Theorems · Theorem · measure theory

MeasureTheory.integral_prod_symm

∀ {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] (f : α × β → E),
  MeasureTheory.Integrable f (μ.prod ν) → ∫ (z : α × β), f z ∂μ.prod ν = ∫ (y : β), ∫ (x : α), f (x, y) ∂μ ∂ν

Symmetric version of Fubini's Theorem: For integrable functions on α × β, the Bochner integral of f is equal to the iterated Bochner integral. This version has the integrals on the right-hand side in the other order.

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.SFinite