Let $G$ be a Lie group, and let $\mathbb R_+=(0,\infty)$ be regarded as a multiplicative group. Fix a left Haar measure $\mu$ on $G$. Define the modular function $\Delta:G\to\mathbb R_+$ by the requirement that, for every $g\in G$,
where $R_g:G\to G$ is the right translation map $R_g(x)=xg$. Then $\Delta$ is a continuous [group homomorphism](/page/Group%20Homomorphism). Moreover, $G$ admits a Haar measure that is both left invariant and right invariant if and only if $\Delta(g)=1$ for every $g\in G$.