Theorems · Theorem · measure theory

MeasureTheory.setLIntegral_prod

∀ {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] {s : Set α} {t : Set β}
  (f : α × β → ENNReal),
  AEMeasurable f ((μ.prod ν).restrict (s ×ˢ t)) →
    ∫⁻ (z : α × β) in s ×ˢ t, f z ∂μ.prod ν = ∫⁻ (x : α) in s, ∫⁻ (y : β) in t, f (x, y) ∂ν ∂μ

Tonelli's Theorem for set integrals: For ℝ≥0∞-valued almost everywhere measurable functions on s ×ˢ t, the integral of f on s ×ˢ t is equal to the iterated integral on s and t respectively.

Defined in

Mathlib.MeasureTheory.Measure.Prod

Cited by

3 results in Mathlib

Foundations

Depth 223 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceMeasurableSpaceMeasureTheory.SFiniteMeasureTheory.SFinite