Invariant measures. Let \((X, \mathscr{A}, \mu)\) be a finite measure space where \(\mathscr{A}=\sigma(\mathscr{G})\) for some \(\cap\)-stable generator \(\mathscr{G}\). Assume that \(\theta: X ightarrow X\) is a map such that \(\theta^{-1}(A) \in \mathscr{A}\) for all \(A \in \mathscr{A}\). Prove that

\[\mu(G)=\mu\left(\theta^{-1}(G)ight) \quad \forall G \in \mathscr{G} \Longrightarrow \mu(A)=\mu\left(\theta^{-1}(A)ight) \quad \forall A \in \mathscr{A}\]

(A measure \(\mu\) with this property is called invariant w.r.t. the map \(\theta\).)