integral_kernel_comp — Mathlib

English

Continuity of the map sending a kernel-valued function to its integral over a product kernel under suitable norms.

Русский

Постоянность отображения ядра в интеграл по произведению при подходящем нормировании.

LaTeX

$$$Continuous (f \mapsto \int f)\; \, (κ,a) $$$

Lean4

theorem _root_.MeasureTheory.AEStronglyMeasurable.integral_kernel_comp [NormedSpace ℝ E] ⦃f : γ → E⦄
    (hf : AEStronglyMeasurable f ((η ∘ₖ κ) a)) : AEStronglyMeasurable (fun x ↦ ∫ y, f y ∂η x) (κ a) :=
  ⟨fun x ↦ ∫ y, hf.mk f y ∂η x, hf.stronglyMeasurable_mk.integral_kernel, by
    filter_upwards [ae_ae_of_ae_comp hf.ae_eq_mk] with _ hx using integral_congr_ae hx⟩

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