Theorems · Theorem · measure theory

MeasureTheory.Measure.measure_isMulInvariant_eq_smul_of_isCompact_closure

∀ {G : Type u_1} [inst : TopologicalSpace G] [inst_1 : Group G] [inst_2 : IsTopologicalGroup G]
  [inst_3 : MeasurableSpace G] [inst_4 : BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G)
  [inst_6 : μ.IsHaarMeasure] [inst_7 : MeasureTheory.IsFiniteMeasureOnCompacts μ'] [inst_8 : μ'.IsMulLeftInvariant]
  {s : Set G}, IsCompact (closure s) → μ' s = μ'.haarScalarFactor μ • μ s

Uniqueness of left-invariant measures: Given two left-invariant measures which are finite on compacts, they coincide in the following sense: they give the same value to sets with compact closure, up to the multiplicative constant haarScalarFactor μ' μ.

Defined in

Mathlib.MeasureTheory.Measure.Haar.Unique

Cited by

4 results in Mathlib

Foundations

Depth 276 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceGroupIsTopologicalGroupMeasurableSpaceBorelSpaceLocallyCompactSpaceMeasureTheory.Measure.IsHaarMeasureMeasureTheory.IsFiniteMeasureOnCompactsMeasureTheory.Measure.IsMulLeftInvariant