://github.com/leanprover-community/mathlib4/issues/5164
-  erw [← integrable_map_measure hf.aestronglyMeasurable (HasPDF.measurable X ℙ μ).aemeasurable,
-    map_eq_withDensity_pdf X ℙ μ, integrable_withDensity_iff (measurable_pdf _ _ _) ae_lt_top]
-#align measure_theory.pdf.integrable_iff_integrable_mul_pdf MeasureTheory.pdf.integrable_iff_integrable_mul_pdf
+  erw [← integrable_map_measure (hf.mono' HasPDF.absolutelyContinuous) (HasPDF.aemeasurable X ℙ μ),
+    map_eq_withDensity_pdf X ℙ μ, pdf_def, integrable_rnDeriv_smul_iff HasPDF.absolutelyContinuous]
+  eta_reduce
+  rw [withDensity_rnDeriv_eq _ _ HasPDF.absolutelyContinuous]
+#align measure_theory.pdf.integrable_iff_integrable_mul_pdf MeasureTheory.pdf.integrable_pdf_smul_iff
 
 /-- **The Law of the Unconscious Statistician**: Given a random variable `X` and a measurable
-function `f`, `f ∘ X` is a random variable with expectation `∫ x, f x * pdf X ∂μ`
+function `f`, `f ∘ X` is a random variable with expectation `∫ x, pdf X x • f x ∂μ`
 where `μ` is a measure on the codomain of `X`. -/
-theorem integral_fun_mul_eq_integral [IsFiniteMeasure ℙ] {X : Ω → E} [HasPDF X ℙ μ] {f : E → ℝ}
-    (hf : Measurable f) : ∫ x, f x * (pdf X ℙ μ x).toReal ∂μ = ∫ x, f (X x) ∂ℙ := by
-  by_cases hpdf : Integrable (fun x => f x * (pdf X ℙ μ x).toReal) μ
-  · rw [← integral_map (HasPDF.measurable X ℙ μ).aemeasurable hf.aestronglyMeasurable,
-      map_eq_withDensity_pdf X ℙ μ, integral_eq_lintegral_pos_part_sub_lintegral_neg_part hpdf,
-      integral_eq_lintegral_pos_part_sub_lintegral_neg_part,
-      lintegral_withDensity_eq_lintegral_mul _ (measurable_pdf X ℙ μ) hf.neg.ennreal_ofReal,
-      lintegral_withDensity_eq_lintegral_mul _ (measurable_pdf X ℙ μ) hf.ennreal_ofReal]
-    · congr 2
-      · have : ∀ x, ENNReal.ofReal (f x * (pdf X ℙ μ x).toReal) =
-            ENNReal.ofReal (pdf X ℙ μ x).toReal * ENNReal.ofReal (f x) := fun x ↦ by
-          rw [mul_comm, ENNReal.ofReal_mul ENNReal.toReal_nonneg]
-        simp_rw [this]
-        exact lintegral_congr_ae (Filter.EventuallyEq.mul ofReal_toReal_ae_eq (ae_eq_refl _))
-      · have :
-          ∀ x,
-            ENNReal.ofReal (-(f x * (pdf X ℙ μ x).toReal)) =
-              ENNReal.ofReal (pdf X ℙ μ x).toReal * ENNReal.ofReal (-f x) := by
-          intro x
-          rw [neg_mul_eq_neg_mul, mul_comm, ENNReal.ofReal_mul ENNReal.toReal_nonneg]
-        simp_rw [this]
-        exact lintegral_congr_ae (Filter.EventuallyEq.mul ofReal_toReal_ae_eq (ae_eq_refl _))
-    · refine' ⟨hf.aestronglyMeasurable, _⟩
-      rw [HasFiniteIntegral,
-        lintegral_withDensity_eq_lintegral_mul _ (measurable_pdf _ _ _)
-          hf.nnnorm.coe_nnreal_ennreal]
-      have : (fun x => (pdf X ℙ μ * fun x => (‖f x‖₊ : ℝ≥0∞)) x) =ᵐ[μ]
-          fun x => ‖f x * (pdf X ℙ μ x).toReal‖₊ := by
-        simp_rw [← smul_eq_mul, nnnorm_smul, ENNReal.coe_mul]
-        rw [smul_eq_mul, mul_comm]
-        refine' Filter.EventuallyEq.mul (ae_eq_refl _) (ae_eq_trans ofReal_toReal_ae_eq.symm _)
-        simp only [Real.ennnorm_eq_ofReal ENNReal.toReal_nonneg, ae_eq_refl]
-      rw [lintegral_congr_ae this]
-      exact hpdf.2
-  · rw [integral_undef hpdf, integral_undef]
-    rwa [← integrable_iff_integrable_mul_pdf hf] at hpdf
-#align measure_theory.pdf.integral_fun_mul_eq_integral MeasureTheory.pdf.integral_fun_mul_eq_integral
-
-theorem map_absolutelyContinuous {X : Ω → E} [HasPDF X ℙ μ] : map X ℙ ≪ μ := by
-  rw [map_eq_withDensity_pdf X ℙ μ]; exact withDensity_absolutelyContinuous _ _
-#align measure_theory.pdf.map_absolutely_continuous MeasureTheory.pdf.map_absolutelyContinuous
+theorem integral_pdf_smul [IsFiniteMeasure ℙ] {X : Ω → E} [HasPDF X ℙ μ] {f : E → F}
+    (hf : AEStronglyMeasurable f μ) : ∫ x, (pdf X ℙ μ x).toReal • f x ∂μ = ∫ x, f (X x) ∂ℙ := by
+  rw [← integral_map (HasPDF.aemeasurable X ℙ μ) (hf.mono' HasPDF.absolutelyContinuous),
+    map_eq_withDensity_pdf X ℙ μ, pdf_def, integral_rnDeriv_smul HasPDF.absolutelyContinuous,
+    withDensity_rnDeriv_eq _ _ HasPDF.absolutelyContinuous]
+#align measure_theory.pdf.integral_fun_mul_eq_integral MeasureTheory.pdf.integral_pdf_smul
 
