Theorem - 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. $\square$