Theorems · Theorem · measure theory

MeasureTheory.Measure.eventually_nonempty_inter_smul_of_density_one

∀ {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 1) →
    ∀ (t : Set E), MeasurableSet t → μ t ≠ 0 → ∀ᶠ (r : ℝ) in nhdsWithin 0 (Set.Ioi 0), (s ∩ ({x} + r • t)).Nonempty

Consider a point x at which a set s has density one, with respect to closed balls (i.e., a Lebesgue density point of s). Then s intersects the rescaled copies {x} + r • t of a given set t with positive measure, for any small enough r.

Defined in

Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Cited by

1 results in Mathlib

Foundations

Depth 264 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceFiniteDimensionalMeasureTheory.Measure.IsAddHaarMeasure