Theorems · Theorem · measure theory

MeasureTheory.integrable_condExpL2_indicator

∀ {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'] [NormedSpace ℝ E'] {m m0 : MeasurableSpace α}
  {μ : MeasureTheory.Measure α} {s : Set α} (hm : m ≤ m0) [MeasureTheory.SigmaFinite (μ.trim hm)] (hs : MeasurableSet s)
  (hμs : μ s ≠ ⊤) (x : E'),
  MeasureTheory.Integrable (↑↑↑((MeasureTheory.condExpL2 E' 𝕜 hm) (MeasureTheory.indicatorConstLp 2 hs hμs x))) μ

If the measure μ.trim hm is sigma-finite, then the conditional expectation of a measurable set with finite measure is integrable.

Defined in

Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Cited by

1 results in Mathlib

Foundations

Depth 277 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceCompleteSpaceNormedSpaceMeasureTheory.SigmaFinite