File: Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean

full_name: MeasureTheory.Measure.withDensity_rnDeriv_eq
code: theorem withDensity_rnDeriv_eq (μ ν : Measure α) [HaveLebesgueDecomposition μ ν] (h : μ ≪ ν) :
    ν.withDensity (rnDeriv μ ν) = μ 
goal: ν.withDensity (μ.rnDeriv ν) = μ

full_name: MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq
code: /-- **The Radon-Nikodym theorem**: Given two measures `μ` and `ν`, if
`HaveLebesgueDecomposition μ ν`, then `μ` is absolutely continuous to `ν` if and only if
`ν.withDensity (rnDeriv μ ν) = μ`. -/
theorem absolutelyContinuous_iff_withDensity_rnDeriv_eq
    [HaveLebesgueDecomposition μ ν] : μ ≪ ν ↔ ν.withDensity (rnDeriv μ ν) = μ 
goal: μ ≪ ν ↔ ν.withDensity (μ.rnDeriv ν) = μ

full_name: MeasureTheory.Measure.rnDeriv_pos
code: lemma rnDeriv_pos [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) :
    ∀ᵐ x ∂μ, 0 < μ.rnDeriv ν x 
goal: ∀ᵐ (x : α) ∂μ, 0 < μ.rnDeriv ν x

full_name: MeasureTheory.Measure.rnDeriv_pos'
code: lemma rnDeriv_pos' [SigmaFinite μ] [SFinite ν] (hμν : μ ≪ ν) :
    ∀ᵐ x ∂μ, 0 < ν.rnDeriv μ x 
goal: ∀ᵐ (x : α) ∂μ, 0 < ν.rnDeriv μ x

full_name: MeasureTheory.Measure.rnDeriv_withDensity_withDensity_rnDeriv_left
code: /-- Auxiliary lemma for `rnDeriv_withDensity_left`. -/
lemma rnDeriv_withDensity_withDensity_rnDeriv_left (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν]
    (hf_ne_top : ∀ᵐ x ∂μ, f x ≠ ∞) :
    ((ν.withDensity (μ.rnDeriv ν)).withDensity f).rnDeriv ν =ᵐ[ν] (μ.withDensity f).rnDeriv ν 
goal: ((ν.withDensity (μ.rnDeriv ν)).withDensity f).rnDeriv ν =ᵐ[ν] (μ.withDensity f).rnDeriv ν

full_name: MeasureTheory.Measure.rnDeriv_withDensity_withDensity_rnDeriv_right
code: /-- Auxiliary lemma for `rnDeriv_withDensity_right`. -/
lemma rnDeriv_withDensity_withDensity_rnDeriv_right (μ ν : Measure α) [SigmaFinite μ]
    [SigmaFinite ν] (hf : AEMeasurable f ν) (hf_ne_zero : ∀ᵐ x ∂ν, f x ≠ 0)
    (hf_ne_top : ∀ᵐ x ∂ν, f x ≠ ∞) :
    (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν.withDensity f) =ᵐ[ν] μ.rnDeriv (ν.withDensity f) 
goal: (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν.withDensity f) =ᵐ[ν] μ.rnDeriv (ν.withDensity f)

full_name: MeasureTheory.Measure.rnDeriv_withDensity_left_of_absolutelyContinuous
code: lemma rnDeriv_withDensity_left_of_absolutelyContinuous {ν : Measure α} [SigmaFinite μ]
    [SigmaFinite ν] (hμν : μ ≪ ν) (hf : AEMeasurable f ν) :
    (μ.withDensity f).rnDeriv ν =ᵐ[ν] fun x ↦ f x * μ.rnDeriv ν x 
goal: (μ.withDensity f).rnDeriv ν =ᵐ[ν] fun x => f x * μ.rnDeriv ν x

full_name: MeasureTheory.Measure.rnDeriv_withDensity_left
code: lemma rnDeriv_withDensity_left {μ ν : Measure α} [SigmaFinite μ] [SigmaFinite ν]
    (hfν : AEMeasurable f ν) (hf_ne_top : ∀ᵐ x ∂μ, f x ≠ ∞) :
    (μ.withDensity f).rnDeriv ν =ᵐ[ν] fun x ↦ f x * μ.rnDeriv ν x 
goal: (μ.withDensity f).rnDeriv ν =ᵐ[ν] fun x => f x * μ.rnDeriv ν x

