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Mathlib.MeasureTheory.Decomposition.Lebesgue

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 #

Main results #

Tags #

Lebesgue decomposition theorem

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.

Instances
    theorem MeasureTheory.Measure.HaveLebesgueDecomposition.lebesgue_decomposition {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} [self : μ.HaveLebesgueDecomposition ν] :
    ∃ (p : MeasureTheory.Measure α × (αENNReal)), Measurable p.2 p.1.MutuallySingular ν μ = p.1 + ν.withDensity p.2
    theorem MeasureTheory.Measure.singularPart_def {α : Type u_3} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) :
    μ.singularPart ν = if h : μ.HaveLebesgueDecomposition ν then (Classical.choose ).1 else 0
    @[irreducible]

    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
    Instances For
      @[irreducible]
      noncomputable def MeasureTheory.Measure.rnDeriv {α : Type u_3} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) :
      αENNReal

      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
      Instances For
        theorem MeasureTheory.Measure.rnDeriv_def {α : Type u_3} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) :
        μ.rnDeriv ν = if h : μ.HaveLebesgueDecomposition 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 : μ.HaveLebesgueDecomposition ν] :
        Measurable (μ.rnDeriv ν) (μ.singularPart ν).MutuallySingular ν μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
        theorem MeasureTheory.Measure.rnDeriv_of_not_haveLebesgueDecomposition {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} (h : ¬μ.HaveLebesgueDecomposition ν) :
        μ.rnDeriv 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 : ¬μ.HaveLebesgueDecomposition ν) :
        μ.singularPart ν = 0
        theorem MeasureTheory.Measure.mutuallySingular_singularPart {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) :
        (μ.singularPart ν).MutuallySingular ν
        theorem MeasureTheory.Measure.haveLebesgueDecomposition_add {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν] :
        μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
        theorem MeasureTheory.Measure.singularPart_add_rnDeriv {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν] :
        μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) = μ
        theorem MeasureTheory.Measure.rnDeriv_add_singularPart {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν] :
        ν.withDensity (μ.rnDeriv ν) + μ.singularPart ν = μ
        instance MeasureTheory.Measure.instHaveLebesgueDecompositionZeroRight {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} :
        μ.HaveLebesgueDecomposition 0
        Equations
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        instance MeasureTheory.Measure.instHaveLebesgueDecompositionSelf {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} :
        μ.HaveLebesgueDecomposition μ
        Equations
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        instance MeasureTheory.Measure.HaveLebesgueDecomposition.sum_left {α : Type u_1} {m : MeasurableSpace α} {ν : MeasureTheory.Measure α} {ι : Type u_3} [Countable ι] (μ : ιMeasureTheory.Measure α) [∀ (i : ι), (μ i).HaveLebesgueDecomposition ν] :
        (MeasureTheory.Measure.sum μ).HaveLebesgueDecomposition ν
        Equations
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        instance MeasureTheory.Measure.HaveLebesgueDecomposition.add_left {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} {μ' : MeasureTheory.Measure α} [μ.HaveLebesgueDecomposition ν] [μ'.HaveLebesgueDecomposition ν] :
        (μ + μ').HaveLebesgueDecomposition ν
        Equations
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        instance MeasureTheory.Measure.haveLebesgueDecompositionSMul' {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν] (r : ENNReal) :
        (r μ).HaveLebesgueDecomposition ν
        Equations
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        instance MeasureTheory.Measure.haveLebesgueDecompositionSMul {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν] (r : NNReal) :
        (r μ).HaveLebesgueDecomposition ν
        Equations
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        instance MeasureTheory.Measure.haveLebesgueDecompositionSMulRight {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν] (r : NNReal) :
        μ.HaveLebesgueDecomposition (r ν)
        Equations
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        theorem MeasureTheory.Measure.haveLebesgueDecomposition_withDensity {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f : αENNReal} (hf : Measurable f) :
        (μ.withDensity f).HaveLebesgueDecomposition μ
        instance MeasureTheory.Measure.haveLebesgueDecompositionRnDeriv {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) :
        (ν.withDensity (μ.rnDeriv ν)).HaveLebesgueDecomposition ν
        Equations
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        instance MeasureTheory.Measure.instHaveLebesgueDecompositionSingularPart {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} :
        (μ.singularPart ν).HaveLebesgueDecomposition ν
        Equations
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        theorem MeasureTheory.Measure.singularPart_le {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) :
