Theorems · Theorem · measure theory

VitaliFamily.ae_tendsto_average

∀ {α : Type u_1} [inst : PseudoMetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
  (v : VitaliFamily μ) {E : Type u_2} [inst_1 : NormedAddCommGroup E] [SecondCountableTopology α] [BorelSpace α]
  [MeasureTheory.IsLocallyFiniteMeasure μ] [inst_5 : NormedSpace ℝ E] [CompleteSpace E] {f : α → E},
  MeasureTheory.LocallyIntegrable f μ →
    ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun a => ⨍ (y : α) in a, f y ∂μ) (v.filterAt x) (nhds (f x))

Lebesgue differentiation theorem: for almost every point x, the average of f on a tends to f x as a shrinks to x along a Vitali family.

Defined in

Mathlib.MeasureTheory.Covering.Differentiation

Cited by

2 results in Mathlib

Foundations

Depth 256 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PseudoMetricSpaceNormedAddCommGroupSecondCountableTopologyBorelSpaceMeasureTheory.IsLocallyFiniteMeasureNormedSpaceCompleteSpace