Theorems · Theorem · measure theory

MeasureTheory.measurePreserving_prod_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] [ν.IsMulLeftInvariant],
  MeasureTheory.MeasurePreserving (fun z => (z.1, z.1 * z.2)) (μ.prod ν) (μ.prod ν)

The multiplicative shear mapping (x, y) ↦ (x, xy) preserves the measure μ × ν. This condition is part of the definition of a measurable group in [Halmos, §59]. There, the map in this lemma is called S.

Defined in

Mathlib.MeasureTheory.Group.Prod

Cited by

3 results in Mathlib

Foundations

Depth 217 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceGroupMeasurableMul₂MeasureTheory.SFiniteMeasureTheory.SFiniteMeasureTheory.Measure.IsMulLeftInvariant