Let \(X\) be a set, let \(\left(X_{i}, \mathscr{A}_{i}ight), i \in I\), be arbitrarily many measurable spaces and let \(T_{i}: X ightarrow X_{i}\) be a family of maps.

(i) Show that for every \(i \in I\) the smallest \(\sigma\)-algebra in \(X\) that makes \(T_{i}\) measurable is given by \(T_{i}^{-1}\left(\mathscr{A}_{i}ight)\).

(ii) Show that \(\sigma\left(\bigcup_{i \in I} T_{i}^{-1}\left(\mathscr{A}_{i}ight)ight)\) is the smallest \(\sigma\)-algebra in \(X\) that makes all \(T_{i}, i \in I\), simultaneously measurable.