Theorems · Theorem · measure theory

ContinuousLinearMap.integral_comp_commSL

∀ {X : Type u_1} {E : Type u_3} {F : Type u_4} [inst : MeasurableSpace X] {μ : MeasureTheory.Measure X} {𝕜 : Type u_6}
  {𝕜' : Type u_7} [inst_1 : RCLike 𝕜] [inst_2 : RCLike 𝕜'] [inst_3 : NormedAddCommGroup E] [inst_4 : NormedSpace 𝕜 E]
  [inst_5 : NormedAddCommGroup F] [inst_6 : NormedSpace 𝕜' F] [inst_7 : NormedSpace ℝ F] {σ : 𝕜 →+* 𝕜'}
  [RingHomIsometric σ] [CompleteSpace F] [inst_10 : NormedSpace ℝ E] [CompleteSpace E],
  (∀ (r : ℝ) (x : 𝕜), σ (r • x) = r • σ x) →
    ∀ (L : E →SL[σ] F) {φ : X → E}, MeasureTheory.Integrable φ μ → ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)

Defined in

Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Cited by

2 results in Mathlib

Foundations

Depth 258 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceRCLikeRCLikeNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedSpaceRingHomIsometricCompleteSpaceNormedSpaceCompleteSpace