.Let \((X, \mathscr{A})\) be a measurable space.

(i). Let \(\mu, u\) be two measures on \((X, \mathscr{A})\). Show that the set function \(ho(A):=a \mu(A)+\) \(b u(A), A \in \mathscr{A}\), for all \(a, b \geqslant 0\) is again a measure.

(ii). Let \(\mu_{1}, \mu_{2}, \ldots\) be countably many measures on \((X, \mathscr{A})\) and let \(\left(\alpha_{i}ight)_{i \in \mathbb{N}}\) be a sequence of positive numbers. Show that \(\mu(A):=\sum_{i=1}^{\infty} \alpha_{i} \mu_{i}(A), A \in \mathscr{A}\), is again a measure.

[ to show \(\sigma\)-additivity use (and prove) the following helpful lemma. For any double sequence \(\beta_{i k}, i, k \in \mathbb{N}\), of real numbers we have

\[\sup _{i \in \mathbb{N}} \sup _{k \in \mathbb{N}} \beta_{i k}=\sup _{k \in \mathbb{N}} \sup _{i \in \mathbb{N}} \beta_{i k}\]

Thus \(\lim _{i ightarrow \infty} \lim _{k ightarrow \infty} \beta_{i k}=\lim _{k ightarrow \infty} \lim _{i ightarrow \infty} \beta_{i k}\) if \(i \mapsto \beta_{i k}\), and \(k \mapsto \beta_{i k}\) increases when the other index is fixed.]