Lebesgue decomposition

This file proves the Lebesgue decomposition theorem. The Lebesgue decomposition theorem states that, given two σ-finite measures μ and ν, there exists a σ-finite measure ξ and a measurable function f such that μ = ξ + fν and ξ is mutually singular with respect to ν.

The Lebesgue decomposition provides the Radon-Nikodym theorem readily.

Main definitions

• MeasureTheory.Measure.HaveLebesgueDecomposition : A pair of measures μ and ν is said to HaveLebesgueDecomposition if there exist a measure ξ and a measurable function f, such that ξ is mutually singular with respect to ν and μ = ξ + ν.withDensity f
• MeasureTheory.Measure.singularPart : If a pair of measures HaveLebesgueDecomposition, then singularPart chooses the measure from HaveLebesgueDecomposition, otherwise it returns the zero measure.
• MeasureTheory.Measure.rnDeriv: If a pair of measures HaveLebesgueDecomposition, then rnDeriv chooses the measurable function from HaveLebesgueDecomposition, otherwise it returns the zero function.

Main results

• MeasureTheory.Measure.haveLebesgueDecomposition_of_sigmaFinite : the Lebesgue decomposition theorem.
• MeasureTheory.Measure.eq_singularPart : Given measures μ and ν, if s is a measure mutually singular to ν and f is a measurable function such that μ = s + fν, then s = μ.singularPart ν.
• MeasureTheory.Measure.eq_rnDeriv : Given measures μ and ν, if s is a measure mutually singular to ν and f is a measurable function such that μ = s + fν, then f = μ.rnDeriv ν.

Tags

Lebesgue decomposition theorem

class MeasureTheory.Measure.HaveLebesgueDecomposition {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) : Prop

A pair of measures μ and ν is said to HaveLebesgueDecomposition if there exists a measure ξ and a measurable function f, such that ξ is mutually singular with respect to ν and μ = ξ + ν.withDensity f.

• lebesgue_decomposition : ∃ (p : MeasureTheory.Measure α × (α → ENNReal)), Measurable p.2 ∧ MeasureTheory.Measure.MutuallySingular p.1 ν ∧ μ = p.1 + MeasureTheory.Measure.withDensity ν p.2

@[irreducible]

def MeasureTheory.Measure.singularPart {α : Type u_3} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) : MeasureTheory.Measure α

If a pair of measures HaveLebesgueDecomposition, then singularPart chooses the measure from HaveLebesgueDecomposition, otherwise it returns the zero measure. For sigma-finite measures, μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν).

Equations

• @MeasureTheory.Measure.singularPart = MeasureTheory.Measure.wrapped✝.1

theorem MeasureTheory.Measure.singularPart_def {α : Type u_3} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) : MeasureTheory.Measure.singularPart μ ν = if h : MeasureTheory.Measure.HaveLebesgueDecomposition μ ν then (Classical.choose ⋯).1 else 0

@[irreducible]

def MeasureTheory.Measure.rnDeriv {α : Type u_3} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) : α → ENNReal

If a pair of measures HaveLebesgueDecomposition, then rnDeriv chooses the measurable function from HaveLebesgueDecomposition, otherwise it returns the zero function. For sigma-finite measures, μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν).

Equations

• @MeasureTheory.Measure.rnDeriv = MeasureTheory.Measure.wrapped✝.1

theorem MeasureTheory.Measure.rnDeriv_def {α : Type u_3} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) : MeasureTheory.Measure.rnDeriv μ ν = if h : MeasureTheory.Measure.HaveLebesgueDecomposition μ ν then (Classical.choose ⋯).2 else 0

theorem MeasureTheory.Measure.haveLebesgueDecomposition_spec {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [h : MeasureTheory.Measure.HaveLebesgueDecomposition μ ν] : Measurable (MeasureTheory.Measure.rnDeriv μ ν) ∧ MeasureTheory.Measure.MutuallySingular (MeasureTheory.Measure.singularPart μ ν) ν ∧ μ = MeasureTheory.Measure.singularPart μ ν + MeasureTheory.Measure.withDensity ν (MeasureTheory.Measure.rnDeriv μ ν)

