Theorems · Theorem · measure theory

MeasureTheory.condExp_ae_eq_restrict_of_measurableSpace_eq_on

∀ {α : Type u_1} {E : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E] {f : α → E}
  {s : Set α} {m m₂ m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (hm₂ : m₂ ≤ m0)
  [MeasureTheory.SigmaFinite (μ.trim hm)] [MeasureTheory.SigmaFinite (μ.trim hm₂)],
  MeasurableSet s → (∀ (t : Set α), MeasurableSet (s ∩ t) ↔ MeasurableSet (s ∩ t)) → μ[f | m] =ᵐ[μ.restrict s] μ[f | m₂]

If the restriction to an m-measurable set s of a σ-algebra m is equal to the restriction to s of another σ-algebra m₂ (hypothesis hs), then μ[f | m] =ᵐ[μ.restrict s] μ[f | m₂].

Defined in

Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Cited by

6 results in Mathlib

Foundations

Depth 301 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpaceMeasureTheory.SigmaFiniteMeasureTheory.SigmaFinite