-/-- A random variable that `HasPDF` is quasi-measure preserving. -/
-theorem to_quasiMeasurePreserving {X : Ω → E} [HasPDF X ℙ μ] : QuasiMeasurePreserving X ℙ μ :=
-  { measurable := HasPDF.measurable X ℙ μ
-    absolutelyContinuous := map_absolutelyContinuous }
-#align measure_theory.pdf.to_quasi_measure_preserving MeasureTheory.pdf.to_quasiMeasurePreserving
-
-theorem haveLebesgueDecomposition_of_hasPDF {X : Ω → E} [hX' : HasPDF X ℙ μ] :
-    (map X ℙ).HaveLebesgueDecomposition μ :=
-  ⟨⟨⟨0, pdf X ℙ μ⟩, by
-      simp only [zero_add, measurable_pdf X ℙ μ, true_and_iff, MutuallySingular.zero_left,
-        map_eq_withDensity_pdf X ℙ μ]⟩⟩
-#align measure_theory.pdf.have_lebesgue_decomposition_of_has_pdf MeasureTheory.pdf.haveLebesgueDecomposition_of_hasPDF
-
-theorem hasPDF_iff {X : Ω → E} :
-    HasPDF X ℙ μ ↔ Measurable X ∧ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ := by
-  constructor
-  · intro hX'
-    exact ⟨hX'.pdf'.1, haveLebesgueDecomposition_of_hasPDF, map_absolutelyContinuous⟩
-  · rintro ⟨hX, h_decomp, h⟩
-    haveI := h_decomp
-    refine' ⟨⟨hX, (Measure.map X ℙ).rnDeriv μ, measurable_rnDeriv _ _, _⟩⟩
-    rwa [withDensity_rnDeriv_eq]
-#align measure_theory.pdf.has_pdf_iff MeasureTheory.pdf.hasPDF_iff
-
-theorem hasPDF_iff_of_measurable {X : Ω → E} (hX : Measurable X) :
-    HasPDF X ℙ μ ↔ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ := by
-  rw [hasPDF_iff]
-  simp only [hX, true_and]
-#align measure_theory.pdf.has_pdf_iff_of_measurable MeasureTheory.pdf.hasPDF_iff_of_measurable
+end IntegralPDFMul
 
 section
 
@@ -257,11 +270,16 @@ map also `HasPDF` if `(map g (map X ℙ)).HaveLebesgueDecomposition μ`.
 
 `quasiMeasurePreserving_hasPDF` is more useful in the case we are working with a
 probability measure and a real-valued random variable. -/
-theorem quasiMeasurePreserving_hasPDF {X : Ω → E} [HasPDF X ℙ μ] {g : E → F}
+theorem quasiMeasurePreserving_hasPDF {X : Ω → E} [HasPDF X ℙ μ] (hX : AEMeasurable X ℙ) {g : E → F}
     (hg : QuasiMeasurePreserving g μ ν) (hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν) :
     HasPDF (g ∘ X) ℙ ν := by
-  rw [hasPDF_iff, ← map_map hg.measurable (HasPDF.measurable X ℙ μ)]
-  refine' ⟨hg.measurable.comp (HasPDF.measurable X ℙ μ), hmap, _⟩
+  wlog hmX : Measurable X
+  · have hae : g ∘ X =ᵐ[ℙ] g ∘ hX.mk := hX.ae_eq_mk.mono fun x h ↦ by dsimp; rw [h]
+    have hXmk : HasPDF hX.mk ℙ μ := HasPDF.congr hX.ae_eq_mk
+    apply (HasPDF.congr' hae).mpr
+    exact this hX.measurable_mk.aemeasurable hg (map_congr hX.ae_eq_mk ▸ hmap) hX.measurable_mk
+  rw [hasPDF_iff, ← map_map hg.measurable hmX]
+  refine' ⟨(hg.measurable.comp hmX).aemeasurable, hmap, _⟩
   rw [map_eq_withDensity_pdf X ℙ μ]
   refine' AbsolutelyContinuous.mk fun s hsm hs => _
   rw [map_apply hg.measurable hsm, withDensity_apply _ (hg.measurable hsm)]
@@ -271,8 +289,9 @@ theorem quasiMeasurePreserving_hasPDF {X : Ω → E} [HasPDF X ℙ μ] {g : E 
 #align measure_theory.pdf.quasi_measure_preserving_has_pdf MeasureTheory.pdf.quasiMeasurePreserving_hasPDF
 
