Theorems · Theorem · probability

ProbabilityTheory.integrable_comp_iff

∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {E : Type u_4} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
  {mγ : MeasurableSpace γ} [inst : NormedAddCommGroup E] {a : α} {κ : ProbabilityTheory.Kernel α β}
  {η : ProbabilityTheory.Kernel β γ} ⦃f : γ → E⦄,
  MeasureTheory.AEStronglyMeasurable f ((η.comp κ) a) →
    (MeasureTheory.Integrable f ((η.comp κ) a) ↔
      (∀ᵐ (y : β) ∂κ a, MeasureTheory.Integrable f (η y)) ∧
        MeasureTheory.Integrable (fun y => ∫ (z : γ), ‖f z‖ ∂η y) (κ a))

Defined in

Mathlib.Probability.Kernel.Composition.IntegralCompProd

Cited by

6 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroup