Theorems · Theorem · measure theory

ContinuousLinearMap.integral_compLp

∀ {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] {p : ENNReal} [inst_7 : NormedSpace ℝ F] {σ : 𝕜 →+* 𝕜'}
  [inst_8 : RingHomIsometric σ] (L : E →SL[σ] F) (φ : ↥(MeasureTheory.Lp E p μ)),
  ∫ (x : X), ↑↑(L.compLp φ) x ∂μ = ∫ (x : X), L (↑↑φ x) ∂μ

Defined in

Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Cited by

1 results in Mathlib

Foundations

Depth 253 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceRCLikeRCLikeNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedSpaceRingHomIsometric