measurableSet_kernel_integrable — Mathlib

English

Under suitable hypotheses, condExp of a trajectory functional can be represented via a conditional trajectory decomposition with respect to a filtration and a product kernel.

Русский

При подходящих гипотезах условное ожидание функционала траектории представимо через разложение траектории относительно фильтрации и произведения ядер.

LaTeX

$$$\\text{condExp}_{\\mathcal{F}}(f) = \\text{comp}(\\text{traj}, \\text{partialTraj})$$$

Lean4

theorem measurableSet_kernel_integrable ⦃f : α → β → E⦄ (hf : StronglyMeasurable (uncurry f)) :
    MeasurableSet {x | Integrable (f x) (κ x)} :=
  by
  simp_rw [Integrable, hf.of_uncurry_left.aestronglyMeasurable, true_and]
  exact measurableSet_lt (Measurable.lintegral_kernel_prod_right hf.enorm) measurable_const

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