measurable_prod — Mathlib

English

For GroupWithZero G₀, if c ≠ 0, then AEMeasurable (λ x, c • f x) μ is equivalent to AEMeasurable f μ.

Русский

Для GroupWithZero G₀, при c ≠ 0, AEMeasurable(λx. c•f(x))μ эквивалентно AEMeasurable f μ.

LaTeX

$$$c \\neq 0 \\Rightarrow (\\text{AEMeasurable } (\\lambda x. c \\cdot f(x)) \\mu \\iff \\text{AEMeasurable } f \\mu)$$$

Lean4

@[to_additive (attr := measurability, fun_prop)]
theorem measurable_prod (l : List (α → M)) (hl : ∀ f ∈ l, Measurable f) : Measurable l.prod := by
  induction l with
  | nil => exact measurable_one
  | cons f l ihl =>
    rw [List.forall_mem_cons] at hl
    rw [List.prod_cons]
    exact hl.1.mul (ihl hl.2)

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