lintegral_mconv — Mathlib

English

Let s be finite and each f_i be AEMeasurable. Then the function a ↦ ∏_{i∈s} f_i(a) is AEMeasurable.

Русский

Пусть s конечное и каждый f_i — AEMeasurable. Тогда функция a ↦ ∏_{i∈s} f_i(a) является AEMeasurable.

LaTeX

$$$\forall s : \mathrm{Finset}\,\iota,\ (\forall i \in s,\ AEMeasurable (f i) ) \rightarrow AEMeasurable (\prod_{i \in s} f i) $$

Lean4

@[to_additive]
theorem lintegral_mconv [MeasurableMul₂ M] {μ ν : Measure M} [SFinite ν] {f : M → ℝ≥0∞} (hf : Measurable f) :
    ∫⁻ z, f z ∂(μ ∗ₘ ν) = ∫⁻ x, ∫⁻ y, f (x * y) ∂ν ∂μ := by
  rw [lintegral_mconv_eq_lintegral_prod hf, lintegral_prod _ (by fun_prop)]

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