Theorems · Theorem · measure theory

IsCompact.measure_eq_iInf_isOpen

∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} [inst_1 : TopologicalSpace α]
  [μ.InnerRegularCompactLTTop] [MeasureTheory.IsLocallyFiniteMeasure μ] [R1Space α] [BorelSpace α] {K : Set α},
  IsCompact K → μ K = ⨅ U, ⨅ (_ : K ⊆ U), ⨅ (_ : IsOpen U), μ U

If μ is inner regular for finite measure sets with respect to compact sets and is locally finite in an R₁ space, then any compact set can be approximated from outside by open sets.

Defined in

Mathlib.MeasureTheory.Measure.Regular

Cited by

0 results in Mathlib

Foundations

Depth 201 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceTopologicalSpaceMeasureTheory.Measure.InnerRegularCompactLTTopMeasureTheory.IsLocallyFiniteMeasureR1SpaceBorelSpace