Theorems · Theorem · measure theory

VitaliFamily.ae_tendsto_average_norm_sub

∀ {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] {f : α → E},
  MeasureTheory.LocallyIntegrable f μ →
    ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (v.filterAt x) (nhds 0)

Lebesgue differentiation theorem: for almost every point x, the average of ‖f y - f x‖ on a tends to 0 as a shrinks to x along a Vitali family.

Defined in

Mathlib.MeasureTheory.Covering.Differentiation

Cited by

3 results in Mathlib

Foundations

Depth 255 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PseudoMetricSpaceNormedAddCommGroupSecondCountableTopologyBorelSpaceMeasureTheory.IsLocallyFiniteMeasure