theorem MeasureTheory.Measure.rnDeriv_of_not_haveLebesgueDecomposition {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} (h : ¬MeasureTheory.Measure.HaveLebesgueDecomposition μ ν) : MeasureTheory.Measure.rnDeriv μ ν = 0

theorem MeasureTheory.Measure.singularPart_of_not_haveLebesgueDecomposition {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} (h : ¬MeasureTheory.Measure.HaveLebesgueDecomposition μ ν) : MeasureTheory.Measure.singularPart μ ν = 0

theorem MeasureTheory.Measure.measurable_rnDeriv {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) : Measurable (MeasureTheory.Measure.rnDeriv μ ν)

theorem MeasureTheory.Measure.mutuallySingular_singularPart {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) : MeasureTheory.Measure.MutuallySingular (MeasureTheory.Measure.singularPart μ ν) ν

theorem MeasureTheory.Measure.haveLebesgueDecomposition_add {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.Measure.HaveLebesgueDecomposition μ ν] : μ = MeasureTheory.Measure.singularPart μ ν + MeasureTheory.Measure.withDensity ν (MeasureTheory.Measure.rnDeriv μ ν)

theorem MeasureTheory.Measure.singularPart_add_rnDeriv {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.Measure.HaveLebesgueDecomposition μ ν] : MeasureTheory.Measure.singularPart μ ν + MeasureTheory.Measure.withDensity ν (MeasureTheory.Measure.rnDeriv μ ν) = μ

theorem MeasureTheory.Measure.rnDeriv_add_singularPart {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.Measure.HaveLebesgueDecomposition μ ν] : MeasureTheory.Measure.withDensity ν (MeasureTheory.Measure.rnDeriv μ ν) + MeasureTheory.Measure.singularPart μ ν = μ

instance MeasureTheory.Measure.instHaveLebesgueDecompositionZeroLeft {α : Type u_1} {m : MeasurableSpace α} {ν : MeasureTheory.Measure α} : MeasureTheory.Measure.HaveLebesgueDecomposition 0 ν

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.instHaveLebesgueDecompositionZeroRight {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} : MeasureTheory.Measure.HaveLebesgueDecomposition μ 0

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.instHaveLebesgueDecompositionSelf {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} : MeasureTheory.Measure.HaveLebesgueDecomposition μ μ

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.haveLebesgueDecompositionSmul' {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.Measure.HaveLebesgueDecomposition μ ν] (r : ENNReal) : MeasureTheory.Measure.HaveLebesgueDecomposition (r • μ) ν

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.haveLebesgueDecompositionSmul {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.Measure.HaveLebesgueDecomposition μ ν] (r : NNReal) : MeasureTheory.Measure.HaveLebesgueDecomposition (r • μ) ν

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.haveLebesgueDecompositionSmulRight {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.Measure.HaveLebesgueDecomposition μ ν] (r : NNReal) : MeasureTheory.Measure.HaveLebesgueDecomposition μ (r • ν)

Equations

• ⋯ = ⋯

theorem MeasureTheory.Measure.haveLebesgueDecomposition_withDensity {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f : α → ENNReal} (hf : Measurable f) : MeasureTheory.Measure.HaveLebesgueDecomposition (MeasureTheory.Measure.withDensity μ f) μ

instance MeasureTheory.Measure.haveLebesgueDecompositionRnDeriv {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) : MeasureTheory.Measure.HaveLebesgueDecomposition (MeasureTheory.Measure.withDensity ν (MeasureTheory.Measure.rnDeriv μ ν)) ν

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.instHaveLebesgueDecompositionSingularPart {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} : MeasureTheory.Measure.HaveLebesgueDecomposition (MeasureTheory.Measure.singularPart μ ν) ν

