Unimodular group

Remarks

Example: every compact group $K$ is unimodular. The modular function is a continuous homomorphism, so $\Delta(K)$ is a compact subgroup of $(0, \infty)$. The only compact subgroup of $(0, \infty)$ is $\set{1}$, so $\Delta \equiv 1$.

Non-example: the group of maps $x \mapsto ax + b$ with $a > 0$ and $b \in \R$ has modular function $\Delta(a, b) = 1/a$, which is not identically $1$.

Every abelian group is unimodular, since $Ag = gA$ for all $g$. Every discrete group is unimodular, since counting measure is invariant on both sides. On a unimodular group, $\int_G f(x^{-1}) \, d\mu(x) = \int_G f(x) \, d\mu(x)$ for every integrable $f$.