Let \(\mu, u\) be \(\sigma\)-finite measures on \((\mathbb{R}, \mathscr{B}(\mathbb{R}))\). Show that

(i) The set \(D:=\{x \in \mathbb{R}: \mu\{x\}>0\}\) is at most countable.

(ii) The diagonal \(\Delta \subset \mathbb{R}^{2}\) has measure \(\mu \times u(\Delta)=\sum_{x \in D} \mu\{x\} u\{x\}\).