Theorems · Theorem · measure theory

MeasureTheory.Measure.haarMeasure_unique

∀ {G : Type u_1} [inst : Group G] [inst_1 : TopologicalSpace G] [inst_2 : IsTopologicalGroup G]
  [inst_3 : MeasurableSpace G] [inst_4 : BorelSpace G] [SecondCountableTopology G] (μ : MeasureTheory.Measure G)
  [MeasureTheory.SigmaFinite μ] [μ.IsMulLeftInvariant] (K₀ : TopologicalSpace.PositiveCompacts G),
  μ = μ ↑K₀ • MeasureTheory.Measure.haarMeasure K₀

Uniqueness of left-invariant measures: In a second-countable locally compact group, any σ-finite left-invariant measure is a scalar multiple of the Haar measure. This is slightly weaker than assuming that μ is a Haar measure (in particular we don't require μ ≠ 0). See also isMulLeftInvariant_eq_smul_of_regular for a statement not assuming second-countability.

Defined in

Mathlib.MeasureTheory.Measure.Haar.Basic

Cited by

4 results in Mathlib

Foundations

Depth 234 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

GroupTopologicalSpaceIsTopologicalGroupMeasurableSpaceBorelSpaceSecondCountableTopologyMeasureTheory.SigmaFiniteMeasureTheory.Measure.IsMulLeftInvariant