Theorems · Theorem · measure theory

MeasureTheory.integral_prod_smul

∀ {α : Type u_1} {β : Type u_2} {E : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β]
  {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [inst_2 : NormedAddCommGroup E] [MeasureTheory.SFinite ν]
  [inst_4 : NormedSpace ℝ E] [MeasureTheory.SFinite μ] {𝕜 : Type u_5} [inst_6 : RCLike 𝕜] [inst_7 : NormedSpace 𝕜 E]
  (f : α → 𝕜) (g : β → E), ∫ (z : α × β), f z.1 • g z.2 ∂μ.prod ν = (∫ (x : α), f x ∂μ) • ∫ (y : β), g y ∂ν

Defined in

Mathlib.MeasureTheory.Integral.Prod

Cited by

2 results in Mathlib

Foundations

Depth 265 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceMeasurableSpaceNormedAddCommGroupMeasureTheory.SFiniteNormedSpaceMeasureTheory.SFiniteRCLikeNormedSpace