Theorems · Theorem · measure theory

MeasureTheory.integral_condExp_indicator

∀ {α : Type u_1} {β : Type u_2} {m₀ : MeasurableSpace α} {μ : MeasureTheory.Measure α} [mβ : MeasurableSpace β]
  {Y : α → β} (hY : Measurable Y) [MeasureTheory.SigmaFinite (μ.trim ⋯)] {A : Set α},
  MeasurableSet A → ∫ (x : α), μ[A.indicator fun x => 1 | MeasurableSpace.comap Y mβ] x ∂μ = μ.real A

Law of total probability using condExp as conditional probability.

Defined in

Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Cited by

0 results in Mathlib

Foundations

Depth 299 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceMeasureTheory.SigmaFinite