Equations

• ⋯ = ⋯

theorem MeasureTheory.Measure.singularPart_le {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) : MeasureTheory.Measure.singularPart μ ν ≤ μ

theorem MeasureTheory.Measure.withDensity_rnDeriv_le {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) : MeasureTheory.Measure.withDensity ν (MeasureTheory.Measure.rnDeriv μ ν) ≤ μ

theorem AEMeasurable.singularPart {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_3} : ∀ {x : MeasurableSpace β} {f : α → β}, AEMeasurable f μ → ∀ (ν : MeasureTheory.Measure α), AEMeasurable f (MeasureTheory.Measure.singularPart μ ν)

theorem AEMeasurable.withDensity_rnDeriv {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_3} : ∀ {x : MeasurableSpace β} {f : α → β}, AEMeasurable f μ → ∀ (ν : MeasureTheory.Measure α), AEMeasurable f (MeasureTheory.Measure.withDensity ν (MeasureTheory.Measure.rnDeriv μ ν))

theorem MeasureTheory.Measure.MutuallySingular.singularPart {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} (h : MeasureTheory.Measure.MutuallySingular μ ν) (ν' : MeasureTheory.Measure α) : MeasureTheory.Measure.MutuallySingular (MeasureTheory.Measure.singularPart μ ν') ν

theorem MeasureTheory.Measure.absolutelyContinuous_withDensity_rnDeriv {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} [MeasureTheory.Measure.HaveLebesgueDecomposition ν μ] (hμν : MeasureTheory.Measure.AbsolutelyContinuous μ ν) : MeasureTheory.Measure.AbsolutelyContinuous μ (MeasureTheory.Measure.withDensity μ (MeasureTheory.Measure.rnDeriv ν μ))

theorem MeasureTheory.Measure.singularPart_eq_zero_of_ac {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} (h : MeasureTheory.Measure.AbsolutelyContinuous μ ν) : MeasureTheory.Measure.singularPart μ ν = 0

@[simp]

theorem MeasureTheory.Measure.singularPart_zero {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α) : MeasureTheory.Measure.singularPart 0 ν = 0

theorem MeasureTheory.Measure.singularPart_eq_zero {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.Measure.HaveLebesgueDecomposition μ ν] : MeasureTheory.Measure.singularPart μ ν = 0 ↔ MeasureTheory.Measure.AbsolutelyContinuous μ ν

@[simp]

theorem MeasureTheory.Measure.withDensity_rnDeriv_eq_zero {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.Measure.HaveLebesgueDecomposition μ ν] : MeasureTheory.Measure.withDensity ν (MeasureTheory.Measure.rnDeriv μ ν) = 0 ↔ MeasureTheory.Measure.MutuallySingular μ ν

@[simp]

theorem MeasureTheory.Measure.rnDeriv_eq_zero {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.Measure.HaveLebesgueDecomposition μ ν] : MeasureTheory.Measure.rnDeriv μ ν =ᶠ[MeasureTheory.Measure.ae ν] 0 ↔ MeasureTheory.Measure.MutuallySingular μ ν

theorem MeasureTheory.Measure.rnDeriv_zero {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α) : MeasureTheory.Measure.rnDeriv 0 ν =ᶠ[MeasureTheory.Measure.ae ν] 0

theorem MeasureTheory.Measure.MutuallySingular.rnDeriv_ae_eq_zero {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} (hμν : MeasureTheory.Measure.MutuallySingular μ ν) : MeasureTheory.Measure.rnDeriv μ ν =ᶠ[MeasureTheory.Measure.ae ν] 0

@[simp]

theorem MeasureTheory.Measure.singularPart_withDensity {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α) (f : α → ENNReal) : MeasureTheory.Measure.singularPart (MeasureTheory.Measure.withDensity ν f) ν = 0

theorem MeasureTheory.Measure.rnDeriv_singularPart {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) : MeasureTheory.Measure.rnDeriv (MeasureTheory.Measure.singularPart μ ν) ν =ᶠ[MeasureTheory.Measure.ae ν] 0

@[simp]

theorem MeasureTheory.Measure.singularPart_self {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) : MeasureTheory.Measure.singularPart μ μ = 0

theorem MeasureTheory.Measure.rnDeriv_self {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] : MeasureTheory.Measure.rnDeriv μ μ =ᶠ[MeasureTheory.Measure.ae μ] fun (x : α) => 1

theorem MeasureTheory.Measure.singularPart_eq_self {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} [MeasureTheory.Measure.HaveLebesgueDecomposition μ ν] : MeasureTheory.Measure.singularPart μ ν = μ ↔ MeasureTheory.Measure.MutuallySingular μ ν

instance MeasureTheory.Measure.singularPart.instIsFiniteMeasure {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] : MeasureTheory.IsFiniteMeasure (MeasureTheory.Measure.singularPart μ ν)

