continuous epimorphism of compact groups preserves Haar measure


right \PMlinkescapephrasepreserves

Theorem - Let G,H be compactPlanetmathPlanetmath HausdorffPlanetmathPlanetmath topological groupsMathworldPlanetmath. If ϕ:GH is a continuousPlanetmathPlanetmath surjectivePlanetmathPlanetmath homomorphismPlanetmathPlanetmathPlanetmathPlanetmath, then ϕ is a measure preserving transformation, in the sense that it preserves the normalized Haar measure.

: Let μ be the Haar measure in G (normalized, i.e. μ(G)=1). Let ν be defined for measurable subsets E of H by


It is easy to see that ν defines a measureMathworldPlanetmath in H. Let us now see that ν is invariant under right translations. For every sG and every measurable subset EH we have that

ϕ-1(ϕ(s)E)=sϕ-1(E) (1)

The inclusion is obvious. To prove the other inclusion notice that if zϕ-1(ϕ(s)E) then ϕ(z)=ϕ(s)t for some tE. Hence, ϕ(s-1z)=t, i.e s-1zϕ-1(E). It now follows that z=s(s-1z)sϕ-1(E).

Thus, equality (1) and the fact that μ is a Haar measure imply that


Since ϕ is surjective it follows that ν is right invariant. It is not difficult to see that ν is regularPlanetmathPlanetmathPlanetmathPlanetmath, finite on compact sets and ν(H)=1. Hence, ν is the normalized Haar measure in H and, by definition, we have that


Thus, ϕ preserves the Haar measure.

Title continuous epimorphism of compact groups preserves Haar measure
Canonical name ContinuousEpimorphismOfCompactGroupsPreservesHaarMeasure
Date of creation 2013-03-22 17:59:06
Last modified on 2013-03-22 17:59:06
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 9
Author asteroid (17536)
Entry type Theorem
Classification msc 37A05
Classification msc 28C10
Classification msc 22C05