Just looked at a result saying that for a probablity measure $\mu$ defined on $Q$, and for a monotone increasing sequence $\lbrace E_n \rbrace $: $E_n \subset E_{n+1}$, $E_n \in Q$,
(1) $\lim\limits_{n}\mu(E_n)=\mu(\lim\limits_{n}E_n)$.
By another elementary result, one also has that $\mu(E_n) \leq \mu(E_{n+1})$ (have been shown prior to above on occasions). In showing (1), none seem to employ the latter, for I thought it was not much more to say (and once the inequalities been established) than $E=\lim\limits_{n}E_n$ exists and that $\lbrace \mu(E_n) \rbrace$ is an increasing sequence of real numbers bounded by $\mu(E)$, so by the "monotone convergence theorem for real numbers" it converges to $\mu(E)$. Am I missing some?
Edit: As pointed out by saz in the comments, $\mu(E_n)$ is bounded by $\mu(E)$, but we don't know that $\mu(E)=\text{sup} \lbrace \mu(E_n): n=1,2,3... \rbrace$. All in all, the suggestion fails.