Theorems · Theorem · measure theory

MeasureTheory.Measure.measure_isMulLeftInvariant_eq_smul_of_ne_top

∀ {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]
  [μ.InnerRegularCompactLTTop] [μ'.InnerRegularCompactLTTop] {s : Set G},
  μ s ≠ ⊤ → μ' s ≠ ⊤ → μ' s = μ'.haarScalarFactor μ • μ s

Uniqueness of left-invariant measures: Given two left-invariant measures which are finite on compacts and inner regular for finite measure sets with respect to compact sets, they coincide in the following sense: they give the same value to finite measure sets, up to a multiplicative constant.

Defined in

Mathlib.MeasureTheory.Measure.Haar.Unique

Cited by

2 results in Mathlib

Foundations

Depth 277 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceGroupIsTopologicalGroupMeasurableSpaceBorelSpaceLocallyCompactSpaceMeasureTheory.Measure.IsHaarMeasureMeasureTheory.IsFiniteMeasureOnCompactsMeasureTheory.Measure.IsMulLeftInvariantMeasureTheory.Measure.InnerRegularCompactLTTopMeasureTheory.Measure.InnerRegularCompactLTTop