Suppose I have a collection of subsets, $\{ A_i \}_{i \ge 1}$, all of which are subsets of some set $S$.
Suppose I have a measure on subsets of $S$: a non-negative function $f$ of the form $f(A)=\sum_{a \in A} g(a)$ where $g$ in non-negative. $f$ can attain infinity, but $g$ can't.
If $f(A_i)$ is monotone increasing, what is the relationship between $\lim f(A_i)$ and $f(\limsup A_i), f(\liminf A_i)$? What about the case where $\lim A_i$ exists?