mem_support_restrict — Mathlib

English

For an OpensMeasurableSpace, the support of μ restricted to s is characterized by frequent small sets inside nhds[x] restricted to s.

Русский

При OpensMeasurableSpace опора меры μ ограничена до s описывается через частые малые множества внутри nhds[x] ограниченного до s.

LaTeX

$$x \\in (\\mu \\restriction s).\\mathrm{support} \\iff \\mathrm{Frequent}(u \\in (\\mathcal{N}[s] x).smallSets, 0 < \\mu(u))$$

Lean4

theorem mem_support_restrict {s : Set X} {x : X} : x ∈ (μ.restrict s).support ↔ ∃ᶠ u in (𝓝[s] x).smallSets, 0 < μ u :=
  by
  rw [nhds_basis_opens x |>.mem_measureSupport, (nhdsWithin_basis_open x s).frequently_smallSets pos_mono]
  grind [IsOpen.measurableSet, restrict_apply]

Похожие утверждения