Let \(\mu\) be a bounded measure on the measure space \(([0, \infty), \mathscr{B}[0, \infty))\).

(i) Show that \(A \in \mathscr{B}[0, \infty) \otimes \mathscr{P}(\mathbb{N})\) if, and only if, \(A=\bigcup_{n \in \mathbb{N}} B_{n} \times\{n\}\), where \(\left(B_{n}ight)_{n \in \mathbb{N}} \subset\) \(\mathscr{B}[0, \infty)\).

(ii) Show that there exists a unique measure \(\pi\) on \(\mathscr{B}[0, \infty) \otimes \mathscr{P}(\mathbb{N})\) satisfying

\[\pi(B \times\{n\})=\int_{B} e^{-t} \frac{t^{n}}{n !} \mu(d t)\]