mapₗ_congr — Mathlib

English

Under IsFiniteMeasure μ and SigmaFinite ν, absolute continuity μ ≪ ν ensures the tilted llr is μ-integrable given integrability hypotheses on f and llr.

Русский

Пусть μ конечна, ν сигма-конечна и μ ≪ ν. При условии интегрируемости f и llr выполняется интегрируемость llr(μ.tilted f, ν) по μ.

LaTeX

$$$\\text{Integrable}(\\mathrm{llr}(\\mu\\tilted f, \\nu))_{\\mu}$$$

Lean4

theorem mapₗ_congr {f g : α → β} (hf : Measurable f) (hg : Measurable g) (h : f =ᵐ[μ] g) : mapₗ f μ = mapₗ g μ :=
  by
  ext1 s hs
  simpa only [mapₗ, hf, hg, hs, dif_pos, liftLinear_apply, OuterMeasure.map_apply] using measure_congr (h.preimage s)

Похожие утверждения