Theorems · Theorem · measure theory

VitaliFamily.ae_tendsto_div

∀ {α : Type u_1} [inst : PseudoMetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
  (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ]
  {ρ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure ρ],
  ρ.AbsolutelyContinuous μ → ∀ᵐ (x : α) ∂μ, ∃ c, Filter.Tendsto (fun a => ρ a / μ a) (v.filterAt x) (nhds c)

If ρ is absolutely continuous with respect to μ, then for almost every x, the ratio ρ a / μ a converges as a shrinks to x along a Vitali family for μ.

Defined in

Mathlib.MeasureTheory.Covering.Differentiation

Cited by

2 results in Mathlib

Foundations

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

Assumes

PseudoMetricSpaceSecondCountableTopologyBorelSpaceMeasureTheory.IsLocallyFiniteMeasureMeasureTheory.IsLocallyFiniteMeasure