        μ.singularPart ν μ
        theorem MeasureTheory.Measure.withDensity_rnDeriv_le {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) :
        ν.withDensity (μ.rnDeriv ν) μ
        theorem AEMeasurable.singularPart {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_3} :
        ∀ {x : MeasurableSpace β} {f : αβ}, AEMeasurable f μ∀ (ν : MeasureTheory.Measure α), AEMeasurable f (μ.singularPart ν)
        theorem AEMeasurable.withDensity_rnDeriv {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_3} :
        ∀ {x : MeasurableSpace β} {f : αβ}, AEMeasurable f μ∀ (ν : MeasureTheory.Measure α), AEMeasurable f (ν.withDensity (μ.rnDeriv ν))
        theorem MeasureTheory.Measure.MutuallySingular.singularPart {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} (h : μ.MutuallySingular ν) (ν' : MeasureTheory.Measure α) :
        (μ.singularPart ν').MutuallySingular ν
        theorem MeasureTheory.Measure.absolutelyContinuous_withDensity_rnDeriv {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} [ν.HaveLebesgueDecomposition μ] (hμν : μ.AbsolutelyContinuous ν) :
        μ.AbsolutelyContinuous (μ.withDensity (ν.rnDeriv 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 : μ.AbsolutelyContinuous ν) :
        μ.singularPart ν = 0
        @[simp]
        theorem MeasureTheory.Measure.singularPart_zero_right {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) :
        μ.singularPart 0 = μ
        theorem MeasureTheory.Measure.singularPart_eq_zero {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν] :
        μ.singularPart ν = 0 μ.AbsolutelyContinuous ν
        @[simp]
        theorem MeasureTheory.Measure.withDensity_rnDeriv_eq_zero {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν] :
        ν.withDensity (μ.rnDeriv ν) = 0 μ.MutuallySingular ν
        @[simp]
        theorem MeasureTheory.Measure.rnDeriv_eq_zero {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν] :
        μ.rnDeriv ν =ᵐ[ν] 0 μ.MutuallySingular ν
        theorem MeasureTheory.Measure.MutuallySingular.rnDeriv_ae_eq_zero {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} (hμν : μ.MutuallySingular ν) :
        μ.rnDeriv ν =ᵐ[ν] 0
        @[simp]
        theorem MeasureTheory.Measure.singularPart_withDensity {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α) (f : αENNReal) :
        (ν.withDensity f).singularPart ν = 0
        theorem MeasureTheory.Measure.rnDeriv_singularPart {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) :
        (μ.singularPart ν).rnDeriv ν =ᵐ[ν] 0
        @[simp]
        theorem MeasureTheory.Measure.singularPart_self {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) :
        μ.singularPart μ = 0
        theorem MeasureTheory.Measure.rnDeriv_self {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] :
        μ.rnDeriv μ =ᵐ[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} [μ.HaveLebesgueDecomposition ν] :
        μ.singularPart ν = μ μ.MutuallySingular ν
        @[simp]
        theorem MeasureTheory.Measure.singularPart_singularPart {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) :
        (μ.singularPart ν).singularPart ν = μ.singularPart ν
        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, μ.rnDeriv ν xν <
        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), μ.rnDeriv ν xν <
        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, μ.rnDeriv 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, μ.rnDeriv 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 : αENNReal} (hf : Measurable f) (hs : s.MutuallySingular ν) (hadd : μ = s + ν.withDensity f) :
        s = μ.singularPart 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) :
        (r μ).singularPart ν = r μ.singularPart 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) :
        μ.singularPart (r ν) = μ.singularPart 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) [μ₁.HaveLebesgueDecomposition ν] [μ₂.HaveLebesgueDecomposition ν] :
        (μ₁ + μ₂).singularPart ν = μ₁.singularPart ν + μ₂.singularPart ν
        theorem MeasureTheory.Measure.singularPart_restrict {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν] {s : Set α} (hs : MeasurableSet s) :
        (μ.restrict s).singularPart ν = (μ.singularPart ν).restrict s
        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 : αENNReal} (hf : Measurable f) (hs : s.MutuallySingular ν) (hadd : μ = s + ν.withDensity f) :
        ν.withDensity f = ν.withDensity (μ.rnDeriv 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 : αENNReal} (hf : AEMeasurable f ν) (hs : s.MutuallySingular ν) (hadd : μ = s + ν.withDensity f) :
        ν.withDensity f = ν.withDensity (μ.rnDeriv 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 : αENNReal} (hf : AEMeasurable f ν) (hs : s.MutuallySingular ν) (hadd : μ = s + ν.withDensity f) :
        f =ᵐ[ν] μ.rnDeriv 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 : αENNReal} (hf : Measurable f) (hs : s.MutuallySingular ν) (hadd : μ = s + ν.withDensity f) :
        f =ᵐ[ν] μ.rnDeriv 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 : αENNReal} (hf : AEMeasurable f ν) :
        (ν.withDensity f).rnDeriv ν =ᵐ[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 : αENNReal} (hf : Measurable f) :
        (ν.withDensity f).rnDeriv ν =ᵐ[ν] f

