integral_prod_right' — Mathlib

English

The Bochner integral is measurable; the map x ↦ ∫ f(x, y) dν is measurable when f is measurable in the first argument.

Русский

Бо́хнеров интеграл измерим; отображение x ↦ ∫ f(x,y) dν измеримо при шестой переменной.

LaTeX

$$$$\\text{StronglyMeasurable}\\left( x \\mapsto \\int_y f(x,y)\\,d\\nu\\right)$$$$

Lean4

/-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
  Fubini's theorem is measurable. -/
theorem integral_prod_right' [SFinite ν] ⦃f : α × β → E⦄ (hf : StronglyMeasurable f) :
    StronglyMeasurable fun x => ∫ y, f (x, y) ∂ν := by rw [← uncurry_curry f] at hf; exact hf.integral_prod_right

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