Theorems · Theorem · measure theory

VitaliFamily.ae_tendsto_rnDeriv_of_absolutelyContinuous

∀ {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, Filter.Tendsto (fun a => ρ a / μ a) (v.filterAt x) (nhds (ρ.rnDeriv μ x))

Weak version of the main theorem on differentiation of measures: given a Vitali family v for a locally finite measure μ, and another locally finite measure ρ, then for μ-almost every x the ratio ρ a / μ a converges, when a shrinks to x along the Vitali family, towards the Radon-Nikodym derivative of ρ with respect to μ. This version assumes that ρ is absolutely continuous with respect to μ. The general version without this superfluous assumption is VitaliFamily.ae_tendsto_rnDeriv.

Defined in

Mathlib.MeasureTheory.Covering.Differentiation

Cited by

1 results in Mathlib

Foundations

Depth 215 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PseudoMetricSpaceSecondCountableTopologyBorelSpaceMeasureTheory.IsLocallyFiniteMeasureMeasureTheory.IsLocallyFiniteMeasure