full_name: MeasureTheory.Measure.rnDeriv_withDensity_right_of_absolutelyContinuous
code: /-- Auxiliary lemma for `rnDeriv_withDensity_right`. -/
lemma rnDeriv_withDensity_right_of_absolutelyContinuous {ν : Measure α} [SigmaFinite μ]
    [SigmaFinite ν] (hμν : μ ≪ ν) (hf : AEMeasurable f ν)
    (hf_ne_zero : ∀ᵐ x ∂ν, f x ≠ 0) (hf_ne_top : ∀ᵐ x ∂ν, f x ≠ ∞) :
    μ.rnDeriv (ν.withDensity f) =ᵐ[ν] fun x ↦ (f x)⁻¹ * μ.rnDeriv ν x 
goal: μ.rnDeriv (ν.withDensity f) =ᵐ[ν] fun x => (f x)⁻¹ * μ.rnDeriv ν x

full_name: MeasureTheory.Measure.rnDeriv_withDensity_right
code: lemma rnDeriv_withDensity_right (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν]
    (hf : AEMeasurable f ν) (hf_ne_zero : ∀ᵐ x ∂ν, f x ≠ 0) (hf_ne_top : ∀ᵐ x ∂ν, f x ≠ ∞) :
    μ.rnDeriv (ν.withDensity f) =ᵐ[ν] fun x ↦ (f x)⁻¹ * μ.rnDeriv ν x 
goal: μ.rnDeriv (ν.withDensity f) =ᵐ[ν] fun x => (f x)⁻¹ * μ.rnDeriv ν x

full_name: MeasureTheory.Measure.rnDeriv_eq_zero_of_mutuallySingular
code: lemma rnDeriv_eq_zero_of_mutuallySingular [SigmaFinite μ] {ν' : Measure α}
    [SigmaFinite ν'] (h : μ ⟂ₘ ν) (hνν' : ν ≪ ν') :
    μ.rnDeriv ν' =ᵐ[ν] 0 
goal: μ.rnDeriv ν' =ᵐ[ν] 0

full_name: MeasureTheory.Measure.rnDeriv_add_right_of_absolutelyContinuous_of_mutuallySingular
code: /-- Auxiliary lemma for `rnDeriv_add_right_of_mutuallySingular`. -/
lemma rnDeriv_add_right_of_absolutelyContinuous_of_mutuallySingular {ν' : Measure α}
    [SigmaFinite μ] [SigmaFinite ν] [SigmaFinite ν']
    (hμν : μ ≪ ν) (hνν' : ν ⟂ₘ ν') :
    μ.rnDeriv (ν + ν') =ᵐ[ν] μ.rnDeriv ν 
goal: μ.rnDeriv (ν + ν') =ᵐ[ν] μ.rnDeriv ν

full_name: MeasureTheory.Measure.rnDeriv_add_right_of_mutuallySingular'
code: /-- Auxiliary lemma for `rnDeriv_add_right_of_mutuallySingular`. -/
lemma rnDeriv_add_right_of_mutuallySingular' {ν' : Measure α}
    [SigmaFinite μ] [SigmaFinite ν] [SigmaFinite ν']
    (hμν' : μ ⟂ₘ ν') (hνν' : ν ⟂ₘ ν') :
    μ.rnDeriv (ν + ν') =ᵐ[ν] μ.rnDeriv ν 
goal: μ.rnDeriv (ν + ν') =ᵐ[ν] μ.rnDeriv ν

full_name: MeasureTheory.Measure.rnDeriv_add_right_of_mutuallySingular
code: lemma rnDeriv_add_right_of_mutuallySingular {ν' : Measure α}
    [SigmaFinite μ] [SigmaFinite ν] [SigmaFinite ν'] (hνν' : ν ⟂ₘ ν') :
    μ.rnDeriv (ν + ν') =ᵐ[ν] μ.rnDeriv ν 
goal: μ.rnDeriv (ν + ν') =ᵐ[ν] μ.rnDeriv ν

