Theorems · Theorem · measure theory

VitaliFamily.ae_eventually_measure_pos

∀ {α : Type u_1} [inst : PseudoMetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
  (v : VitaliFamily μ) [SecondCountableTopology α], ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a

For almost every point x, sufficiently small sets in a Vitali family around x have positive measure. (This is a nontrivial result, following from the covering property of Vitali families).

Defined in

Mathlib.MeasureTheory.Covering.Differentiation

Cited by

3 results in Mathlib

Foundations

Depth 190 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PseudoMetricSpaceSecondCountableTopology