 theorem quasiMeasurePreserving_hasPDF' [IsFiniteMeasure ℙ] [SigmaFinite ν] {X : Ω → E}
-    [HasPDF X ℙ μ] {g : E → F} (hg : QuasiMeasurePreserving g μ ν) : HasPDF (g ∘ X) ℙ ν :=
-  quasiMeasurePreserving_hasPDF hg inferInstance
+    [HasPDF X ℙ μ] (hX : AEMeasurable X ℙ) {g : E → F} (hg : QuasiMeasurePreserving g μ ν) :
+    HasPDF (g ∘ X) ℙ ν :=
+  quasiMeasurePreserving_hasPDF hX hg inferInstance
 #align measure_theory.pdf.quasi_measure_preserving_has_pdf' MeasureTheory.pdf.quasiMeasurePreserving_hasPDF'
 
 end
@@ -283,15 +302,15 @@ variable [IsFiniteMeasure ℙ] {X : Ω → ℝ}
 
 /-- A real-valued random variable `X` `HasPDF X ℙ λ` (where `λ` is the Lebesgue measure) if and
 only if the push-forward measure of `ℙ` along `X` is absolutely continuous with respect to `λ`. -/
-nonrec theorem _root_.Real.hasPDF_iff_of_measurable (hX : Measurable X) :
+nonrec theorem _root_.Real.hasPDF_iff_of_aemeasurable (hX : AEMeasurable X ℙ) :
     HasPDF X ℙ ↔ map X ℙ ≪ volume := by
-  rw [hasPDF_iff_of_measurable hX]
+  rw [hasPDF_iff_of_aemeasurable hX]
   exact and_iff_right inferInstance
-#align measure_theory.pdf.real.has_pdf_iff_of_measurable Real.hasPDF_iff_of_measurable
+#align measure_theory.pdf.real.has_pdf_iff_of_measurable Real.hasPDF_iff_of_aemeasurable
 
-theorem _root_.Real.hasPDF_iff : HasPDF X ℙ ↔ Measurable X ∧ map X ℙ ≪ volume := by
-  by_cases hX : Measurable X
-  · rw [Real.hasPDF_iff_of_measurable hX, iff_and_self]
+theorem _root_.Real.hasPDF_iff : HasPDF X ℙ ↔ AEMeasurable X ℙ ∧ map X ℙ ≪ volume := by
+  by_cases hX : AEMeasurable X ℙ
+  · rw [Real.hasPDF_iff_of_aemeasurable hX, iff_and_self]
     exact fun _ => hX
   · exact ⟨fun h => False.elim (hX h.pdf'.1), fun h => False.elim (hX h.1)⟩
 #align measure_theory.pdf.real.has_pdf_iff Real.hasPDF_iff
@@ -299,7 +318,9 @@ theorem _root_.Real.hasPDF_iff : HasPDF X ℙ ↔ Measurable X ∧ map X ℙ ≪
 /-- If `X` is a real-valued random variable that has pdf `f`, then the expectation of `X` equals
 `∫ x, x * f x ∂λ` where `λ` is the Lebesgue measure. -/
 theorem integral_mul_eq_integral [HasPDF X ℙ] : ∫ x, x * (pdf X ℙ volume x).toReal = ∫ x, X x ∂ℙ :=
-  integral_fun_mul_eq_integral measurable_id
+  calc
+    _ = ∫ x, (pdf X ℙ volume x).toReal * x := by congr with x; exact mul_comm _ _
+    _ = _ := integral_pdf_smul measurable_id.aestronglyMeasurable
 #align measure_theory.pdf.integral_mul_eq_integral MeasureTheory.pdf.integral_mul_eq_integral
 
 theorem hasFiniteIntegral_mul {f : ℝ → ℝ} {g : ℝ → ℝ≥0∞} (hg : pdf X ℙ =ᵐ[volume] g)
@@ -321,92 +342,104 @@ section
 
 /-! **Uniform Distribution** -/
 
-/-- A random variable `X` has uniform distribution if it has a probability density function `f`
-with support `s` such that `f = (μ s)⁻¹ 1ₛ` a.e. where `1ₛ` is the indicator function for `s`. -/
-def IsUniform {_ : MeasurableSpace Ω} (X : Ω → E) (support : Set E) (ℙ : Measure Ω)
-    (μ : Measure E := by volume_tac) :=
-  pdf X ℙ μ =ᵐ[μ] support.indicator ((μ support)⁻¹ • (1 : E → ℝ≥0∞))
+/-- A random variable `X` has uniform distribution on `s` if its push-forward measure is
+`(μ s)⁻¹ • μ.restrict s`. -/
+def IsUniform (X : Ω → E) (support : Set E) (ℙ : Measure Ω) (μ : Measure E := by volume_tac) :=
+  map X ℙ = (μ support)⁻¹ • μ.restrict support
 #align measure_theory.pdf.is_uniform MeasureTheory.pdf.IsUniform
 