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.singularPart.instSigmaFinite {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] : MeasureTheory.SigmaFinite (MeasureTheory.Measure.singularPart μ ν)

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.singularPart.instIsLocallyFiniteMeasure {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} [TopologicalSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] : MeasureTheory.IsLocallyFiniteMeasure (MeasureTheory.Measure.singularPart μ ν)

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.withDensity.instIsFiniteMeasure {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] : MeasureTheory.IsFiniteMeasure (MeasureTheory.Measure.withDensity ν (MeasureTheory.Measure.rnDeriv μ ν))

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.withDensity.instSigmaFinite {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] : MeasureTheory.SigmaFinite (MeasureTheory.Measure.withDensity ν (MeasureTheory.Measure.rnDeriv μ ν))

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.withDensity.instIsLocallyFiniteMeasure {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} [TopologicalSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] : MeasureTheory.IsLocallyFiniteMeasure (MeasureTheory.Measure.withDensity ν (MeasureTheory.Measure.rnDeriv μ ν))

Equations

• ⋯ = ⋯

theorem MeasureTheory.Measure.lintegral_rnDeriv_lt_top_of_measure_ne_top {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (ν : MeasureTheory.Measure α) {s : Set α} (hs : ↑↑μ s ≠ ⊤) : ∫⁻ (x : α) in s, MeasureTheory.Measure.rnDeriv μ ν x ∂ν < ⊤

theorem MeasureTheory.Measure.lintegral_rnDeriv_lt_top {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] : ∫⁻ (x : α), MeasureTheory.Measure.rnDeriv μ ν x ∂ν < ⊤

theorem MeasureTheory.Measure.integrable_toReal_rnDeriv {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] : MeasureTheory.Integrable (fun (x : α) => (MeasureTheory.Measure.rnDeriv μ ν x).toReal) ν

theorem MeasureTheory.Measure.rnDeriv_lt_top {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] : ∀ᵐ (x : α) ∂ν, MeasureTheory.Measure.rnDeriv μ ν x < ⊤

The Radon-Nikodym derivative of a sigma-finite measure μ with respect to another measure ν is ν-almost everywhere finite.

theorem MeasureTheory.Measure.rnDeriv_ne_top {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] : ∀ᵐ (x : α) ∂ν, MeasureTheory.Measure.rnDeriv μ ν x ≠ ⊤

theorem MeasureTheory.Measure.eq_singularPart {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} {s : MeasureTheory.Measure α} {f : α → ENNReal} (hf : Measurable f) (hs : MeasureTheory.Measure.MutuallySingular s ν) (hadd : μ = s + MeasureTheory.Measure.withDensity ν f) : s = MeasureTheory.Measure.singularPart μ ν

Given measures μ and ν, if s is a measure mutually singular to ν and f is a measurable function such that μ = s + fν, then s = μ.singularPart μ.

This theorem provides the uniqueness of the singularPart in the Lebesgue decomposition theorem, while MeasureTheory.Measure.eq_rnDeriv provides the uniqueness of the rnDeriv.

theorem MeasureTheory.Measure.singularPart_smul {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) (r : NNReal) : MeasureTheory.Measure.singularPart (r • μ) ν = r • MeasureTheory.Measure.singularPart μ ν

theorem MeasureTheory.Measure.singularPart_smul_right {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) (r : NNReal) (hr : r ≠ 0) : MeasureTheory.Measure.singularPart μ (r • ν) = MeasureTheory.Measure.singularPart μ ν

theorem MeasureTheory.Measure.singularPart_add {α : Type u_1} {m : MeasurableSpace α} (μ₁ : MeasureTheory.Measure α) (μ₂ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.Measure.HaveLebesgueDecomposition μ₁ ν] [MeasureTheory.Measure.HaveLebesgueDecomposition μ₂ ν] : MeasureTheory.Measure.singularPart (μ₁ + μ₂) ν = MeasureTheory.Measure.singularPart μ₁ ν + MeasureTheory.Measure.singularPart μ₂ ν

theorem MeasureTheory.Measure.eq_withDensity_rnDeriv {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} {s : MeasureTheory.Measure α} {f : α → ENNReal} (hf : Measurable f) (hs : MeasureTheory.Measure.MutuallySingular s ν) (hadd : μ = s + MeasureTheory.Measure.withDensity ν f) : MeasureTheory.Measure.withDensity ν f = MeasureTheory.Measure.withDensity ν (MeasureTheory.Measure.rnDeriv μ ν)

Given measures μ and ν, if s is a measure mutually singular to ν and f is a measurable function such that μ = s + fν, then f = μ.rnDeriv ν.

