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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?

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If $S=\Bbb N$ and $g\equiv 1$ (counting measure), if we allow $f(A_i)=+\infty$ consider $A_i=\{k,k\geq i\}$. $\limsup A_i=\liminf A_i=\emptyset$ but $\lim f(A_i)=+\infty$. –  Davide Giraudo Jul 21 '12 at 10:34
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Even if $f(A_i)$ is finite (take $A_i=\{i,\dots,2i\}$, $\lim f(A_i)$ is infinite, but the measure of $\limsup$ and $\liminf$ is $0$. –  Davide Giraudo Jul 21 '12 at 11:28

1 Answer 1

To get at least some positive result, think of $f(A_i)$ as the integral of characteristic function $\chi_{A_i}$; the convergence of sets is readily interpreted as convergence of characteristic functions. Fatou's Lemma, Monotone convergence theorem and Dominated convergence theorem give you some information, though you probably knew that already. The monotonicity of $f(A_i)$ does not appear to help much.

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