Theorems · Theorem · measure theory

MeasureTheory.Measure.integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport

∀ {G : Type u_1} [inst : TopologicalSpace G] [inst_1 : Group G] [inst_2 : IsTopologicalGroup G]
  [inst_3 : MeasurableSpace G] [inst_4 : BorelSpace G] (μ' μ : MeasureTheory.Measure G) [inst_5 : μ.IsHaarMeasure]
  [inst_6 : MeasureTheory.IsFiniteMeasureOnCompacts μ'] [inst_7 : μ'.IsMulLeftInvariant] {f : G → ℝ},
  Continuous f → HasCompactSupport f → ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂μ'.haarScalarFactor μ • μ

Two left invariant measures integrate in the same way continuous compactly supported functions, up to the scalar haarScalarFactor μ' μ. See also measure_isMulInvariant_eq_smul_of_isCompact_closure, which gives the same result for compact sets, and measure_isHaarMeasure_eq_smul_of_isOpen for open sets.

Defined in

Mathlib.MeasureTheory.Measure.Haar.Unique

Cited by

4 results in Mathlib

Foundations

Depth 271 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceGroupIsTopologicalGroupMeasurableSpaceBorelSpaceMeasureTheory.Measure.IsHaarMeasureMeasureTheory.IsFiniteMeasureOnCompactsMeasureTheory.Measure.IsMulLeftInvariant