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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 $\lim f(A_i)=+\infty$.2012-07-21
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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 $0$.2012-07-21

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