Editing Set-theoretic limit (section)

==Properties==
* If the limit of <math>\mathbb{1}_{A_n}(x),</math> as <math>n</math> goes to infinity, exists for all <math>x</math> then <math display=block>\lim_{n \to \infty} A_n = \left\{ x \in X : \lim_{n \to \infty} \mathbb{1}_{A_n}(x) = 1 \right\}.</math> Otherwise, the limit for <math>\left(A_n\right)</math> does not exist.
* It can be shown that the limit infimum is contained in the limit supremum: <math display=block>\liminf_{n\to\infty} A_n \subseteq \limsup_{n\to\infty} A_n,</math> for example, simply by observing that <math>x \in A_n</math> all but finitely often implies <math>x \in A_n</math> infinitely often.
* Using the [[Set-theoretic limit#Monotone Sequences|monotonicity]] of <math display=inline> B_n = \bigcap_{j \geq n} A_j</math> and of <math display=inline> C_n = \bigcup_{j \geq n} A_j,</math> <math display=block>\liminf_{n\to\infty} A_n = \lim_{n\to\infty}\bigcap_{j \geq n} A_j \quad \text{ and } \quad \limsup_{n\to\infty} A_n = \lim_{n\to\infty} \bigcup_{j \geq n} A_j.</math>
* By using [[De Morgan's law]] twice, with [[set complement]] <math>A^c := X \setminus A,</math> <math display=block>\liminf_{n \to \infty} A_n = \bigcup_n \left(\bigcup_{j \geq n} A_j^c\right)^c
= \left(\bigcap_n \bigcup_{j \geq n} A_j^c\right)^c
= \left(\limsup_{n \to \infty} A_n^c\right)^c.</math> That is, <math>x \in A_n</math> all but finitely often is the same as <math>x \not\in A_n</math> finitely often.
* From the second definition above and the definitions for limit infimum and limit supremum of a real-valued sequence, <math display="block">\mathbb{1}_{\liminf_{n \to \infty} A_n}(x) = \liminf_{n \to \infty}\mathbb{1}_{A_n}(x) = \sup_{n \geq 1} \inf_{j \geq n} \mathbb{1}_{A_j}(x)</math> and <math display="block">\mathbb{1}_{\limsup_{n \to \infty} A_n}(x) = \limsup_{n \to \infty} \mathbb{1}_{A_n}(x) = \inf_{n \geq 1} \sup_{j \geq n} \mathbb{1}_{A_j}(x).</math>
* Suppose <math>\mathcal{F}</math> is a [[Sigma algebra|{{sigma}}-algebra]] of subsets of <math>X.</math> That is, <math>\mathcal{F}</math> is [[Empty set|nonempty]] and is closed under complement and under unions and intersections of [[countably many]] sets. Then, by the first definition above, if each <math>A_n \in \mathcal{F}</math> then both <math>\liminf_{n \to \infty} A_n</math> and <math>\limsup_{n \to \infty} A_n</math> are elements of <math>\mathcal{F}.</math>