# Lebesgue decomposition #

This file proves the Lebesgue decomposition theorem for signed measures. 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.

## Main definitions #

• MeasureTheory.SignedMeasure.HaveLebesgueDecomposition : A signed measure s and a measure μ is said to HaveLebesgueDecomposition if both the positive part and negative part of s HaveLebesgueDecomposition with respect to μ.
• MeasureTheory.SignedMeasure.singularPart : The singular part between a signed measure s and a measure μ is simply the singular part of the positive part of s with respect to μ minus the singular part of the negative part of s with respect to μ.
• MeasureTheory.SignedMeasure.rnDeriv : The Radon-Nikodym derivative of a signed measure s with respect to a measure μ is the Radon-Nikodym derivative of the positive part of s with respect to μ minus the Radon-Nikodym derivative of the negative part of s with respect to μ.

## Main results #

• MeasureTheory.SignedMeasure.singularPart_add_withDensity_rnDeriv_eq : the Lebesgue decomposition theorem between a signed measure and a σ-finite positive measure.

## Tags #

Lebesgue decomposition theorem

class MeasureTheory.SignedMeasure.HaveLebesgueDecomposition {α : Type u_1} {m : } (s : ) (μ : ) :

A signed measure s is said to HaveLebesgueDecomposition with respect to a measure μ if the positive part and the negative part of s both HaveLebesgueDecomposition with respect to μ.

• posPart : s.toJordanDecomposition.posPart.HaveLebesgueDecomposition μ
• negPart : s.toJordanDecomposition.negPart.HaveLebesgueDecomposition μ
Instances
theorem MeasureTheory.SignedMeasure.HaveLebesgueDecomposition.posPart {α : Type u_1} {m : } {s : } {μ : } [self : s.HaveLebesgueDecomposition μ] :
s.toJordanDecomposition.posPart.HaveLebesgueDecomposition μ
theorem MeasureTheory.SignedMeasure.HaveLebesgueDecomposition.negPart {α : Type u_1} {m : } {s : } {μ : } [self : s.HaveLebesgueDecomposition μ] :
s.toJordanDecomposition.negPart.HaveLebesgueDecomposition μ
theorem MeasureTheory.SignedMeasure.not_haveLebesgueDecomposition_iff {α : Type u_1} {m : } (s : ) (μ : ) :
¬s.HaveLebesgueDecomposition μ ¬s.toJordanDecomposition.posPart.HaveLebesgueDecomposition μ ¬s.toJordanDecomposition.negPart.HaveLebesgueDecomposition μ
@[instance 100]
instance MeasureTheory.SignedMeasure.haveLebesgueDecomposition_of_sigmaFinite {α : Type u_1} {m : } (s : ) (μ : ) :
s.HaveLebesgueDecomposition μ
Equations
• =
instance MeasureTheory.SignedMeasure.haveLebesgueDecomposition_neg {α : Type u_1} {m : } (s : ) (μ : ) [s.HaveLebesgueDecomposition μ] :
(-s).HaveLebesgueDecomposition μ
Equations
• =
instance MeasureTheory.SignedMeasure.haveLebesgueDecomposition_smul {α : Type u_1} {m : } (s : ) (μ : ) [s.HaveLebesgueDecomposition μ] (r : NNReal) :
(r s).HaveLebesgueDecomposition μ
Equations
• =
instance MeasureTheory.SignedMeasure.haveLebesgueDecomposition_smul_real {α : Type u_1} {m : } (s : ) (μ : ) [s.HaveLebesgueDecomposition μ] (r : ) :
(r s).HaveLebesgueDecomposition μ
Equations
• =
def MeasureTheory.SignedMeasure.singularPart {α : Type u_1} {m : } (s : ) (μ : ) :

Given a signed measure s and a measure μ, s.singularPart μ is the signed measure such that s.singularPart μ + μ.withDensityᵥ (s.rnDeriv μ) = s and s.singularPart μ is mutually singular with respect to μ.

Equations
• s.singularPart μ = (s.toJordanDecomposition.posPart.singularPart μ).toSignedMeasure - (s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure
Instances For
theorem MeasureTheory.SignedMeasure.singularPart_mutuallySingular {α : Type u_1} {m : } (s : ) (μ : ) :
(s.toJordanDecomposition.posPart.singularPart μ).MutuallySingular (s.toJordanDecomposition.negPart.singularPart μ)
theorem MeasureTheory.SignedMeasure.singularPart_totalVariation {α : Type u_1} {m : } (s : ) (μ : ) :
(s.singularPart μ).totalVariation = s.toJordanDecomposition.posPart.singularPart μ + s.toJordanDecomposition.negPart.singularPart μ
theorem MeasureTheory.SignedMeasure.mutuallySingular_singularPart {α : Type u_1} {m : } (s : ) (μ : ) :
MeasureTheory.VectorMeasure.MutuallySingular (s.singularPart μ) μ.toENNRealVectorMeasure
def MeasureTheory.SignedMeasure.rnDeriv {α : Type u_1} {m : } (s : ) (μ : ) :
α

The Radon-Nikodym derivative between a signed measure and a positive measure.

rnDeriv s μ satisfies μ.withDensityᵥ (s.rnDeriv μ) = s if and only if s is absolutely continuous with respect to μ and this fact is known as MeasureTheory.SignedMeasure.absolutelyContinuous_iff_withDensity_rnDeriv_eq and can be found in MeasureTheory.Decomposition.RadonNikodym.

