Let \(X=\mathbb{Z}=\{0, \pm 1, \pm 2, \ldots\}\). Show that

(i) \(\mathscr{A}:=\{A \subset \mathbb{Z} \mid \forall n>0: 2 n \in A \Longleftrightarrow 2 n+1 \in A\}\) is a \(\sigma\)-algebra;

(ii) \(T: \mathbb{Z} ightarrow \mathbb{Z}, T(n):=n+2\) is \(\mathscr{A} / \mathscr{A}\)-measurable and bijective, but \(T^{-1}\) is not measurable.