English
Left-parametric differentiation reduces to the right-parametric case via universe-equivalences, preserving the derivative structure.
Русский
Левая параметрическая дифференциация редуцируется к правому случаю через эквивалентности вселенных, сохраняя структуру производной.
LaTeX
$$$$\\text{contDiffOn}_{𝕜} n (g_p⋆ f) = \\text{contDiffOn}_{𝕜} n (f⋆ g_p).$$$$
Lean4
theorem integrable_posConvolution {f : ℝ → E} {g : ℝ → E'} {μ ν : Measure ℝ} [SFinite μ] [SFinite ν]
[IsAddRightInvariant μ] [NoAtoms ν] (hf : IntegrableOn f (Ioi 0) ν) (hg : IntegrableOn g (Ioi 0) μ)
(L : E →L[ℝ] E' →L[ℝ] F) : Integrable (posConvolution f g L ν) μ :=
by
rw [← integrable_indicator_iff (measurableSet_Ioi : MeasurableSet (Ioi (0 : ℝ)))] at hf hg
rw [posConvolution_eq_convolution_indicator f g L ν]
exact (hf.convolution_integrand L hg).integral_prod_left