 namespace IsUniform
 
-theorem hasPDF {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E} {s : Set E}
-    (hns : μ s ≠ 0) (hnt : μ s ≠ ∞) (hu : IsUniform X s ℙ μ) : HasPDF X ℙ μ :=
-  hasPDF_of_pdf_ne_zero
-    (by
-      intro hpdf
-      simp only [IsUniform, hpdf] at hu
-      suffices μ (s ∩ Function.support ((μ s)⁻¹ • (1 : E → ℝ≥0∞))) = 0 by
-        have heq : Function.support ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) = Set.univ := by
-          ext x
-          rw [Function.mem_support]
-          simp [hnt]
-        rw [heq, Set.inter_univ] at this
-        exact hns this
-      exact Set.indicator_ae_eq_zero.1 hu.symm)
-#align measure_theory.pdf.is_uniform.has_pdf MeasureTheory.pdf.IsUniform.hasPDF
-
-theorem pdf_toReal_ae_eq {_ : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E}
-    {s : Set E} (hX : IsUniform X s ℙ μ) :
-    (fun x => (pdf X ℙ μ x).toReal) =ᵐ[μ] fun x =>
-      (s.indicator ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) x).toReal :=
-  Filter.EventuallyEq.fun_comp hX ENNReal.toReal
-#align measure_theory.pdf.is_uniform.pdf_to_real_ae_eq MeasureTheory.pdf.IsUniform.pdf_toReal_ae_eq
-
-theorem measure_preimage {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E}
-    {s : Set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞) (hms : MeasurableSet s) (hu : IsUniform X s ℙ μ)
-    {A : Set E} (hA : MeasurableSet A) : ℙ (X ⁻¹' A) = μ (s ∩ A) / μ s := by
-  haveI := hu.hasPDF hns hnt
-  rw [← Measure.map_apply (HasPDF.measurable X ℙ μ) hA, map_eq_set_lintegral_pdf X ℙ μ hA,
-    lintegral_congr_ae hu.restrict]
-  simp only [hms, hA, lintegral_indicator, Pi.smul_apply, Pi.one_apply, Algebra.id.smul_eq_mul,
-    mul_one, lintegral_const, restrict_apply', Set.univ_inter]
-  rw [ENNReal.div_eq_inv_mul]
+theorem aemeasurable {X : Ω → E} {s : Set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞)
+    (hu : IsUniform X s ℙ μ) : AEMeasurable X ℙ := by
+  dsimp [IsUniform] at hu
+  by_contra h
+  rw [map_of_not_aemeasurable h] at hu
+  apply zero_ne_one' ℝ≥0∞
+  calc
+    0 = (0 : Measure E) Set.univ := rfl
+    _ = _ := by rw [hu, smul_apply, restrict_apply MeasurableSet.univ,
+      Set.univ_inter, smul_eq_mul, ENNReal.inv_mul_cancel hns hnt]
+
+theorem absolutelyContinuous {X : Ω → E} {s : Set E} (hu : IsUniform X s ℙ μ) : map X ℙ ≪ μ := by
+  intro t ht
+  rw [hu, smul_apply, smul_eq_mul]
+  apply mul_eq_zero.mpr
+  refine Or.inr (le_antisymm ?_ (zero_le _))
+  exact ht ▸ restrict_apply_le s t
+
+theorem measure_preimage {X : Ω → E} {s : Set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞)
+    (hu : IsUniform X s ℙ μ) {A : Set E} (hA : MeasurableSet A) :
+    ℙ (X ⁻¹' A) = μ (s ∩ A) / μ s := by
+  rw [← map_apply_of_aemeasurable (hu.aemeasurable hns hnt) hA, hu, smul_apply, restrict_apply hA,
+    ENNReal.div_eq_inv_mul, smul_eq_mul, Set.inter_comm]
 #align measure_theory.pdf.is_uniform.measure_preimage MeasureTheory.pdf.IsUniform.measure_preimage
 
-theorem isProbabilityMeasure {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E}
-    {s : Set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞) (hms : MeasurableSet s) (hu : IsUniform X s ℙ μ) :
-    IsProbabilityMeasure ℙ :=
+theorem isProbabilityMeasure {X : Ω → E} {s : Set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞)
+    (hu : IsUniform X s ℙ μ) : IsProbabilityMeasure ℙ :=
   ⟨by
     have : X ⁻¹' Set.univ = Set.univ := by simp only [Set.preimage_univ]
-    rw [← this, hu.measure_preimage hns hnt hms MeasurableSet.univ, Set.inter_univ,
+    rw [← this, hu.measure_preimage hns hnt MeasurableSet.univ, Set.inter_univ,
       ENNReal.div_self hns hnt]⟩
 #align measure_theory.pdf.is_uniform.is_probability_measure MeasureTheory.pdf.IsUniform.isProbabilityMeasure
 
