Theorems · Theorem · measure theory

ContinuousMultilinearMap.integral_apply

∀ {X : Type u_1} {E : Type u_3} [inst : MeasurableSpace X] {μ : MeasureTheory.Measure X} {𝕜 : Type u_6}
  [inst_1 : RCLike 𝕜] [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace 𝕜 E] [inst_4 : NormedSpace ℝ E]
  {ι : Type u_8} [inst_5 : Fintype ι] {M : ι → Type u_9} [inst_6 : (i : ι) → NormedAddCommGroup (M i)]
  [inst_7 : (i : ι) → NormedSpace 𝕜 (M i)] {φ : X → ContinuousMultilinearMap 𝕜 M E},
  MeasureTheory.Integrable φ μ → ∀ (m : (i : ι) → M i), (∫ (x : X), φ x ∂μ) m = ∫ (x : X), (φ x) m ∂μ

Defined in

Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Cited by

1 results in Mathlib

Foundations

Depth 260 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceRCLikeNormedAddCommGroupNormedSpaceNormedSpaceFintypeNormedAddCommGroupNormedSpace