# definition of unimodular group

Suppose $$G$$ is a locally compact group,we call $$G$$ an unimodular group if the modular function $$\Delta =1$$,that is to say,if the left Haar measure is also a right Haar measure.

How to show that "$$\Delta =1$$ if and only if the left Haar measure is also a right Haar measure.

By definition of the modular function, if $$\nu$$ is a right Haar measure, then for every $$g\in G$$, we have $$\nu(g^{-1}S)=\Delta(g)\nu(S)$$ for all open $$S\subseteq G$$.
Now if $$\Delta \equiv 1$$, then $$\nu$$ becomes left invariant, too, since $$\nu(g^{-1}S)=\nu(S)$$.