Theorems · Theorem · probability

Measurable.lintegral_kernel_prod_right

∀ {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α β}
  [ProbabilityTheory.IsSFiniteKernel κ] {f : α → β → ENNReal},
  Measurable (Function.uncurry f) → Measurable fun a => ∫⁻ (b : β), f a b ∂κ a

For an s-finite kernel κ and a function f : α → β → ℝ≥0∞ which is measurable when seen as a map from α × β (hypothesis Measurable (uncurry f)), the integral a ↦ ∫⁻ b, f a b ∂(κ a) is measurable.

Defined in

Mathlib.Probability.Kernel.MeasurableLIntegral

Cited by

5 results in Mathlib

Foundations

Depth 205 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ProbabilityTheory.IsSFiniteKernel