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In my measure theory book I came across the following definition: Let $(A_n)_{n\ge1}$ be a sequence of subsets of some set $X$. Define:

$\limsup_{n\to\infty} A_n:=\bigcap_{n\ge1}\bigcup_{k\ge n}A_k$

$\liminf_{n\to\infty} A_n:=\bigcup_{n\ge1}\bigcap_{k\ge n}A_k$

Call the sequence convergent if $\limsup_{n\to\infty} A_n=\liminf_{n\to\infty} A_n$ , in which case we define $\lim_{n\to\infty} A_n:=\limsup_{n\to\infty} A_n$

My question is, does this notion of convergence correspond to some sort of metric on the set of subsets of $X$, or is it completely unrelated to the usual concept of a limit? Thanks

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Why should it be metric? One can define limits for sequences in topological spaces. –  user59671 Feb 13 '13 at 20:54
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2 Answers 2

The usual real limit can be phrased in terms of this language. Suppose $(a_n)$ is a real sequence and define $A_n := (-\infty,a_n]$. Then we have $$\sup \left ( {\lim \sup}_{n \to \infty} A_n \right) = {\lim \sup}_{n \to \infty} a_n$$ and similarly for limes inferior and limit. Informally, the usual convergence can be formulated in terms of set convergence of rays of real numbers.

Nevertheless, the set convergence is much more general and requires no additional structure. You can formulate it for any collection of sets whatsoever (not just sequences). Actually note that not every notion of convergence is topologizable (and so a fortiori not metrizable). So in general you shouldn't expect that there is a metric involved where convergence is.

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I agree with @Marek's answer that we do not really need to look at it topologically at all (there is a general notion of convergence spaces, e.g.). There is however one context where it does correspond to a metric convergence: the so-called Hausdorff metric on the hyperspace of a compact metric space (which is the set of non-empty closed subsets of that space, a topological analogue of a powerset). IIRC, your notion of convergence and the one induced by the Hausdorff metric coincide. But the set-theoretic one is much more general.

Also, I seem to recall that if we consider $\mathcal{P}(X)$ to be topologized like $\{0,1\}^X$, identifying a set and its characteristic function, and using the product topology, we get this notion of convergence as well. http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior seems to confirm this.

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