Theorems · Theorem · probability

MeasureTheory.measure_eq_measure_of_levyProkhorovEDist_eq_zero_of_isClosed

∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] [inst_1 : PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω]
  {μ ν : MeasureTheory.Measure Ω},
  MeasureTheory.levyProkhorovEDist μ ν = 0 →
    ∀ {s : Set Ω},
      IsClosed s → (∃ δ > 0, μ (Metric.thickening δ s) ≠ ⊤) → (∃ δ > 0, ν (Metric.thickening δ s) ≠ ⊤) → μ s = ν s

Two measures at vanishing Lévy-Prokhorov distance from each other assign the same values to all closed sets.

Defined in

Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Cited by

0 results in Mathlib

Foundations

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

Assumes

MeasurableSpacePseudoEMetricSpaceOpensMeasurableSpace