-variable {X : Ω → ℝ} {s : Set ℝ} (hms : MeasurableSet s) (hns : volume s ≠ 0)
+theorem hasPDF {X : Ω → E} {s : Set E} (hms : MeasurableSet s) (hns : μ s ≠ 0) (hnt : μ s ≠ ∞)
+    (hu : IsUniform X s ℙ μ) : HasPDF X ℙ μ := by
+  apply hasPDF_of_map_eq_withDensity (hu.aemeasurable hns hnt) (s.indicator ((μ s)⁻¹ • 1)) <|
+    (measurable_one.aemeasurable.const_smul (μ s)⁻¹).indicator hms
+  rw [hu, withDensity_indicator hms, withDensity_smul _ measurable_one, withDensity_one]
+#align measure_theory.pdf.is_uniform.has_pdf MeasureTheory.pdf.IsUniform.hasPDF
+
+theorem hasPDF₀ {X : Ω → E} {s : Set E} (hms : NullMeasurableSet s μ) (hns : μ s ≠ 0)
+    (hnt : μ s ≠ ∞) (hu : IsUniform X s ℙ μ) : HasPDF X ℙ μ := by
+  have hms' := measurableSet_toMeasurable μ s
+  apply hasPDF (m := m) (ℙ := ℙ) (μ := μ) hms'
+    (measure_toMeasurable s ▸ hns) (measure_toMeasurable s ▸ hnt) _
+  unfold IsUniform
+  rw [measure_toMeasurable, restrict_congr_set hms.toMeasurable_ae_eq, hu]
+
+theorem pdf_eq {X : Ω → E} {s : Set E} (hms : MeasurableSet s) (hns : μ s ≠ 0) (hnt : μ s ≠ ∞)
+    (hu : IsUniform X s ℙ μ) : pdf X ℙ μ =ᵐ[μ] s.indicator ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) := by
+  have : HasPDF X ℙ μ := hasPDF hms hns hnt hu
+  have : IsProbabilityMeasure ℙ := isProbabilityMeasure hns hnt hu
+  apply (eq_of_map_eq_withDensity _ _).mp
+  · rw [hu, withDensity_indicator hms, withDensity_smul _ measurable_one, withDensity_one]
+  · exact (measurable_one.aemeasurable.const_smul (μ s)⁻¹).indicator hms
+
+theorem pdf_toReal_ae_eq {X : Ω → E}
+    {s : Set E} (hms : MeasurableSet s) (hns : μ s ≠ 0) (hnt : μ s ≠ ∞)
+    (hX : IsUniform X s ℙ μ) :
+    (fun x => (pdf X ℙ μ x).toReal) =ᵐ[μ] fun x =>
+      (s.indicator ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) x).toReal :=
+  Filter.EventuallyEq.fun_comp (pdf_eq hms hns hnt hX) ENNReal.toReal
+#align measure_theory.pdf.is_uniform.pdf_to_real_ae_eq MeasureTheory.pdf.IsUniform.pdf_toReal_ae_eq
+
+variable {X : Ω → ℝ} {s : Set ℝ} (hms : MeasurableSet s) (hns : volume s ≠ 0) (hnt : volume s ≠ ∞)
 
 theorem mul_pdf_integrable [IsFiniteMeasure ℙ] (hcs : IsCompact s) (huX : IsUniform X s ℙ) :
     Integrable fun x : ℝ => x * (pdf X ℙ volume x).toReal := by
-  by_cases hsupp : volume s = ∞
-  · have : pdf X ℙ =ᵐ[volume] 0 := by
-      refine' ae_eq_trans huX _
-      simp [hsupp, ae_eq_refl]
-    refine' Integrable.congr (integrable_zero _ _ _) _
-    rw [(by simp : (fun x => 0 : ℝ → ℝ) = fun x => x * (0 : ℝ≥0∞).toReal)]
-    refine'
-      Filter.EventuallyEq.mul (ae_eq_refl _) (Filter.EventuallyEq.fun_comp this.symm ENNReal.toReal)
   constructor -- porting note: `refine` was failing, don't know why
   · exact aestronglyMeasurable_id.mul
       (measurable_pdf X ℙ).aemeasurable.ennreal_toReal.aestronglyMeasurable
-  refine' hasFiniteIntegral_mul huX _
+  refine' hasFiniteIntegral_mul (pdf_eq hms hns hnt huX) _
   set ind := (volume s)⁻¹ • (1 : ℝ → ℝ≥0∞)
   have : ∀ x, ↑‖x‖₊ * s.indicator ind x = s.indicator (fun x => ‖x‖₊ * ind x) x := fun x =>
     (s.indicator_mul_right (fun x => ↑‖x‖₊) ind).symm
   simp only [this, lintegral_indicator _ hms, mul_one, Algebra.id.smul_eq_mul, Pi.one_apply,
     Pi.smul_apply]
   rw [lintegral_mul_const _ measurable_nnnorm.coe_nnreal_ennreal]
-  refine' (ENNReal.mul_lt_top (set_lintegral_lt_top_of_isCompact hsupp hcs continuous_nnnorm).ne
+  refine' (ENNReal.mul_lt_top (set_lintegral_lt_top_of_isCompact hnt hcs continuous_nnnorm).ne
     (ENNReal.inv_lt_top.2 (pos_iff_ne_zero.mpr hns)).ne).ne
 #align measure_theory.pdf.is_uniform.mul_pdf_integrable MeasureTheory.pdf.IsUniform.mul_pdf_integrable
 
