Theorems · Theorem · measure theory

MeasureTheory.Measure.exists_integral_isAddLeftInvariant_eq_smul_of_hasCompactSupport

∀ {G : Type u_1} [inst : TopologicalSpace G] [inst_1 : AddGroup G] [IsTopologicalAddGroup G]
  [inst_3 : MeasurableSpace G] [BorelSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure]
  [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsAddLeftInvariant],
  ∃ c, ∀ (f : G → ℝ), Continuous f → HasCompactSupport f → ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂c • μ

Given two left-invariant measures which are finite on compacts, they coincide in the following sense: they give the same value to the integral of continuous compactly supported functions, up to a multiplicative constant.

Defined in

Mathlib.MeasureTheory.Measure.Haar.Unique

Cited by

4 results in Mathlib

Foundations

Depth 269 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceAddGroupIsTopologicalAddGroupMeasurableSpaceBorelSpaceMeasureTheory.Measure.IsAddHaarMeasureMeasureTheory.IsFiniteMeasureOnCompactsMeasureTheory.Measure.IsAddLeftInvariant