Theorems · Theorem · measure theory

MeasureTheory.condExp_indicator

∀ {α : Type u_1} {E : Type u_2} {m m0 : MeasurableSpace α} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
  [CompleteSpace E] {μ : MeasureTheory.Measure α} {f : α → E} {s : Set α},
  MeasureTheory.Integrable f μ → MeasurableSet s → μ[s.indicator f | m] =ᵐ[μ] s.indicator μ[f | m]

The conditional expectation of the indicator of a function over an m-measurable set with respect to the σ-algebra m is a.e. equal to the indicator of the conditional expectation.

Defined in

Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Cited by

5 results in Mathlib

Foundations

Depth 300 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpace