Theorems · Theorem · measure theory

tendsto_measure_thickening

∀ {α : Type u_1} [inst : PseudoEMetricSpace α] [inst_1 : MeasurableSpace α] [OpensMeasurableSpace α]
  {μ : MeasureTheory.Measure α} {s : Set α},
  (∃ R > 0, μ (Metric.thickening R s) ≠ ⊤) →
    Filter.Tendsto (fun r => μ (Metric.thickening r s)) (nhdsWithin 0 (Set.Ioi 0)) (nhds (μ (closure s)))

If a set has a thickening with finite measure, then the measures of its r-thickenings converge to the measure of its closure as r > 0 tends to 0.

Defined in

Mathlib.MeasureTheory.Constructions.BorelSpace.Metric

Cited by

2 results in Mathlib

Foundations

Depth 205 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PseudoEMetricSpaceMeasurableSpaceOpensMeasurableSpace