full_name: MeasureTheory.Measure.rnDeriv_withDensity_rnDeriv
code: lemma rnDeriv_withDensity_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
    μ.rnDeriv (μ.withDensity (ν.rnDeriv μ)) =ᵐ[μ] μ.rnDeriv ν 
goal: μ.rnDeriv (μ.withDensity (ν.rnDeriv μ)) =ᵐ[μ] μ.rnDeriv ν

full_name: MeasureTheory.Measure.inv_rnDeriv_aux
code: /-- Auxiliary lemma for `inv_rnDeriv`. -/
lemma inv_rnDeriv_aux [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) (hνμ : ν ≪ μ) :
    (μ.rnDeriv ν)⁻¹ =ᵐ[μ] ν.rnDeriv μ 
goal: (μ.rnDeriv ν)⁻¹ =ᵐ[μ] ν.rnDeriv μ

full_name: MeasureTheory.Measure.inv_rnDeriv
code: lemma inv_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
    (μ.rnDeriv ν)⁻¹ =ᵐ[μ] ν.rnDeriv μ 
goal: (μ.rnDeriv ν)⁻¹ =ᵐ[μ] ν.rnDeriv μ

full_name: MeasureTheory.Measure.inv_rnDeriv'
code: lemma inv_rnDeriv' [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
    (ν.rnDeriv μ)⁻¹ =ᵐ[μ] μ.rnDeriv ν 
goal: (ν.rnDeriv μ)⁻¹ =ᵐ[μ] μ.rnDeriv ν

full_name: MeasureTheory.Measure.setLIntegral_rnDeriv_le
code: lemma setLIntegral_rnDeriv_le (s : Set α) :
    ∫⁻ x in s, μ.rnDeriv ν x ∂ν ≤ μ s 
goal: ∫⁻ (x : α) in s, μ.rnDeriv ν x ∂ν ≤ μ s

full_name: MeasureTheory.Measure.setLIntegral_rnDeriv'
code: lemma setLIntegral_rnDeriv' [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) {s : Set α}
    (hs : MeasurableSet s) :
    ∫⁻ x in s, μ.rnDeriv ν x ∂ν = μ s 
goal: ∫⁻ (x : α) in s, μ.rnDeriv ν x ∂ν = μ s

full_name: MeasureTheory.Measure.setLIntegral_rnDeriv
code: lemma setLIntegral_rnDeriv [HaveLebesgueDecomposition μ ν] [SFinite ν]
    (hμν : μ ≪ ν) (s : Set α) :
    ∫⁻ x in s, μ.rnDeriv ν x ∂ν = μ s 
goal: ∫⁻ (x : α) in s, μ.rnDeriv ν x ∂ν = μ s

full_name: MeasureTheory.Measure.lintegral_rnDeriv
code: lemma lintegral_rnDeriv [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) :
    ∫⁻ x, μ.rnDeriv ν x ∂ν = μ Set.univ 
goal: ∫⁻ (x : α), μ.rnDeriv ν x ∂ν = μ Set.univ

full_name: MeasureTheory.Measure.integrableOn_toReal_rnDeriv
code: lemma integrableOn_toReal_rnDeriv {s : Set α} (hμs : μ s ≠ ∞) :
    IntegrableOn (fun x ↦ (μ.rnDeriv ν x).toReal) s ν 
goal: MeasureTheory.IntegrableOn (fun x => (μ.rnDeriv ν x).toReal) s ν

full_name: MeasureTheory.Measure.setIntegral_toReal_rnDeriv_eq_withDensity'
code: lemma setIntegral_toReal_rnDeriv_eq_withDensity' [SigmaFinite μ]
    {s : Set α} (hs : MeasurableSet s) :
    ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (ν.withDensity (μ.rnDeriv ν) s).toReal 
goal: ∫ (x : α) in s, (μ.rnDeriv ν x).toReal ∂ν = ((ν.withDensity (μ.rnDeriv ν)) s).toReal

full_name: MeasureTheory.Measure.setIntegral_toReal_rnDeriv_eq_withDensity
code: lemma setIntegral_toReal_rnDeriv_eq_withDensity [SigmaFinite μ] [SFinite ν] (s : Set α) :
    ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (ν.withDensity (μ.rnDeriv ν) s).toReal 
goal: ∫ (x : α) in s, (μ.rnDeriv ν x).toReal ∂ν = ((ν.withDensity (μ.rnDeriv ν)) s).toReal

