integrable_rnDeriv_smul_iff — Mathlib

English

For left-invariant groups and Lebesgue-decomposed measures, the RN-derivative of the convolution μ ∗ ν with respect to μ equals the product of RN-derivatives with respect to μ, a.e. under μ.

Русский

Для сводимого к свёртке μ ∗ ν плотности Радона–Никодима относительно μ равны добротности, произведение RN-дерививов даёт плотность конволюции.

LaTeX

$$$$ (ν₁ ∗ₘ ν₂).rnDeriv μ =^\mathrm{ae}_{μ} (ν₁.rnDeriv μ) ⋆ₘₗ[μ] (ν₂.rnDeriv μ) $$$$

Lean4

theorem integrable_rnDeriv_smul_iff (hμν : μ ≪ ν) :
    Integrable (fun x ↦ (μ.rnDeriv ν x).toReal • f x) ν ↔ Integrable f μ :=
  by
  nth_rw 2 [← withDensity_rnDeriv_eq μ ν hμν]
  rw [← integrable_withDensity_iff_integrable_smul' (E := E) (measurable_rnDeriv μ ν) (rnDeriv_lt_top μ ν)]

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