# Are monotone classes closed under $\limsup_{n \to \infty}$ and $\liminf_{n \to \infty}$?

If $X$ is a non-empty set, a family $\mathcal{K} \subseteq \mathcal{P}(X)$ is called a monotone class if for every increasing sequence of subsets of $X$, $\lbrace B_{n} \rbrace_{n\in \mathbb{N}}$, such that $(\forall n \in \mathbb{N}) B_{n} \in \mathcal{K}$, we have $\bigcup_{n=1}^{\infty} B_{n} \in \mathcal{K}$, and for every decreasing sequence of subsets of $X$, $\lbrace C_{n} \rbrace_{n\in \mathbb{N}}$, $\bigcap_{n=1}^{\infty} C_{n} \in \mathcal{K}$.

My professor told us today that monotone classes are closed under $\limsup_{n \to \infty}$ and $\liminf_{n \to \infty}$, and told us to try to prove that. I'm having some difficulties. Here's what I tried:

Let $B_{n} \in \mathcal{K}$ for all $n \in \mathbb{N}$. Then $\limsup_{n \to \infty} B_{n} = \bigcap_{n=1}^{\infty} \bigcup_{k=n}^{\infty} B_{k}$. Denote $C_{n} = \bigcup_{k=n}^{\infty} B_{k}$. Then $C_{n} \downarrow \limsup_{n \to \infty} B_{n}$, so if I prove that $C_{n} \in \mathcal{K}$ for all $n \in \mathbb{N}$, I'm done. However, I can't prove this, and I'm not even sure if it's true.

Is the statement true, and if so, how does the proof go?

If it were true then every monotone class would be closed under arbitrary finite unions, because the $\limsup$ of the sequence $A,B,A,B,\dots$ is $A\cup B$.
And if that were true it would be a theorem in those books. Say $I$ is the class of all (open, half-open or closed, bounded or unbounded, empty or nonempty) intervals. Then $I$ is a monotone class not closed under finite unions.
• Or just $X=\{1,2\}, \mathcal K=\{\{1\},\{2\}\}$. – hmakholm left over Monica Dec 22 '17 at 16:52