Equations
• s.rnDeriv μ x = (s.toJordanDecomposition.posPart.rnDeriv μ x).toReal - (s.toJordanDecomposition.negPart.rnDeriv μ x).toReal
Instances For
theorem MeasureTheory.SignedMeasure.rnDeriv_def {α : Type u_1} {m : } (s : ) (μ : ) :
s.rnDeriv μ = fun (x : α) => (s.toJordanDecomposition.posPart.rnDeriv μ x).toReal - (s.toJordanDecomposition.negPart.rnDeriv μ x).toReal
theorem MeasureTheory.SignedMeasure.measurable_rnDeriv {α : Type u_1} {m : } (s : ) (μ : ) :
Measurable (s.rnDeriv μ)
theorem MeasureTheory.SignedMeasure.integrable_rnDeriv {α : Type u_1} {m : } (s : ) (μ : ) :
MeasureTheory.Integrable (s.rnDeriv μ) μ
theorem MeasureTheory.SignedMeasure.singularPart_add_withDensity_rnDeriv_eq {α : Type u_1} {m : } (μ : ) (s : ) [s.HaveLebesgueDecomposition μ] :
s.singularPart μ + μ.withDensityᵥ (s.rnDeriv μ) = s

The Lebesgue Decomposition theorem between a signed measure and a measure: Given a signed measure s and a σ-finite measure μ, there exist a signed measure t and a measurable and integrable function f, such that t is mutually singular with respect to μ and s = t + μ.withDensityᵥ f. In this case t = s.singularPart μ and f = s.rnDeriv μ.

theorem MeasureTheory.SignedMeasure.jordanDecomposition_add_withDensity_mutuallySingular {α : Type u_1} {m : } {μ : } {t : } {f : α} (hf : ) (htμ : MeasureTheory.VectorMeasure.MutuallySingular t μ.toENNRealVectorMeasure) :
(t.toJordanDecomposition.posPart + μ.withDensity fun (x : α) => ENNReal.ofReal (f x)).MutuallySingular (t.toJordanDecomposition.negPart + μ.withDensity fun (x : α) => ENNReal.ofReal (-f x))
theorem MeasureTheory.SignedMeasure.toJordanDecomposition_eq_of_eq_add_withDensity {α : Type u_1} {m : } {μ : } {s : } {t : } {f : α} (hf : ) (hfi : ) (htμ : MeasureTheory.VectorMeasure.MutuallySingular t μ.toENNRealVectorMeasure) (hadd : s = t + μ.withDensityᵥ f) :
s.toJordanDecomposition = MeasureTheory.JordanDecomposition.mk (t.toJordanDecomposition.posPart + μ.withDensity fun (x : α) => ENNReal.ofReal (f x)) (t.toJordanDecomposition.negPart + μ.withDensity fun (x : α) => ENNReal.ofReal (-f x))
theorem MeasureTheory.SignedMeasure.haveLebesgueDecomposition_mk {α : Type u_1} {m : } {s : } {t : } (μ : ) {f : α} (hf : ) (htμ : MeasureTheory.VectorMeasure.MutuallySingular t μ.toENNRealVectorMeasure) (hadd : s = t + μ.withDensityᵥ f) :
s.HaveLebesgueDecomposition μ
theorem MeasureTheory.SignedMeasure.eq_singularPart {α : Type u_1} {m : } {μ : } {s : } (t : ) (f : α) (htμ : MeasureTheory.VectorMeasure.MutuallySingular t μ.toENNRealVectorMeasure) (hadd : s = t + μ.withDensityᵥ f) :
t = s.singularPart μ

Given a measure μ, signed measures s and t, and a function f such that t is mutually singular with respect to μ and s = t + μ.withDensityᵥ f, we have t = singularPart s μ, i.e. t is the singular part of the Lebesgue decomposition between s and μ.

