setIntegral_comp — Mathlib

English

For deterministic f: X → Y, the kernel composition with copy equals copy composed with deterministic: deterministic f ∥ copy equals copy composed with deterministic f.

Русский

Для детерминированной отображения копирование и композиция ядра commute.

LaTeX

$$$$ (\\text{deterministic } f) \\parallel\\! (\\text{copy } f) = (\\text{copy } f) \\circ (\\text{deterministic } f) $$$$

Lean4

theorem setIntegral_comp {f : γ → E} {s : Set γ} (hs : MeasurableSet s) (hf : IntegrableOn f s ((η ∘ₖ κ) a)) :
    ∫ z in s, f z ∂(η ∘ₖ κ) a = ∫ x, ∫ y in s, f y ∂η x ∂κ a :=
  by
  rw [← restrict_apply (η ∘ₖ κ) hs, ← comp_restrict hs, integral_comp]
  · simp_rw [restrict_apply]
  · rwa [comp_restrict, restrict_apply]

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