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单词 CompactGroupsAreUnimodular
释义

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.

Proof:

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.

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更新时间:2025/5/4 13:07:38