Theorems · Theorem · probability

ProbabilityTheory.IndepFun.integral_fun_smul_eq_smul_integral

∀ {Ω : Type u_1} {𝕜 : Type u_2} [inst : RCLike 𝕜] {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {E : Type u_5}
  [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace ℝ E] [inst_3 : NormedSpace 𝕜 E] [inst_4 : MeasurableSpace E]
  [BorelSpace E] {X : Ω → 𝕜} {Y : Ω → E},
  ProbabilityTheory.IndepFun X Y μ →
    MeasureTheory.AEStronglyMeasurable X μ →
      MeasureTheory.AEStronglyMeasurable Y μ → ∫ (ω : Ω), X ω • Y ω ∂μ = (∫ (ω : Ω), X ω ∂μ) • ∫ (ω : Ω), Y ω ∂μ

Defined in

Mathlib.Probability.Independence.Integration

Cited by

1 results in Mathlib

Foundations

Depth 270 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedSpaceNormedSpaceMeasurableSpaceBorelSpace