# Limit sequence sets

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

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

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.
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.