compact groups are unimodular

Theorem - If G is a compact Hausdorff topological groupMathworldPlanetmath, then G is unimodular, i.e. it’s left and right Haar measures coincide.


Let Δ denote the modular functionMathworldPlanetmath of G. It is enough to prove that Δ is constant and equal to 1, since this proves that every left Haar measure is right invariant.

Since Δ is continuousPlanetmathPlanetmath and G is compact, Δ(G) is a compact subset of +. In particular, Δ(G) is a bounded subset of +.

But if Δ is not identically one, then there is a tG such that Δ(t)>1 (recall that Δ is an homomorphismPlanetmathPlanetmathPlanetmathPlanetmath). Hence, Δ(tn)=Δ(t)n as n increases, which is a contradictionMathworldPlanetmathPlanetmath since Δ(G) is bounded.

Title compact groups are unimodular
Canonical name CompactGroupsAreUnimodular
Date of creation 2013-03-22 17:58:23
Last modified on 2013-03-22 17:58:23
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 4
Author asteroid (17536)
Entry type Theorem
Classification msc 22C05
Classification msc 28C10