compact groups are unimodular

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

Proof:

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

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

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

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