Theorems · Theorem · measure theory

tendsto_measure_cthickening_of_isCompact

∀ {α : Type u_1} [inst : MetricSpace α] [inst_1 : MeasurableSpace α] [OpensMeasurableSpace α] [ProperSpace α]
  {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasureOnCompacts μ] {s : Set α},
  IsCompact s → Filter.Tendsto (fun r => μ (Metric.cthickening r s)) (nhds 0) (nhds (μ s))

Given a compact set in a proper space, the measure of its r-closed thickenings converges to its measure as r tends to 0.

Defined in

Mathlib.MeasureTheory.Constructions.BorelSpace.Metric

Cited by

1 results in Mathlib

Foundations

Depth 207 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MetricSpaceMeasurableSpaceOpensMeasurableSpaceProperSpaceMeasureTheory.IsFiniteMeasureOnCompacts