Theorems · Theorem · measure theory

ContinuousMap.integral_apply

∀ {X : Type u_1} {Y : Type u_2} {E : Type u_3} [inst : MeasurableSpace X] {μ : MeasureTheory.Measure X}
  [inst_1 : NormedAddCommGroup E] [inst_2 : TopologicalSpace Y] [inst_3 : CompactSpace Y] [inst_4 : NormedSpace ℝ E]
  [CompleteSpace E] {f : X → C(Y, E)},
  MeasureTheory.Integrable f μ → ∀ (y : Y), (∫ (x : X), f x ∂μ) y = ∫ (x : X), (f x) y ∂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

MeasurableSpaceNormedAddCommGroupTopologicalSpaceCompactSpaceNormedSpaceCompleteSpace