Theorems · Theorem · measure theory

MeasureTheory.setIntegral_condExp

∀ {α : Type u_1} {E : Type u_3} {m m₀ : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → E} {s : Set α}
  [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E] (hm : m ≤ m₀)
  [MeasureTheory.SigmaFinite (μ.trim hm)],
  MeasureTheory.Integrable f μ → MeasurableSet s → ∫ (x : α) in s, μ[f | m] x ∂μ = ∫ (x : α) in s, f x ∂μ

The integral of the conditional expectation μ[f|hm] over an m-measurable set is equal to the integral of f on that set.

Defined in

Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Cited by

13 results in Mathlib

Foundations

Depth 297 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpaceMeasureTheory.SigmaFinite