This theorem provides the uniqueness of the rnDeriv in the Lebesgue decomposition theorem, while MeasureTheory.Measure.eq_singularPart provides the uniqueness of the singularPart. Here, the uniqueness is given in terms of the measures, while the uniqueness in terms of the functions is given in eq_rnDeriv.

theorem MeasureTheory.Measure.eq_withDensity_rnDeriv₀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} {s : MeasureTheory.Measure α} {f : α → ENNReal} (hf : AEMeasurable f ν) (hs : MeasureTheory.Measure.MutuallySingular s ν) (hadd : μ = s + MeasureTheory.Measure.withDensity ν f) : MeasureTheory.Measure.withDensity ν f = MeasureTheory.Measure.withDensity ν (MeasureTheory.Measure.rnDeriv μ ν)

theorem MeasureTheory.Measure.eq_rnDeriv₀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite ν] {s : MeasureTheory.Measure α} {f : α → ENNReal} (hf : AEMeasurable f ν) (hs : MeasureTheory.Measure.MutuallySingular s ν) (hadd : μ = s + MeasureTheory.Measure.withDensity ν f) : f =ᶠ[MeasureTheory.Measure.ae ν] MeasureTheory.Measure.rnDeriv μ ν

theorem MeasureTheory.Measure.eq_rnDeriv {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite ν] {s : MeasureTheory.Measure α} {f : α → ENNReal} (hf : Measurable f) (hs : MeasureTheory.Measure.MutuallySingular s ν) (hadd : μ = s + MeasureTheory.Measure.withDensity ν f) : f =ᶠ[MeasureTheory.Measure.ae ν] MeasureTheory.Measure.rnDeriv μ ν

Given measures μ and ν, if s is a measure mutually singular to ν and f is a measurable function such that μ = s + fν, then f = μ.rnDeriv ν.

This theorem provides the uniqueness of the rnDeriv in the Lebesgue decomposition theorem, while MeasureTheory.Measure.eq_singularPart provides the uniqueness of the singularPart. Here, the uniqueness is given in terms of the functions, while the uniqueness in terms of the functions is given in eq_withDensity_rnDeriv.

theorem MeasureTheory.Measure.rnDeriv_withDensity₀ {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite ν] {f : α → ENNReal} (hf : AEMeasurable f ν) : MeasureTheory.Measure.rnDeriv (MeasureTheory.Measure.withDensity ν f) ν =ᶠ[MeasureTheory.Measure.ae ν] f

The Radon-Nikodym derivative of f ν with respect to ν is f.

theorem MeasureTheory.Measure.rnDeriv_withDensity {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite ν] {f : α → ENNReal} (hf : Measurable f) : MeasureTheory.Measure.rnDeriv (MeasureTheory.Measure.withDensity ν f) ν =ᶠ[MeasureTheory.Measure.ae ν] f

The Radon-Nikodym derivative of f ν with respect to ν is f.

theorem MeasureTheory.Measure.rnDeriv_restrict_self {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite ν] {s : Set α} (hs : MeasurableSet s) : MeasureTheory.Measure.rnDeriv (MeasureTheory.Measure.restrict ν s) ν =ᶠ[MeasureTheory.Measure.ae ν] Set.indicator s 1

The Radon-Nikodym derivative of the restriction of a measure to a measurable set is the indicator function of this set.

theorem MeasureTheory.Measure.rnDeriv_smul_left {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α) (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure ν] [MeasureTheory.Measure.HaveLebesgueDecomposition ν μ] (r : NNReal) : MeasureTheory.Measure.rnDeriv (r • ν) μ =ᶠ[MeasureTheory.Measure.ae μ] r • MeasureTheory.Measure.rnDeriv ν μ

Radon-Nikodym derivative of the scalar multiple of a measure. See also rnDeriv_smul_left', which requires sigma-finite ν and μ.

theorem MeasureTheory.Measure.rnDeriv_smul_left_of_ne_top {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α) (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure ν] [MeasureTheory.Measure.HaveLebesgueDecomposition ν μ] {r : ENNReal} (hr : r ≠ ⊤) : MeasureTheory.Measure.rnDeriv (r • ν) μ =ᶠ[MeasureTheory.Measure.ae μ] r • MeasureTheory.Measure.rnDeriv ν μ

Radon-Nikodym derivative of the scalar multiple of a measure. See also rnDeriv_smul_left_of_ne_top', which requires sigma-finite ν and μ.

theorem MeasureTheory.Measure.rnDeriv_smul_right {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α) (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure ν] [MeasureTheory.Measure.HaveLebesgueDecomposition ν μ] {r : NNReal} (hr : r ≠ 0) : MeasureTheory.Measure.rnDeriv ν (r • μ) =ᶠ[MeasureTheory.Measure.ae μ] r⁻¹ • MeasureTheory.Measure.rnDeriv ν μ

Radon-Nikodym derivative with respect to the scalar multiple of a measure. See also rnDeriv_smul_right', which requires sigma-finite ν and μ.

theorem MeasureTheory.Measure.rnDeriv_smul_right_of_ne_top {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α) (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure ν] [MeasureTheory.Measure.HaveLebesgueDecomposition ν μ] {r : ENNReal} (hr : r ≠ 0) (hr_ne_top : r ≠ ⊤) : MeasureTheory.Measure.rnDeriv ν (r • μ) =ᶠ[MeasureTheory.Measure.ae μ] r⁻¹ • MeasureTheory.Measure.rnDeriv ν μ

Radon-Nikodym derivative with respect to the scalar multiple of a measure. See also rnDeriv_smul_right_of_ne_top', which requires sigma-finite ν and μ.

theorem MeasureTheory.Measure.rnDeriv_add {α : Type u_1} {m : MeasurableSpace α} (ν₁ : MeasureTheory.Measure α) (ν₂ : MeasureTheory.Measure α) (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure ν₁] [MeasureTheory.IsFiniteMeasure ν₂] [MeasureTheory.Measure.HaveLebesgueDecomposition ν₁ μ] [MeasureTheory.Measure.HaveLebesgueDecomposition ν₂ μ] [MeasureTheory.Measure.HaveLebesgueDecomposition (ν₁ + ν₂) μ] : MeasureTheory.Measure.rnDeriv (ν₁ + ν₂) μ =ᶠ[MeasureTheory.Measure.ae μ] MeasureTheory.Measure.rnDeriv ν₁ μ + MeasureTheory.Measure.rnDeriv ν₂ μ

Radon-Nikodym derivative of a sum of two measures. See also rnDeriv_add', which requires sigma-finite ν₁, ν₂ and μ.

theorem MeasureTheory.Measure.exists_positive_of_not_mutuallySingular {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν] (h : ¬MeasureTheory.Measure.MutuallySingular μ ν) : ∃ (ε : NNReal), 0 < ε ∧ ∃ (E : Set α), MeasurableSet E ∧ 0 < ↑↑ν E ∧ MeasureTheory.VectorMeasure.restrict 0 E ≤ MeasureTheory.VectorMeasure.restrict (MeasureTheory.Measure.toSignedMeasure μ - MeasureTheory.Measure.toSignedMeasure (ε • ν)) E

If two finite measures μ and ν are not mutually singular, there exists some ε > 0 and a measurable set E, such that ν(E) > 0 and E is positive with respect to μ - εν.

This lemma is useful for the Lebesgue decomposition theorem.

def MeasureTheory.Measure.LebesgueDecomposition.measurableLE {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) : Set (α → ENNReal)

Given two measures μ and ν, measurableLE μ ν is the set of measurable functions f, such that, for all measurable sets A, ∫⁻ x in A, f x ∂μ ≤ ν A.

This is useful for the Lebesgue decomposition theorem.

