Claim: Let $Z$, $Z_{1}$, $Z_{2}$, $\dots$ be random variables. Suppose $P(|Z_{n}-Z|\geq\varepsilon\text{ i.o.})=0$ for each $\varepsilon>0$. Then $P(Z_{n}\to Z)=1$.
Proof: $P(Z_{n}\to Z)=P(\forall\varepsilon>0,|Z_{n}-Z|<\varepsilon\text{ a.a.})=\cdots$
The argument of $P$ is a set. How do I interpret "$\forall\varepsilon>0,|Z_{n}-Z|<\varepsilon\text{ a.a.}$" as a set?
$"\forall \varepsilon>0, |Z_n-Z|<\varepsilon a.a."$ is really the set
$$ \cap_{\varepsilon>0} \cup_{N=1}^{\infty} \cap_{n=N}^{\infty} (|Z_n-Z|\leq \varepsilon)= \cap_{m=1}^{\infty} \cup_{N=1}^{\infty} \cap_{n=N}^{\infty} (|Z_n-Z|\leq \frac{1}{m}), $$
with the latter rewriting to emphasise that this is, in fact, a measurable set.