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

        theorem MeasureTheory.Measure.rnDeriv_restrict {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν] [MeasureTheory.SigmaFinite ν] {s : Set α} (hs : MeasurableSet s) :
        (μ.restrict s).rnDeriv ν =ᵐ[ν] s.indicator (μ.rnDeriv ν)
        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) :
        (ν.restrict s).rnDeriv ν =ᵐ[ν] s.indicator 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] [ν.HaveLebesgueDecomposition μ] (r : NNReal) :
        (r ν).rnDeriv μ =ᵐ[μ] r ν.rnDeriv 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] [ν.HaveLebesgueDecomposition μ] {r : ENNReal} (hr : r ) :
        (r ν).rnDeriv μ =ᵐ[μ] r ν.rnDeriv 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] [ν.HaveLebesgueDecomposition μ] {r : NNReal} (hr : r 0) :
        ν.rnDeriv (r μ) =ᵐ[μ] r⁻¹ ν.rnDeriv 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] [ν.HaveLebesgueDecomposition μ] {r : ENNReal} (hr : r 0) (hr_ne_top : r ) :
        ν.rnDeriv (r μ) =ᵐ[μ] r⁻¹ ν.rnDeriv 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] [ν₁.HaveLebesgueDecomposition μ] [ν₂.HaveLebesgueDecomposition μ] [(ν₁ + ν₂).HaveLebesgueDecomposition μ] :
        (ν₁ + ν₂).rnDeriv μ =ᵐ[μ] ν₁.rnDeriv μ + ν₂.rnDeriv 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 : ¬μ.MutuallySingular ν) :
        ∃ (ε : NNReal), 0 < ε ∃ (E : Set α), MeasurableSet E 0 < ν E MeasureTheory.VectorMeasure.restrict 0 E MeasureTheory.VectorMeasure.restrict (μ.toSignedMeasure - (ε ν).toSignedMeasure) 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.

        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
        Instances For
          theorem MeasureTheory.Measure.LebesgueDecomposition.iSup_succ_eq_sup {α : Sort u_3} (f : αENNReal) (m : ) (a : α) :
          ⨆ (k : ), ⨆ (_ : k m + 1), f k a = f m.succ a ⨆ (k : ), ⨆ (_ : k m), f k a
          theorem MeasureTheory.Measure.LebesgueDecomposition.iSup_monotone {α : Type u_3} (f : αENNReal) :
          Monotone fun (n : ) (x : α) => ⨆ (k : ), ⨆ (_ : k n), f k x
          theorem MeasureTheory.Measure.LebesgueDecomposition.iSup_monotone' {α : Type u_3} (f : αENNReal) (x : α) :
          Monotone fun (n : ) => ⨆ (k : ), ⨆ (_ : k n), f k x
          theorem MeasureTheory.Measure.LebesgueDecomposition.iSup_le_le {α : Type u_3} (f : αENNReal) (n : ) (k : ) (hk : k n) :
          f k fun (x : α) => ⨆ (k : ), ⨆ (_ : k n), f k x

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

          Equations
          Instances For

            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.

            theorem MeasureTheory.Measure.HaveLebesgueDecomposition.sfinite_of_isFiniteMeasure {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} [MeasureTheory.SFinite μ] (_h : ∀ (μ : MeasureTheory.Measure α) [inst : MeasureTheory.IsFiniteMeasure μ], μ.HaveLebesgueDecomposition ν) :
            μ.HaveLebesgueDecomposition ν

            If any finite measure has a Lebesgue decomposition with respect to ν, then the same is true for any s-finite measure.

            @[instance 100]

            The Lebesgue decomposition theorem: Any s-finite measure μ has Lebesgue decomposition with respect to any σ-finite measure ν. 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
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            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.

            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) :
            ν.rnDeriv (r μ) =ᵐ[μ] r⁻¹ ν.rnDeriv 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 ) :
            ν.rnDeriv (r μ) =ᵐ[μ] r⁻¹ ν.rnDeriv 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] :
            (ν₁ + ν₂).rnDeriv μ =ᵐ[μ] ν₁.rnDeriv μ + ν₂.rnDeriv 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 : ν₂.MutuallySingular μ) :
            (ν₁ + ν₂).rnDeriv μ =ᵐ[μ] ν₁.rnDeriv μ