Theorems · Theorem · measure theory

MeasureTheory.integral_condExpL2_eq

∀ {α : Type u_1} {E' : Type u_3} {𝕜 : Type u_7} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E']
  [inst_2 : InnerProductSpace 𝕜 E'] [inst_3 : CompleteSpace E'] [inst_4 : NormedSpace ℝ E'] {m m0 : MeasurableSpace α}
  {μ : MeasureTheory.Measure α} {s : Set α} (hm : m ≤ m0) (f : ↥(MeasureTheory.Lp E' 2 μ)),
  MeasurableSet s → μ s ≠ ⊤ → ∫ (x : α) in s, ↑↑↑((MeasureTheory.condExpL2 E' 𝕜 hm) f) x ∂μ = ∫ (x : α) in s, ↑↑f x ∂μ

condExpL2 verifies the equality of integrals defining the conditional expectation.

Defined in

Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Cited by

3 results in Mathlib

Foundations

Depth 273 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceCompleteSpaceNormedSpace