Theorems · Theorem · measure theory

MeasureTheory.integrable_prod_iff

∀ {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]
  ⦃f : α × β → E⦄,
  MeasureTheory.AEStronglyMeasurable f (μ.prod ν) →
    (MeasureTheory.Integrable f (μ.prod ν) ↔
      (∀ᵐ (x : α) ∂μ, MeasureTheory.Integrable (fun y => f (x, y)) ν) ∧
        MeasureTheory.Integrable (fun x => ∫ (y : β), ‖f (x, y)‖ ∂ν) μ)

A binary function is integrable if the function y ↦ f (x, y) is integrable for almost every x and the function x ↦ ∫ ‖f (x, y)‖ dy is integrable.

Defined in

Mathlib.MeasureTheory.Integral.Prod

Cited by

5 results in Mathlib

Foundations

Depth 257 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceMeasurableSpaceNormedAddCommGroupMeasureTheory.SFinite