full_name: MeasureTheory.Measure.setIntegral_toReal_rnDeriv_le
code: lemma setIntegral_toReal_rnDeriv_le [SigmaFinite μ] {s : Set α} (hμs : μ s ≠ ∞) :
    ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν ≤ (μ s).toReal 
goal: ∫ (x : α) in s, (μ.rnDeriv ν x).toReal ∂ν ≤ (μ s).toReal

full_name: MeasureTheory.Measure.setIntegral_toReal_rnDeriv'
code: lemma setIntegral_toReal_rnDeriv' [SigmaFinite μ] [HaveLebesgueDecomposition μ ν]
    (hμν : μ ≪ ν) {s : Set α} (hs : MeasurableSet s) :
    ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (μ s).toReal 
goal: ∫ (x : α) in s, (μ.rnDeriv ν x).toReal ∂ν = (μ s).toReal

full_name: MeasureTheory.Measure.setIntegral_toReal_rnDeriv
code: lemma setIntegral_toReal_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) (s : Set α) :
    ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (μ s).toReal 
goal: ∫ (x : α) in s, (μ.rnDeriv ν x).toReal ∂ν = (μ s).toReal

full_name: MeasureTheory.Measure.integral_toReal_rnDeriv
code: lemma integral_toReal_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
    ∫ x, (μ.rnDeriv ν x).toReal ∂ν = (μ Set.univ).toReal 
goal: ∫ (x : α), (μ.rnDeriv ν x).toReal ∂ν = (μ Set.univ).toReal

full_name: MeasureTheory.Measure.rnDeriv_mul_rnDeriv
code: lemma rnDeriv_mul_rnDeriv {κ : Measure α} [SigmaFinite μ] [SigmaFinite ν] [SigmaFinite κ]
    (hμν : μ ≪ ν) :
    μ.rnDeriv ν * ν.rnDeriv κ =ᵐ[κ] μ.rnDeriv κ 
goal: μ.rnDeriv ν * ν.rnDeriv κ =ᵐ[κ] μ.rnDeriv κ

full_name: MeasureTheory.Measure.rnDeriv_le_one_of_le
code: lemma rnDeriv_le_one_of_le (hμν : μ ≤ ν) [SigmaFinite ν] : μ.rnDeriv ν ≤ᵐ[ν] 1 
goal: μ.rnDeriv ν ≤ᵐ[ν] 1

full_name: MeasurableEmbedding.rnDeriv_map_aux
code: lemma _root_.MeasurableEmbedding.rnDeriv_map_aux (hf : MeasurableEmbedding f)
    (hμν : μ ≪ ν) [SigmaFinite μ] [SigmaFinite ν] :
    (fun x ↦ (μ.map f).rnDeriv (ν.map f) (f x)) =ᵐ[ν] μ.rnDeriv ν 
goal: (fun x => (MeasureTheory.Measure.map f μ).rnDeriv (MeasureTheory.Measure.map f ν) (f x)) =ᵐ[ν] μ.rnDeriv ν

full_name: MeasurableEmbedding.rnDeriv_map
code: lemma _root_.MeasurableEmbedding.rnDeriv_map (hf : MeasurableEmbedding f)
    (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν] :
    (fun x ↦ (μ.map f).rnDeriv (ν.map f) (f x)) =ᵐ[ν] μ.rnDeriv ν 
goal: (fun x => (MeasureTheory.Measure.map f μ).rnDeriv (MeasureTheory.Measure.map f ν) (f x)) =ᵐ[ν] μ.rnDeriv ν

full_name: MeasurableEmbedding.map_withDensity_rnDeriv
code: lemma _root_.MeasurableEmbedding.map_withDensity_rnDeriv (hf : MeasurableEmbedding f)
    (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν] :
    (ν.withDensity (μ.rnDeriv ν)).map f = (ν.map f).withDensity ((μ.map f).rnDeriv (ν.map f)) 
goal: MeasureTheory.Measure.map f (ν.withDensity (μ.rnDeriv ν)) =
  (MeasureTheory.Measure.map f ν).withDensity ((MeasureTheory.Measure.map f μ).rnDeriv (MeasureTheory.Measure.map f ν))

