In a proof of the Borel-Cantelli lemma in the stochastic process textbook, the author used the following.
$$\limsup_{n\to\infty}A_n=\bigcap_{n\ge1}\bigcup_{k\ge n} A_k$$
Can someone explain why lim sup is intersection and union? Thank you
borel-cantelli-lemmaslimsup-and-liminfprobability theory
In a proof of the Borel-Cantelli lemma in the stochastic process textbook, the author used the following.
$$\limsup_{n\to\infty}A_n=\bigcap_{n\ge1}\bigcup_{k\ge n} A_k$$
Can someone explain why lim sup is intersection and union? Thank you
Best Answer
I find it very helpful to think of the limit superior and limit inferior of a sequence of real numbers and a sequence of sets as special cases of limit superior and limit inferior in so called complete lattices:
You probably already know the following notions for the special case of the ordered set $(\mathbb{R},\leq)$:
If $(S, \leq)$ is a partially ordered set and $A \subseteq S$ a subset then neither a supremum of $A$, nor an infimum of $A$ need to exist. If it does, however, then it is unique, and is denoted by $\sup A$ and $\inf A$ respectively.
Notice that in the special case of $(\mathbb{R},\leq)$ the above definition coincides with the usual notion of the supremum and infimum of a set of real numbers. In the case of the extended real numbers $\mathbb{R}\cup \{-\infty,\infty\} = [-\infty,\infty]$ we have the nice property that each subset $S \subseteq [-\infty,\infty]$ has a supremum (possibly $\infty$) and an infimum (possibly $-\infty$). Such ordered sets are called complete lattices.
Aside from the extended real numbers $[-\infty,\infty]$ another complete lattice which we commonly encounter are power sets:
Notice that this result is not very suprising: The smallest set containing all sets of $\mathcal{A}$ in naturally the union of these sets. In the same way the biggest set which is contained in all sets of $\mathcal{A}$ is naturally the intersection of these sets.
Since in complete lattices we have suprema and infima we have all that we need to define the limit superior and limit inferior.
Notice that for the extended real line $[-\infty,\infty]$ this is the usual definition of limit superior and limit inferior. But what about power sets?
So we see that the definiton of the limit superior and limit inferior of a sequence of sets really comes down to what the supremum and infimum of a collections of sets is, which naturally is their union and intersection respectively.