Theorems · Theorem · measure theory

MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : MeasurableSpace E] [BorelSpace E]
  [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (s : Set E) (x : E),
  Filter.Tendsto (fun r => μ (s ∩ Metric.closedBall x r) / μ (Metric.closedBall x r)) (nhdsWithin 0 (Set.Ioi 0))
      (nhds 0) →
    ∀ (t : Set E),
      MeasurableSet t →
        μ t ≠ ⊤ → Filter.Tendsto (fun r => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (nhdsWithin 0 (Set.Ioi 0)) (nhds 0)

Consider a point x at which a set s has density zero, with respect to closed balls. Then it also has density zero with respect to any measurable set t: the proportion of points in s belonging to a rescaled copy {x} + r • t of t tends to zero as r tends to zero.

Defined in

Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Cited by

1 results in Mathlib

Foundations

Depth 261 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceFiniteDimensionalMeasureTheory.Measure.IsAddHaarMeasure