outerRegular — Mathlib

English

The measure μ.measure is outer regular: for every measurable A, μ.measure(A) equals the infimum over open supersets.

Русский

Мера μ.measure является внешне регулярной: для каждого измеримого A мEинфимум по открытым надмножествах.

LaTeX

$$$$\\text{For all } A,\\ μ.measure(A) = \\inf\\{ μ.measure(U) : A \\subseteq U, U \\text{ open}\\}.$$$$

Lean4

instance outerRegular : μ.measure.OuterRegular :=
  by
  refine ⟨fun A hA r (hr : _ < _) ↦ ?_⟩
  rw [μ.measure_apply hA, outerMeasure_eq_iInf] at hr
  simp only [iInf_lt_iff] at hr
  rcases hr with ⟨U, hUo, hAU, hr⟩
  rw [← μ.outerMeasure_of_isOpen U hUo, ← μ.measure_apply hUo.measurableSet] at hr
  exact ⟨U, hAU, hUo, hr⟩

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