Theorems · Theorem · measure theory

MeasureTheory.measurePreserving_prod_mul_swap

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

The map (x, y) ↦ (y, yx) sends the measure μ × ν to ν × μ. This is the map SR in [Halmos, §59]. S is the map (x, y) ↦ (x, xy) and R is Prod.swap.

Defined in

Mathlib.MeasureTheory.Group.Prod

Cited by

2 results in Mathlib

Foundations

Depth 224 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceGroupMeasurableMul₂MeasureTheory.SFiniteMeasureTheory.SFiniteMeasureTheory.Measure.IsMulLeftInvariant