 /-- A real uniform random variable `X` with support `s` has expectation
 `(λ s)⁻¹ * ∫ x in s, x ∂λ` where `λ` is the Lebesgue measure. -/
-theorem integral_eq (hnt : volume s ≠ ∞) (huX : IsUniform X s ℙ) :
+theorem integral_eq (huX : IsUniform X s ℙ) :
     ∫ x, X x ∂ℙ = (volume s)⁻¹.toReal * ∫ x in s, x := by
-  haveI := hasPDF hns hnt huX
-  haveI := huX.isProbabilityMeasure hns hnt hms
+  haveI := hasPDF hms hns hnt huX
+  haveI := huX.isProbabilityMeasure hns hnt
   rw [← integral_mul_eq_integral]
-  rw [integral_congr_ae (Filter.EventuallyEq.mul (ae_eq_refl _) (pdf_toReal_ae_eq huX))]
+  rw [integral_congr_ae (Filter.EventuallyEq.mul (ae_eq_refl _) (pdf_toReal_ae_eq hms hns hnt huX))]
   have :
     ∀ x,
       x * (s.indicator ((volume s)⁻¹ • (1 : ℝ → ℝ≥0∞)) x).toReal =
@@ -435,17 +468,17 @@ theorem indepFun_iff_pdf_prod_eq_pdf_mul_pdf
     [IsFiniteMeasure ℙ] [SigmaFinite μ] [SigmaFinite ν] [HasPDF (fun ω ↦ (X ω, Y ω)) ℙ (μ.prod ν)] :
     IndepFun X Y ℙ ↔
       pdf (fun ω ↦ (X ω, Y ω)) ℙ (μ.prod ν) =ᵐ[μ.prod ν] fun z ↦ pdf X ℙ μ z.1 * pdf Y ℙ ν z.2 := by
-  have : HasPDF X ℙ μ := quasiMeasurePreserving_hasPDF' (μ := μ.prod ν) (X := fun ω ↦ (X ω, Y ω))
-    quasiMeasurePreserving_fst
-  have : HasPDF Y ℙ ν := quasiMeasurePreserving_hasPDF' (μ := μ.prod ν) (X := fun ω ↦ (X ω, Y ω))
-    quasiMeasurePreserving_snd
+  have : HasPDF X ℙ μ := quasiMeasurePreserving_hasPDF' (μ := μ.prod ν)
+    (HasPDF.aemeasurable (fun ω ↦ (X ω, Y ω)) ℙ (μ.prod ν)) quasiMeasurePreserving_fst
+  have : HasPDF Y ℙ ν := quasiMeasurePreserving_hasPDF' (μ := μ.prod ν)
+    (HasPDF.aemeasurable (fun ω ↦ (X ω, Y ω)) ℙ (μ.prod ν)) quasiMeasurePreserving_snd
   have h₀ : (ℙ.map X).prod (ℙ.map Y) =
       (μ.prod ν).withDensity fun z ↦ pdf X ℙ μ z.1 * pdf Y ℙ ν z.2 :=
     prod_eq fun s t hs ht ↦ by rw [withDensity_apply _ (hs.prod ht), ← prod_restrict,
       lintegral_prod_mul (measurable_pdf X ℙ μ).aemeasurable (measurable_pdf Y ℙ ν).aemeasurable,
       map_eq_set_lintegral_pdf X ℙ μ hs, map_eq_set_lintegral_pdf Y ℙ ν ht]
-  rw [indepFun_iff_map_prod_eq_prod_map_map (HasPDF.measurable X ℙ μ) (HasPDF.measurable Y ℙ ν),
-    ← unique, h₀]
+  rw [indepFun_iff_map_prod_eq_prod_map_map (HasPDF.aemeasurable X ℙ μ) (HasPDF.aemeasurable Y ℙ ν),
+    ← eq_of_map_eq_withDensity, h₀]
   exact (((measurable_pdf X ℙ μ).comp measurable_fst).mul
     ((measurable_pdf Y ℙ ν).comp measurable_snd)).aemeasurable
 

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

@@ -5,6 +5,7 @@ Authors: Kexing Ying
 -/
 import Mathlib.MeasureTheory.Decomposition.RadonNikodym
 import Mathlib.MeasureTheory.Measure.Haar.OfBasis
+import Mathlib.Probability.Independence.Basic
 
 #align_import probability.density from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
 
@@ -136,13 +137,25 @@ theorem lintegral_eq_measure_univ {X : Ω → E} [HasPDF X ℙ μ] :
     Measure.map_apply (HasPDF.measurable X ℙ μ) MeasurableSet.univ, Set.preimage_univ]
 #align measure_theory.pdf.lintegral_eq_measure_univ MeasureTheory.pdf.lintegral_eq_measure_univ
 
