(i) State Tonelli's and Fubini's theorems for spaces of sequences, i.e. for the measure space \((\mathbb{N}, \mathscr{P}(\mathbb{N}), \mu)\), where \(\mu:=\sum_{n \in \mathbb{N}} \delta_{n}\), and obtain criteria specifying when one can interchange two infinite summations.

(ii) Using similar considerations to those in part (i) deduce the following.

Lemma. Let \(\left(A_{n}ight)_{n}\) be countably many mutually disjoint sets whose union is \(\mathbb{N}\), and let \(\left(x_{k}ight)_{k \in \mathbb{N}} \subset \mathbb{R}\) be a sequence. Then

\[\sum_{k \in \mathbb{N}} x_{k}=\sum_{n} \sum_{k \in A_{n}} x_{k}\]