Equations

• MeasureTheory.Measure.LebesgueDecomposition.measurableLE μ ν = {f : α → ENNReal | Measurable f ∧ ∀ (A : Set α), MeasurableSet A → ∫⁻ (x : α) in A, f x ∂μ ≤ ↑↑ν A}

theorem MeasureTheory.Measure.LebesgueDecomposition.zero_mem_measurableLE {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} : 0 ∈ MeasureTheory.Measure.LebesgueDecomposition.measurableLE μ ν

theorem MeasureTheory.Measure.LebesgueDecomposition.sup_mem_measurableLE {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} {f : α → ENNReal} {g : α → ENNReal} (hf : f ∈ MeasureTheory.Measure.LebesgueDecomposition.measurableLE μ ν) (hg : g ∈ MeasureTheory.Measure.LebesgueDecomposition.measurableLE μ ν) : (fun (a : α) => f a ⊔ g a) ∈ MeasureTheory.Measure.LebesgueDecomposition.measurableLE μ ν

theorem MeasureTheory.Measure.LebesgueDecomposition.iSup_succ_eq_sup {α : Sort u_3} (f : ℕ → α → ENNReal) (m : ℕ) (a : α) : ⨆ (k : ℕ), ⨆ (_ : k ≤ m + 1), f k a = f (Nat.succ m) a ⊔ ⨆ (k : ℕ), ⨆ (_ : k ≤ m), f k a

theorem MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} (f : ℕ → α → ENNReal) (hf : ∀ (n : ℕ), f n ∈ MeasureTheory.Measure.LebesgueDecomposition.measurableLE μ ν) (n : ℕ) : (fun (x : α) => ⨆ (k : ℕ), ⨆ (_ : k ≤ n), f k x) ∈ MeasureTheory.Measure.LebesgueDecomposition.measurableLE μ ν

theorem MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE' {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} (f : ℕ → α → ENNReal) (hf : ∀ (n : ℕ), f n ∈ MeasureTheory.Measure.LebesgueDecomposition.measurableLE μ ν) (n : ℕ) : ⨆ (k : ℕ), ⨆ (_ : k ≤ n), f k ∈ MeasureTheory.Measure.LebesgueDecomposition.measurableLE μ ν

theorem MeasureTheory.Measure.LebesgueDecomposition.iSup_monotone {α : Type u_3} (f : ℕ → α → ENNReal) : Monotone fun (n : ℕ) (x : α) => ⨆ (k : ℕ), ⨆ (_ : k ≤ n), f k x

theorem MeasureTheory.Measure.LebesgueDecomposition.iSup_monotone' {α : Type u_3} (f : ℕ → α → ENNReal) (x : α) : Monotone fun (n : ℕ) => ⨆ (k : ℕ), ⨆ (_ : k ≤ n), f k x

theorem MeasureTheory.Measure.LebesgueDecomposition.iSup_le_le {α : Type u_3} (f : ℕ → α → ENNReal) (n : ℕ) (k : ℕ) (hk : k ≤ n) : f k ≤ fun (x : α) => ⨆ (k : ℕ), ⨆ (_ : k ≤ n), f k x

def MeasureTheory.Measure.LebesgueDecomposition.measurableLEEval {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) : Set ENNReal

measurableLEEval μ ν is the set of ∫⁻ x, f x ∂μ for all f ∈ measurableLE μ ν.

Equations

• MeasureTheory.Measure.LebesgueDecomposition.measurableLEEval μ ν = (fun (f : α → ENNReal) => ∫⁻ (x : α), f x ∂μ) '' MeasureTheory.Measure.LebesgueDecomposition.measurableLE μ ν

theorem MeasureTheory.Measure.haveLebesgueDecomposition_of_finiteMeasure {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν] : MeasureTheory.Measure.HaveLebesgueDecomposition μ ν

Any pair of finite measures μ and ν, HaveLebesgueDecomposition. That is to say, there exist a measure ξ and a measurable function f, such that ξ is mutually singular with respect to ν and μ = ξ + ν.withDensity f.