+theorem unique [IsFiniteMeasure ℙ] {X : Ω → E} [HasPDF X ℙ μ] (f : E → ℝ≥0∞)
+    (hmf : AEMeasurable f μ) : ℙ.map X = μ.withDensity f ↔ pdf X ℙ μ =ᵐ[μ] f := by
+  rw [map_eq_withDensity_pdf X ℙ μ]
+  apply withDensity_eq_iff (measurable_pdf X ℙ μ).aemeasurable hmf
+  rw [lintegral_eq_measure_univ]
+  exact measure_ne_top _ _
+
+theorem unique' [SigmaFinite μ] {X : Ω → E} [HasPDF X ℙ μ] (f : E → ℝ≥0∞) (hmf : AEMeasurable f μ) :
+    ℙ.map X = μ.withDensity f ↔ pdf X ℙ μ =ᵐ[μ] f :=
+  map_eq_withDensity_pdf X ℙ μ ▸
+    withDensity_eq_iff_of_sigmaFinite (measurable_pdf X ℙ μ).aemeasurable hmf
+
 nonrec theorem ae_lt_top [IsFiniteMeasure ℙ] {μ : Measure E} {X : Ω → E} :
     ∀ᵐ x ∂μ, pdf X ℙ μ x < ∞ := by
   by_cases hpdf : HasPDF X ℙ μ
   · haveI := hpdf
     refine' ae_lt_top (measurable_pdf X ℙ μ) _
     rw [lintegral_eq_measure_univ]
-    exact (measure_lt_top _ _).ne
+    exact measure_ne_top _ _
   · simp [pdf, hpdf]
 #align measure_theory.pdf.ae_lt_top MeasureTheory.pdf.ae_lt_top
 
@@ -411,6 +424,33 @@ end IsUniform
 
 end
 
+section TwoVariables
+
+open ProbabilityTheory
+
+variable {F : Type*} [MeasurableSpace F] {ν : Measure F} {X : Ω → E} {Y : Ω → F}
+
+/-- Random variables are independent iff their joint density is a product of marginal densities. -/
+theorem indepFun_iff_pdf_prod_eq_pdf_mul_pdf
+    [IsFiniteMeasure ℙ] [SigmaFinite μ] [SigmaFinite ν] [HasPDF (fun ω ↦ (X ω, Y ω)) ℙ (μ.prod ν)] :
+    IndepFun X Y ℙ ↔
+      pdf (fun ω ↦ (X ω, Y ω)) ℙ (μ.prod ν) =ᵐ[μ.prod ν] fun z ↦ pdf X ℙ μ z.1 * pdf Y ℙ ν z.2 := by
+  have : HasPDF X ℙ μ := quasiMeasurePreserving_hasPDF' (μ := μ.prod ν) (X := fun ω ↦ (X ω, Y ω))
+    quasiMeasurePreserving_fst
+  have : HasPDF Y ℙ ν := quasiMeasurePreserving_hasPDF' (μ := μ.prod ν) (X := fun ω ↦ (X ω, Y ω))
+    quasiMeasurePreserving_snd
+  have h₀ : (ℙ.map X).prod (ℙ.map Y) =
+      (μ.prod ν).withDensity fun z ↦ pdf X ℙ μ z.1 * pdf Y ℙ ν z.2 :=
+    prod_eq fun s t hs ht ↦ by rw [withDensity_apply _ (hs.prod ht), ← prod_restrict,
+      lintegral_prod_mul (measurable_pdf X ℙ μ).aemeasurable (measurable_pdf Y ℙ ν).aemeasurable,
+      map_eq_set_lintegral_pdf X ℙ μ hs, map_eq_set_lintegral_pdf Y ℙ ν ht]
+  rw [indepFun_iff_map_prod_eq_prod_map_map (HasPDF.measurable X ℙ μ) (HasPDF.measurable Y ℙ ν),
+    ← unique, h₀]
+  exact (((measurable_pdf X ℙ μ).comp measurable_fst).mul
+    ((measurable_pdf Y ℙ ν).comp measurable_snd)).aemeasurable
+
+end TwoVariables
+
 end pdf
 
 end MeasureTheory

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

This has nice performance benefits.

@@ -59,7 +59,7 @@ noncomputable section
 
 namespace MeasureTheory
 
-variable {Ω E : Type _} [MeasurableSpace E]
+variable {Ω E : Type*} [MeasurableSpace E]
 
 /-- A random variable `X : Ω → E` is said to `HasPDF` with respect to the measure `ℙ` on `Ω` and
 `μ` on `E` if there exists a measurable function `f` such that the push-forward measure of `ℙ`
@@ -237,7 +237,7 @@ theorem hasPDF_iff_of_measurable {X : Ω → E} (hX : Measurable X) :
 
 section
 
-variable {F : Type _} [MeasurableSpace F] {ν : Measure F}
+variable {F : Type*} [MeasurableSpace F] {ν : Measure F}
 
 /-- A random variable that `HasPDF` transformed under a `QuasiMeasurePreserving`
 map also `HasPDF` if `(map g (map X ℙ)).HaveLebesgueDecomposition μ`.

