# compact groups are unimodular

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

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Proof:

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

Since $\Delta$ is continuous and $G$ is compact, $\Delta(G)$ is a compact subset of $\mathbb{R}^{+}$. In particular, $\Delta(G)$ is a bounded subset of $\mathbb{R}^{+}$.

But if $\Delta$ is not identically one, then there is a $t\in G$ such that $\Delta(t)>1$ (recall that $\Delta$ is an homomorphism). Hence, $\Delta(t^{n})=\Delta(t)^{n}\longrightarrow\infty$ as $n\in\mathbb{N}$ increases, which is a contradiction since $\Delta(G)$ is bounded. $\square$

Title compact groups are unimodular CompactGroupsAreUnimodular 2013-03-22 17:58:23 2013-03-22 17:58:23 asteroid (17536) asteroid (17536) 4 asteroid (17536) Theorem msc 22C05 msc 28C10