Theorems · Theorem · measure theory

MeasureTheory.Measure.div_mem_nhds_one_of_haar_pos

∀ {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.InnerRegular] (E : Set G),
  MeasurableSet E → 0 < μ E → E / E ∈ nhds 1

Steinhaus Theorem. In any locally compact group G with an inner regular Haar measure μ, for any measurable set E of positive measure, the set E / E is a neighbourhood of 1.

Defined in

Mathlib.MeasureTheory.Measure.Haar.Basic

Cited by

0 results in Mathlib

Foundations

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

Assumes

GroupTopologicalSpaceIsTopologicalGroupMeasurableSpaceBorelSpaceMeasureTheory.Measure.IsHaarMeasureLocallyCompactSpaceMeasureTheory.Measure.InnerRegular