Theorems · Theorem · measure theory

MeasureTheory.condExpL2_const_inner

∀ {α : Type u_1} {E : Type u_2} {𝕜 : Type u_7} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : InnerProductSpace 𝕜 E] [inst_3 : CompleteSpace E] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
  (hm : m ≤ m0) (f : ↥(MeasureTheory.Lp E 2 μ)) (c : E),
  ↑↑↑((MeasureTheory.condExpL2 𝕜 𝕜 hm) (MeasureTheory.MemLp.toLp (fun a => inner 𝕜 c (↑↑f a)) ⋯)) =ᵐ[μ] fun a =>
    inner 𝕜 c (↑↑↑((MeasureTheory.condExpL2 E 𝕜 hm) f) a)

condExpL2 commutes with taking inner products with constants. See the lemma condExpL2_comp_continuousLinearMap for a more general result about commuting with continuous linear maps.

Defined in

Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Cited by

1 results in Mathlib

Foundations

Depth 272 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceCompleteSpace