Unimodular locally compact group

Assume AC and let $G$ be an LCH group with fixed left Haar measure $\mu$ and modular function ${\mathrm{\Delta }}_G$ (Modular function of a locally compact group). The group $G$ is unimodular when ${\mathrm{\Delta }}_G(g)=1\text{for every }g\in G\mathrm{.}$ Equivalently, by the defining identity of the modular function, $G$ is unimodular exactly when the fixed left Haar measure $\mu$ satisfies ${\int }_{G}f(xg)d\mu (x)={\int }_{G}fd\mu (x)(f\in C_c(G),g\in G),$ that is, exactly when $\mu$ is right invariant and hence is also a right Haar measure. Unimodularity is a property of the group alone, because the modular function does not depend on the normalisation of $\mu$.

• Right invariance is the same condition. A left Haar measure $\mu$ is right invariant precisely when $c(g)=1$ for all $g$, since $c(g)$ was defined as the unique scalar with $\int f(xg)d\mu =c(g)\int fd\mu$. Thus the group is unimodular exactly when some, equivalently every, left Haar measure is right invariant.
• Choice cost. The equivalence above is a rewriting of the defining identity and needs no selection; the existence of $\mu$, and with it of ${\mathrm{\Delta }}_G$, is what carries AC, through the declared dependency.
• Naming. A nonunimodular group is one with ${\mathrm{\Delta }}_G\equiv ̸1$, and a left Haar measure of a nonunimodular group is never right invariant. The companion page exhibits the positive affine group as such an example.