theorem MeasureTheory.SignedMeasure.singularPart_zero {α : Type u_1} {m : } (μ : ) :
theorem MeasureTheory.SignedMeasure.singularPart_neg {α : Type u_1} {m : } (s : ) (μ : ) :
(-s).singularPart μ = -s.singularPart μ
theorem MeasureTheory.SignedMeasure.singularPart_smul_nnreal {α : Type u_1} {m : } (s : ) (μ : ) (r : NNReal) :
(r s).singularPart μ = r s.singularPart μ
theorem MeasureTheory.SignedMeasure.singularPart_smul {α : Type u_1} {m : } (s : ) (μ : ) (r : ) :
(r s).singularPart μ = r s.singularPart μ
theorem MeasureTheory.SignedMeasure.singularPart_add {α : Type u_1} {m : } (s : ) (t : ) (μ : ) [s.HaveLebesgueDecomposition μ] [t.HaveLebesgueDecomposition μ] :
(s + t).singularPart μ = s.singularPart μ + t.singularPart μ
theorem MeasureTheory.SignedMeasure.singularPart_sub {α : Type u_1} {m : } (s : ) (t : ) (μ : ) [s.HaveLebesgueDecomposition μ] [t.HaveLebesgueDecomposition μ] :
(s - t).singularPart μ = s.singularPart μ - t.singularPart μ
theorem MeasureTheory.SignedMeasure.eq_rnDeriv {α : Type u_1} {m : } {μ : } {s : } (t : ) (f : α) (hfi : ) (htμ : MeasureTheory.VectorMeasure.MutuallySingular t μ.toENNRealVectorMeasure) (hadd : s = t + μ.withDensityᵥ f) :
f =ᵐ[μ] s.rnDeriv μ

Given a measure μ, signed measures s and t, and a function f such that t is mutually singular with respect to μ and s = t + μ.withDensityᵥ f, we have f = rnDeriv s μ, i.e. f is the Radon-Nikodym derivative of s and μ.

theorem MeasureTheory.SignedMeasure.rnDeriv_neg {α : Type u_1} {m : } (s : ) (μ : ) [s.HaveLebesgueDecomposition μ] :
(-s).rnDeriv μ =ᵐ[μ] -s.rnDeriv μ
theorem MeasureTheory.SignedMeasure.rnDeriv_smul {α : Type u_1} {m : } (s : ) (μ : ) [s.HaveLebesgueDecomposition μ] (r : ) :
(r s).rnDeriv μ =ᵐ[μ] r s.rnDeriv μ
theorem MeasureTheory.SignedMeasure.rnDeriv_add {α : Type u_1} {m : } (s : ) (t : ) (μ : ) [s.HaveLebesgueDecomposition μ] [t.HaveLebesgueDecomposition μ] [(s + t).HaveLebesgueDecomposition μ] :
(s + t).rnDeriv μ =ᵐ[μ] s.rnDeriv μ + t.rnDeriv μ
theorem MeasureTheory.SignedMeasure.rnDeriv_sub {α : Type u_1} {m : } (s : ) (t : ) (μ : ) [s.HaveLebesgueDecomposition μ] [t.HaveLebesgueDecomposition μ] [hst : (s - t).HaveLebesgueDecomposition μ] :
(s - t).rnDeriv μ =ᵐ[μ] s.rnDeriv μ - t.rnDeriv μ
class MeasureTheory.ComplexMeasure.HaveLebesgueDecomposition {α : Type u_1} {m : } (c : ) (μ : ) :

A complex measure is said to HaveLebesgueDecomposition with respect to a positive measure if both its real and imaginary part HaveLebesgueDecomposition with respect to that measure.

• rePart : (MeasureTheory.ComplexMeasure.re c).HaveLebesgueDecomposition μ
• imPart : (MeasureTheory.ComplexMeasure.im c).HaveLebesgueDecomposition μ
Instances
theorem MeasureTheory.ComplexMeasure.HaveLebesgueDecomposition.rePart {α : Type u_1} {m : } {c : } {μ : } [self : c.HaveLebesgueDecomposition μ] :
(MeasureTheory.ComplexMeasure.re c).HaveLebesgueDecomposition μ
theorem MeasureTheory.ComplexMeasure.HaveLebesgueDecomposition.imPart {α : Type u_1} {m : } {c : } {μ : } [self : c.HaveLebesgueDecomposition μ] :
(MeasureTheory.ComplexMeasure.im c).HaveLebesgueDecomposition μ
def MeasureTheory.ComplexMeasure.singularPart {α : Type u_1} {m : } (c : ) (μ : ) :

The singular part between a complex measure c and a positive measure μ is the complex measure satisfying c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c. This property is given by MeasureTheory.ComplexMeasure.singularPart_add_withDensity_rnDeriv_eq.

Equations
• c.singularPart μ = ((MeasureTheory.ComplexMeasure.re c).singularPart μ).toComplexMeasure ((MeasureTheory.ComplexMeasure.im c).singularPart μ)
Instances For
def MeasureTheory.ComplexMeasure.rnDeriv {α : Type u_1} {m : } (c : ) (μ : ) :
α

The Radon-Nikodym derivative between a complex measure and a positive measure.

Equations
• c.rnDeriv μ x = { re := (MeasureTheory.ComplexMeasure.re c).rnDeriv μ x, im := (MeasureTheory.ComplexMeasure.im c).rnDeriv μ x }
Instances For
theorem MeasureTheory.ComplexMeasure.integrable_rnDeriv {α : Type u_1} {m : } (c : ) (μ : ) :
MeasureTheory.Integrable (c.rnDeriv μ) μ
theorem MeasureTheory.ComplexMeasure.singularPart_add_withDensity_rnDeriv_eq {α : Type u_1} {m : } {μ : } {c : } [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c