Theorems · Theorem · measure theory

MeasureTheory.Measure.div_mem_nhds_one_of_haar_pos_ne_top

∀ {G : Type u_1} [inst : Group G] [inst_1 : TopologicalSpace G] [IsTopologicalGroup G] [inst_3 : MeasurableSpace G]
  [BorelSpace G] (μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [LocallyCompactSpace G] [μ.InnerRegularCompactLTTop]
  (E : Set G), MeasurableSet E → 0 < μ E → μ E ≠ ⊤ → E / E ∈ nhds 1

Steinhaus Theorem for finite mass sets. In any locally compact group G with a Haar measure μ that's inner regular on finite measure sets, for any measurable set E of finite positive measure, the set E / E is a neighbourhood of 1.

Defined in

Mathlib.MeasureTheory.Measure.Haar.Basic

Cited by

1 results in Mathlib

Foundations

Depth 204 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

GroupTopologicalSpaceIsTopologicalGroupMeasurableSpaceBorelSpaceMeasureTheory.Measure.IsHaarMeasureLocallyCompactSpaceMeasureTheory.Measure.InnerRegularCompactLTTop