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Given a squence of sets $\{S_{n}\}_{n=1}^{\infty}$, where each $S_{n}$ is countable set. I came over this statement in some article, it says: "Let $S$ be the weak limit of $S_{n}$". But I couldn't find a good definition for weak limit of a sequence of sets.

Any one know anything about this!

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    Is the question then "how would you define something called the weak limit of a sequence of sets"? Well, I'm going to define it to be $\varnothing$.2012-06-04

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I'll assume the $S_n$ are subsets of a Banach space $\cal B$. Then one possible interpretation of "weak limit of $S_n$" would be a set $S$ such that for every finite $\{f_1, \ldots, f_k\} \subset B^*$, $\{(f_1(s),\ldots,f_k(s)): s \in S\}$ is the limit (in the Hausdorff metric) of $\{(f_1(s),\ldots,f_k(s)): s \in S_n\}$ as $n \to \infty$.