Theorems · Theorem · measure theory

integral_pair

∀ {X : Type u_1} {E : Type u_3} {F : Type u_4} [inst : MeasurableSpace X] {μ : MeasureTheory.Measure X}
  [inst_1 : NormedAddCommGroup E] [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ E] [inst_4 : NormedSpace ℝ F]
  [CompleteSpace E] [CompleteSpace F] {f : X → E} {g : X → F},
  MeasureTheory.Integrable f μ →
    MeasureTheory.Integrable g μ → ∫ (x : X), (f x, g x) ∂μ = (∫ (x : X), f x ∂μ, ∫ (x : X), g x ∂μ)

Defined in

Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Cited by

1 results in Mathlib

Foundations

Depth 264 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceNormedAddCommGroupNormedAddCommGroupNormedSpaceNormedSpaceCompleteSpaceCompleteSpace