Theorems · Theorem · measure theory

MeasureTheory.ae_ae_add_linearMap_mem_iff

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
  [inst_2 : NormedAddCommGroup E] [inst_3 : MeasurableSpace E] [BorelSpace E] [inst_5 : NormedSpace 𝕜 E]
  [inst_6 : NormedAddCommGroup F] [inst_7 : MeasurableSpace F] [BorelSpace F] [inst_9 : NormedSpace 𝕜 F] (L : E →ₗ[𝕜] F)
  (μ : MeasureTheory.Measure E) (ν : MeasureTheory.Measure F) [μ.IsAddHaarMeasure] [ν.IsAddHaarMeasure]
  [LocallyCompactSpace E] [LocallyCompactSpace F] {s : Set F},
  MeasurableSet s → ((∀ᵐ (y : F) ∂ν, ∀ᵐ (x : E) ∂μ, y + L x ∈ s) ↔ ∀ᵐ (y : F) ∂ν, y ∈ s)

Given a linear map L : E → F, a property holds almost everywhere in F if and only if, almost everywhere in F, it holds almost everywhere along the subspace spanned by the image of L. This is an instance of a disintegration argument for additive Haar measures.

Defined in

Mathlib.MeasureTheory.Measure.Haar.Disintegration

Cited by

1 results in Mathlib

Foundations

Depth 283 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldCompleteSpaceNormedAddCommGroupMeasurableSpaceBorelSpaceNormedSpaceNormedAddCommGroupMeasurableSpaceBorelSpaceNormedSpaceMeasureTheory.Measure.IsAddHaarMeasureMeasureTheory.Measure.IsAddHaarMeasureLocallyCompactSpaceLocallyCompactSpace