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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.

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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 $\nu(g^{-1}S)=\nu(S)$.
The converse follow similarly.

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