outerMeasure_lt_top_of_isCompact — Mathlib

English

If G is weakly locally compact and K is compact, then μ.outerMeasure(K) < ∞.

Русский

Если G слаб locally компактно, и K компактно, то μ.outerMeasure(K) < ∞.

LaTeX

$$$$K \\text{ компакт } \\Rightarrow μ.outerMeasure(K) < \\infty.$$$$

Lean4

theorem outerMeasure_lt_top_of_isCompact [WeaklyLocallyCompactSpace G] {K : Set G} (hK : IsCompact K) :
    μ.outerMeasure K < ∞ := by
  rcases exists_compact_superset hK with ⟨F, h1F, h2F⟩
  calc
    μ.outerMeasure K ≤ μ.outerMeasure (interior F) := measure_mono h2F
    _ ≤ μ ⟨F, h1F⟩ := by apply μ.outerMeasure_le ⟨interior F, isOpen_interior⟩ ⟨F, h1F⟩ interior_subset
    _ < ⊤ := μ.lt_top _

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