Properties of limsup and liminf of sequences of sets

Question

I came across the following problems on limit supremums and infimums:

Let $(A_n)$ be a sequence of subsets of $X$. Define $$\text{lim sup} \ A_n = \{x \in X: x \in A_n \ \text{frequently} \}$$ and $$\text{lim inf} \ A_n = \{x \in X: x \in A_n \ \text{ultimately} \}$$

Show that $$\text{lim inf} \ A_n \subset \text{lim sup} \ A_n$$

$$\text{lim inf} \ A_n = \bigcup_{n=1}^{\infty} \bigcap_{k=n}^{\infty} A_k$$

$$(\text{lim sup} \ A_n)' = \text{lim inf} \ A_{n}'$$ $$\text{lim sup} \ A_n = \bigcap_{n=1}^{\infty} \bigcup_{k=n}^{\infty} A_k$$

Note that $x_n \in A_n$ frequently means that $(\forall N) \ \exists n \geq N \ni x_n \in A_n$. Also $x_n \in A_n$ ultimately means that $\exists N \ni n \geq N \Rightarrow x_n \in A$.

The first follows since for $x \in \text{lim inf} \ A_n$ then $x \in A_n$ ultimately which means that it is contained in $\text{lim sup} \ A_n$. For the last three, would I just use DeMorgan's laws and the definition of unions and intersections to deduce that they are the same as the definitions given above?

Answer 1

(1) Your proof that $\liminf A_n\subseteq \limsup A_n$ is correct.

Exercise 1: Under what conditions does equality hold in the above inclusion, i.e., under what conditions is it true that $\liminf A_n=\limsup A_n$?

(2) Note that $x\in \liminf A_n$ if and only if there exists a positive integer $N$ such that $x\in A_n$ for all $n\geq N$ if and only if there exists a positive integer $N$ such that $x\in \bigcap_{k=N}^{\infty} A_k$ if and only if $x\in \bigcup_{n=1}^{\infty} \bigcap_{k=n}^{\infty} A_k$.

I will leave the remaining questions as easy exercises (with similar solutions).

Exercise 2: Let $\{A_n\}$ be a sequence of measurable subsets of a measurable space $(X,\mu)$. Assume that $\Sigma_{n=1}^{\infty} \mu(A_n)<\infty$. Let $A=\limsup A_n$. Prove that $\mu(A)=0$. (Hint: Use the fourth assertion in your question, namely, use the characterization of $\limsup A_n$ in your question.)

Exercise 3: Give an example (in the context of Exercise 2) where $\lim_{n\to\infty} \mu(A_n)=0$ but that $\mu(A)>0$. Do not assume that $\Sigma_{n=1}^{\infty} \mu(A_n)<\infty$. (Hint: let $\{A_n\}$ be an appropriate sequence of intervals in $[0,1]$, for example.)