Theorems · Theorem · measure theory

MeasureTheory.lintegral_prod_symm

∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {μ : MeasureTheory.Measure α}
  {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] (f : α × β → ENNReal),
  AEMeasurable f (μ.prod ν) → ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (y : β), ∫⁻ (x : α), f (x, y) ∂μ ∂ν

The symmetric version of Tonelli's Theorem: For ℝ≥0∞-valued almost everywhere measurable functions on α × β, the integral of f is equal to the iterated integral, in reverse order.

Defined in

Mathlib.MeasureTheory.Measure.Prod

Cited by

6 results in Mathlib

Foundations

Depth 225 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceMeasurableSpaceMeasureTheory.SFiniteMeasureTheory.SFinite