convolution_congr — Mathlib

English

If y is a scalar and the smul commutes appropriately with the space, then convolving f with y g is the same as convolving y f with g.

Русский

Если y — скаляр и умножение по смулам совместимо с пространством, то f ⋆ (y g) = y (f ⋆ g).

LaTeX

$$$f⋆(lsmul\\ L\\ g) = lsmul(y) (f⋆g)$$$

Lean4

theorem convolution_congr [MeasurableAdd₂ G] [MeasurableNeg G] [SFinite μ] [IsAddRightInvariant μ] (h1 : f =ᵐ[μ] f')
    (h2 : g =ᵐ[μ] g') : f ⋆[L, μ] g = f' ⋆[L, μ] g' := by
  ext x
  apply integral_congr_ae
  exact
    (h1.prodMk <| h2.comp_tendsto (quasiMeasurePreserving_sub_left_of_right_invariant μ x).tendsto_ae).fun_comp
      ↿fun x y ↦ L x y

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