convolution_integrand — Mathlib

English

Given f and g measurable, the convolution integrand with swapped arguments is measurable and integrable on the product measure.

Русский

С учетом измеримости f и g, интегрант конволюции со swapped аргументами измерим и интегрируем на произведенной мере.

LaTeX

$$AEStronglyMeasurable (λ p, L (f (p.2)) (g (p.1 − p.2))) (μ.prod ν)$$

Lean4

theorem convolution_integrand (hf : AEStronglyMeasurable f ν) (hg : AEStronglyMeasurable g μ) :
    AEStronglyMeasurable (fun p : G × G => L (f p.2) (g (p.1 - p.2))) (μ.prod ν) :=
  hf.convolution_integrand' L <| hg.mono_ac (quasiMeasurePreserving_sub_of_right_invariant μ ν).absolutelyContinuous

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