• depends on: #5966

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

@@ -2,15 +2,12 @@
 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 probability.density
-! leanprover-community/mathlib commit c14c8fcde993801fca8946b0d80131a1a81d1520
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.MeasureTheory.Decomposition.RadonNikodym
 import Mathlib.MeasureTheory.Measure.Haar.OfBasis
 
+#align_import probability.density from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
+
 /-!
 # Probability density function
 

Match https://github.com/leanprover-community/mathlib/pull/19199

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

@@ -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 probability.density
-! leanprover-community/mathlib commit fd5edc43dc4f10b85abfe544b88f82cf13c5f844
+! leanprover-community/mathlib commit c14c8fcde993801fca8946b0d80131a1a81d1520
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -333,7 +333,7 @@ theorem hasPDF {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ :
           simp [hnt]
         rw [heq, Set.inter_univ] at this
         exact hns this
-      exact MeasureTheory.Set.indicator_ae_eq_zero hu.symm)
+      exact Set.indicator_ae_eq_zero.1 hu.symm)
 #align measure_theory.pdf.is_uniform.has_pdf MeasureTheory.pdf.IsUniform.hasPDF
 
 theorem pdf_toReal_ae_eq {_ : MeasurableSpace Ω} {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E}

@@ -134,7 +134,7 @@ namespace pdf
 variable {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
 
 theorem lintegral_eq_measure_univ {X : Ω → E} [HasPDF X ℙ μ] :
-    (∫⁻ x, pdf X ℙ μ x ∂μ) = ℙ Set.univ := by
+    ∫⁻ x, pdf X ℙ μ x ∂μ = ℙ Set.univ := by
   rw [← set_lintegral_univ, ← map_eq_set_lintegral_pdf X ℙ μ MeasurableSet.univ,
     Measure.map_apply (HasPDF.measurable X ℙ μ) MeasurableSet.univ, Set.preimage_univ]
 #align measure_theory.pdf.lintegral_eq_measure_univ MeasureTheory.pdf.lintegral_eq_measure_univ

@@ -32,7 +32,7 @@ random variables with this distribution.
 * `MeasureTheory.pdf` : If `X` is a random variable that `HasPDF X ℙ μ`, then `pdf X`
   is the measurable function `f` such that the push-forward measure of `ℙ` along `X` equals
   `μ.withDensity f`.
-* `measure_theory.pdf.uniform` : A random variable `X` is said to follow the uniform
+* `MeasureTheory.pdf.IsUniform` : A random variable `X` is said to follow the uniform
   distribution if it has a constant probability density function with a compact, non-null support.
 
 ## Main results
@@ -42,7 +42,7 @@ random variables with this distribution.
   all measurable `g : E → ℝ`.
 * `MeasureTheory.pdf.integral_mul_eq_integral` : A real-valued random variable `X` with
   pdf `f` has expectation `∫ x, x * f x dx`.
-* `measure_theory.pdf.uniform.integral_eq` : If `X` follows the uniform distribution with
+* `MeasureTheory.pdf.IsUniform.integral_eq` : If `X` follows the uniform distribution with
   its pdf having support `s`, then `X` has expectation `(λ s)⁻¹ * ∫ x in s, x dx` where `λ`
   is the Lebesgue measure.
 
@@ -242,10 +242,10 @@ section
 
 variable {F : Type _} [MeasurableSpace F] {ν : Measure F}
 
-/-- A random variable that `HasPDF` transformed under a `quasi_measure_preserving`
-map also `HasPDF` if `(map g (map X ℙ)).have_lebesgue_decomposition μ`.
+/-- A random variable that `HasPDF` transformed under a `QuasiMeasurePreserving`
+map also `HasPDF` if `(map g (map X ℙ)).HaveLebesgueDecomposition μ`.
 
-`quasi_measure_preserving_hasPDF'` is more useful in the case we are working with a
+`quasiMeasurePreserving_hasPDF` is more useful in the case we are working with a
 probability measure and a real-valued random variable. -/
 theorem quasiMeasurePreserving_hasPDF {X : Ω → E} [HasPDF X ℙ μ] {g : E → F}
     (hg : QuasiMeasurePreserving g μ ν) (hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν) :

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au>

@@ -158,6 +158,7 @@ theorem integrable_iff_integrable_mul_pdf [IsFiniteMeasure ℙ] {X : Ω → E} [
     (hf : Measurable f) :
     Integrable (fun x => f (X x)) ℙ ↔ Integrable (fun x => f x * (pdf X ℙ μ x).toReal) μ := by
   -- porting note: using `erw` because `rw` doesn't recognize `(f <| X ·)` as `f ∘ X`
+  -- https://github.com/leanprover-community/mathlib4/issues/5164
   erw [← integrable_map_measure hf.aestronglyMeasurable (HasPDF.measurable X ℙ μ).aemeasurable,
     map_eq_withDensity_pdf X ℙ μ, integrable_withDensity_iff (measurable_pdf _ _ _) ae_lt_top]
 #align measure_theory.pdf.integrable_iff_integrable_mul_pdf MeasureTheory.pdf.integrable_iff_integrable_mul_pdf

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au>