convolution_integrand_fst — Mathlib

English

If f and g are suitable, then the convolution integrand with swapped arguments is measurable: L(f(x−t), g(t)) is measurable as a function of t.

Русский

Если f и g удовлетворяют условиям, то интегранд свёртки со swapped аргументами измерим как функция t: L(f(x−t), g(t)).

LaTeX

$$AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → AEStronglyMeasurable (λ t, L (f (x−t)) (g t)) μ$$

Lean4

theorem _root_.Continuous.convolution_integrand_fst [ContinuousSub G] (hg : Continuous g) (t : G) :
    Continuous fun x => L (f t) (g (x - t)) :=
  L.continuous₂.comp₂ continuous_const <| hg.comp <| continuous_id.sub continuous_const

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