measurable_edist_left — Mathlib

English

If s is closed, and there exists R > 0 with μ(cthickening R s) finite, then μ(cthickening r s) tends to μ(s) as r → 0.

Русский

Если s замкнуто, и существует R > 0 с конечной мерой μ(cthickening(R,s)), то μ(cthickening(r,s)) стремится к μ(s) при r → 0.

LaTeX

$$$\\text{IsClosed}(s) \\Rightarrow \\operatorname{Tendsto}\\left(r \\mapsto μ(\\mathrm{cthickening}(r,s))\\right)(\\mathcal{N}_0)(\\mathcal{N}_{μ(s)})$$$

Lean4

@[measurability, fun_prop]
theorem measurable_edist_left : Measurable fun y => edist y x :=
  (continuous_id.edist continuous_const).measurable

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