full_name: MeasurableEmbedding.singularPart_map
code: lemma _root_.MeasurableEmbedding.singularPart_map (hf : MeasurableEmbedding f)
    (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν] :
    (μ.map f).singularPart (ν.map f) = (μ.singularPart ν).map f 
goal: (MeasureTheory.Measure.map f μ).singularPart (MeasureTheory.Measure.map f ν) =
  MeasureTheory.Measure.map f (μ.singularPart ν)

full_name: MeasureTheory.SignedMeasure.withDensityᵥ_rnDeriv_eq
code: theorem withDensityᵥ_rnDeriv_eq (s : SignedMeasure α) (μ : Measure α) [SigmaFinite μ]
    (h : s ≪ᵥ μ.toENNRealVectorMeasure) : μ.withDensityᵥ (s.rnDeriv μ) = s 
goal: μ.withDensityᵥ (s.rnDeriv μ) = s

full_name: MeasureTheory.SignedMeasure.absolutelyContinuous_iff_withDensityᵥ_rnDeriv_eq
code: /-- The Radon-Nikodym theorem for signed measures. -/
theorem absolutelyContinuous_iff_withDensityᵥ_rnDeriv_eq (s : SignedMeasure α) (μ : Measure α)
    [SigmaFinite μ] : s ≪ᵥ μ.toENNRealVectorMeasure ↔ μ.withDensityᵥ (s.rnDeriv μ) = s 
goal: s ≪ᵥ μ.toENNRealVectorMeasure ↔ μ.withDensityᵥ (s.rnDeriv μ) = s

full_name: MeasureTheory.lintegral_rnDeriv_mul
code: theorem lintegral_rnDeriv_mul [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) {f : α → ℝ≥0∞}
    (hf : AEMeasurable f ν) : ∫⁻ x, μ.rnDeriv ν x * f x ∂ν = ∫⁻ x, f x ∂μ 
goal: ∫⁻ (x : α), μ.rnDeriv ν x * f x ∂ν = ∫⁻ (x : α), f x ∂μ

full_name: MeasureTheory.setLIntegral_rnDeriv_mul
code: lemma setLIntegral_rnDeriv_mul [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) {f : α → ℝ≥0∞}
    (hf : AEMeasurable f ν) {s : Set α} (hs : MeasurableSet s) :
    ∫⁻ x in s, μ.rnDeriv ν x * f x ∂ν = ∫⁻ x in s, f x ∂μ 
goal: ∫⁻ (x : α) in s, μ.rnDeriv ν x * f x ∂ν = ∫⁻ (x : α) in s, f x ∂μ

full_name: MeasureTheory.integrable_rnDeriv_smul_iff
code: theorem integrable_rnDeriv_smul_iff [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν)
    [SigmaFinite μ] {f : α → E} :
    Integrable (fun x ↦ (μ.rnDeriv ν x).toReal • f x) ν ↔ Integrable f μ 
goal: MeasureTheory.Integrable (fun x => (μ.rnDeriv ν x).toReal • f x) ν ↔ MeasureTheory.Integrable f μ

full_name: MeasureTheory.withDensityᵥ_rnDeriv_smul
code: theorem withDensityᵥ_rnDeriv_smul [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν)
    [SigmaFinite μ] {f : α → E} (hf : Integrable f μ) :
    ν.withDensityᵥ (fun x ↦ (rnDeriv μ ν x).toReal • f x) = μ.withDensityᵥ f 
goal: (ν.withDensityᵥ fun x => (μ.rnDeriv ν x).toReal • f x) = μ.withDensityᵥ f

full_name: MeasureTheory.integral_rnDeriv_smul
code: theorem integral_rnDeriv_smul [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν)
    [SigmaFinite μ] {f : α → E} :
    ∫ x, (μ.rnDeriv ν x).toReal • f x ∂ν = ∫ x, f x ∂μ 
goal: ∫ (x : α), (μ.rnDeriv ν x).toReal • f x ∂ν = ∫ (x : α), f x ∂μ