Why the limit of a series of increasing/decreasing sets in a Monotonous class is usually described as their union or intersection?

Question

When referring to the limit of a series of increasing/decreasing sets in Monotonous class, it is common to write this limit as the union/intersection of those sets. For example, when you read the proof of Monotone class theorem, you will find the limit of a set series $A_{1}\subset A_{2}\subset A_{2}\dots$ is $\bigcup^{\infty}_{i=1}A_{i}$ but not $\lim_{n\rightarrow\infty}A_n$. But what is the difference between the two kinds of expressions? Is it just a habit or have something more meaningful? Are there any cases satisfying $\lim_{n\rightarrow\infty}A_n \neq \bigcup^{\infty}_{i=1} A_i$ , or the former does not exist but the later exists?

Answer 1

They are exactly the same thing for monotone increasing series of sets.

For any sequence of sets $\{A_n\}_{n=0}^{\infty}$, the lim sup and lim inf can be defined by

$$\liminf_n A_n = \{ a \mid \exists N : \forall n > N, a \in A_n\}$$ and $$\limsup_n A_n = \{ a \mid \forall N, \exists n > N | a \in A_n\}$$

That is, the lim inf is the set of all objects that are eventually in every $A_n$, while the lim sup is the set of all objects that are in infinitely many $A_n$. The sequence converges if the lim inf and lim sup are the same. That is, every object that is in infinitely many of the $A_n$ is eventually in all of them. And of course the limit is the common value.

Now suppose that $\{A_n\}$ is increasing: $A_0 \subset A_1 \subset A_2 \subset ...$. If $a \in A_N$ for some $N$, then $a \in A_n$ for all $n \ge N$. Therefore $a \in \liminf A_n$ and $a \in \limsup A_n$. Hence $$\liminf A_n = \limsup A_n = \bigcup_{n=0}^\infty A_n$$

A similar argument shows that if $\{A_n\}$ is decreasing, $A_0 \supset A_1 \supset A_2 \supset ...$, then $$\liminf A_n = \limsup A_n = \bigcap_{n=0}^\infty A_n$$