This is not an instance since this is also shown for the more general σ-finite measures with MeasureTheory.Measure.haveLebesgueDecomposition_of_sigmaFinite.

instance MeasureTheory.Measure.restrict.instIsFiniteMeasure {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {S : MeasureTheory.Measure.FiniteSpanningSetsIn μ {s : Set α | MeasurableSet s}} (n : ℕ) : MeasureTheory.IsFiniteMeasure (MeasureTheory.Measure.restrict μ (S.set n))

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.haveLebesgueDecomposition_of_sigmaFinite {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] : MeasureTheory.Measure.HaveLebesgueDecomposition μ ν

The Lebesgue decomposition theorem: Any pair of σ-finite measures μ and ν HaveLebesgueDecomposition. That is to say, there exist a measure ξ and a measurable function f, such that ξ is mutually singular with respect to ν and μ = ξ + ν.withDensity f

Equations

• ⋯ = ⋯

theorem MeasureTheory.Measure.rnDeriv_smul_left' {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α) (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite ν] [MeasureTheory.SigmaFinite μ] (r : NNReal) : MeasureTheory.Measure.rnDeriv (r • ν) μ =ᶠ[MeasureTheory.Measure.ae μ] r • MeasureTheory.Measure.rnDeriv ν μ

Radon-Nikodym derivative of the scalar multiple of a measure. See also rnDeriv_smul_left, which has no hypothesis on μ but requires finite ν.

theorem MeasureTheory.Measure.rnDeriv_smul_left_of_ne_top' {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α) (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite ν] [MeasureTheory.SigmaFinite μ] {r : ENNReal} (hr : r ≠ ⊤) : MeasureTheory.Measure.rnDeriv (r • ν) μ =ᶠ[MeasureTheory.Measure.ae μ] r • MeasureTheory.Measure.rnDeriv ν μ

Radon-Nikodym derivative of the scalar multiple of a measure. See also rnDeriv_smul_left_of_ne_top, which has no hypothesis on μ but requires finite ν.

theorem MeasureTheory.Measure.rnDeriv_smul_right' {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α) (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite ν] [MeasureTheory.SigmaFinite μ] {r : NNReal} (hr : r ≠ 0) : MeasureTheory.Measure.rnDeriv ν (r • μ) =ᶠ[MeasureTheory.Measure.ae μ] r⁻¹ • MeasureTheory.Measure.rnDeriv ν μ

Radon-Nikodym derivative with respect to the scalar multiple of a measure. See also rnDeriv_smul_right, which has no hypothesis on μ but requires finite ν.

theorem MeasureTheory.Measure.rnDeriv_smul_right_of_ne_top' {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α) (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite ν] [MeasureTheory.SigmaFinite μ] {r : ENNReal} (hr : r ≠ 0) (hr_ne_top : r ≠ ⊤) : MeasureTheory.Measure.rnDeriv ν (r • μ) =ᶠ[MeasureTheory.Measure.ae μ] r⁻¹ • MeasureTheory.Measure.rnDeriv ν μ

Radon-Nikodym derivative with respect to the scalar multiple of a measure. See also rnDeriv_smul_right_of_ne_top, which has no hypothesis on μ but requires finite ν.

theorem MeasureTheory.Measure.rnDeriv_add' {α : Type u_1} {m : MeasurableSpace α} (ν₁ : MeasureTheory.Measure α) (ν₂ : MeasureTheory.Measure α) (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite ν₁] [MeasureTheory.SigmaFinite ν₂] [MeasureTheory.SigmaFinite μ] : MeasureTheory.Measure.rnDeriv (ν₁ + ν₂) μ =ᶠ[MeasureTheory.Measure.ae μ] MeasureTheory.Measure.rnDeriv ν₁ μ + MeasureTheory.Measure.rnDeriv ν₂ μ

Radon-Nikodym derivative of a sum of two measures. See also rnDeriv_add, which has no hypothesis on μ but requires finite ν₁ and ν₂.

theorem MeasureTheory.Measure.rnDeriv_add_of_mutuallySingular {α : Type u_1} {m : MeasurableSpace α} (ν₁ : MeasureTheory.Measure α) (ν₂ : MeasureTheory.Measure α) (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite ν₁] [MeasureTheory.SigmaFinite ν₂] [MeasureTheory.SigmaFinite μ] (h : MeasureTheory.Measure.MutuallySingular ν₂ μ) : MeasureTheory.Measure.rnDeriv (ν₁ + ν₂) μ =ᶠ[MeasureTheory.Measure.ae μ] MeasureTheory.Measure.rnDeriv ν₁ μ