Definition 4

The number $a > 0$ such that $\varphi^{-1}(\mu) = a \mu$ is called the modulus of the automorphism $\varphi$ and is denoted $\operatorname{mod}_G \varphi$ or simply $\operatorname{mod} \varphi$.

If $f$ is a $\mu$-integrable function on G, then

If A is a $\mu$-integrable subset of G, then

In particular, for $s \in G$, let $i_s$ be the inner automorphism $x \mapsto s^{-1} x s$. Then $i_s^{-1} = \delta(s) \gamma(s)$, therefore

consequently

If G is either discrete or compact, then its normalized Haar measure is transformed into itself by every automorphism $\varphi$ of G, as one sees immediately by transport of structure. Thus an automorphism of a discrete or compact group has modulus 1.