TITLE: What is meant by limit of sets?
QUESTION [10 upvotes]: I am having a hard time grasping the limit of sets as used in $\sigma$ algebra and probability space. What does the notation $\lim_{n\rightarrow\infty}S_n=S$ mean when all these $S_n$ are sets. I have some intuitive idea. But I understand limit in real analysis with the $\delta$-$\epsilon$ concept. So what is the rigorous definition of limit applied to sets, a definition parallel to the $\delta$-$\epsilon$ notation as applied to real analysis? 
Citing one or two examples will be great.

REPLY [10 votes]: We have (see e. g. Billingsley, Probiability and measure, 3rd edition, page 52), that the limes superior and the limes inferior are defined - as for any ordered space - on the power set: 

Definition. Let $X$ be a set, $(S_n)$ a sequence in $\mathfrak P(X)$. We define
  (1) The limes superior of $(S_n)$ is the set which contains all elements which are frequently in $S_n$, that is in infinitely many $S_n$:
  $$ \limsup S_n := \bigcap_{n\ge 1} \bigcup_{k\ge n} S_k $$
  (2) The limes inferior of $(S_n)$ is the set which contains all elements which are finally in $S_n$, that is in all but finitely many $S_n$:
  $$ \liminf S_n := \bigcup_{n\ge 1} \bigcap_{k\ge n} S_k $$
  (3) If $\liminf S_n$ and $\liminf S_n$ coincide, we say, $(S_n)$ converges and write 
  $$ \lim S_n := \limsup S_n = \liminf S_n. $$

Example: If $S_n$ is montonically increasing, then $\bigcup_{k\ge n} S_k$ is equal for all $n$, hence $\limsup S_n = \bigcup_{k\ge 1} S_k$, on the other side, $\bigcap_{k\ge n} S_k = S_n$, hence $\liminf S_n = \bigcup_{n\ge 1} S_n$, that is $\lim S_n = \bigcup_n S_n$.