Theorems · Theorem · measure theory

MeasureTheory.measurePreserving_prod_inv_mul

∀ {G : Type u_1} [inst : MeasurableSpace G] [inst_1 : Group G] [MeasurableMul₂ G] (μ ν : MeasureTheory.Measure G)
  [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableInv G] [ν.IsMulLeftInvariant],
  MeasureTheory.MeasurePreserving (fun z => (z.1, z.1⁻¹ * z.2)) (μ.prod ν) (μ.prod ν)

The map (x, y) ↦ (x, x⁻¹y) is measure-preserving. This is the function S⁻¹ in [Halmos, §59], where S is the map (x, y) ↦ (x, xy).

Defined in

Mathlib.MeasureTheory.Group.Prod

Cited by

2 results in Mathlib

Foundations

Depth 218 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceGroupMeasurableMul₂MeasureTheory.SFiniteMeasureTheory.SFiniteMeasurableInvMeasureTheory.Measure.IsMulLeftInvariant