Editing Σ-algebra (section)

===Limits of sets===
Many uses of measure, such as the probability concept of [[convergence of random variables|almost sure convergence]], involve [[set-theoretic limit|limits of sequences of sets]]. For this, closure under countable unions and intersections is paramount. Set limits are defined as follows on σ-algebras.

* The {{em|[[limit supremum]]}} or {{em|outer limit}} of a sequence <math>A_1, A_2, A_3, \ldots</math> of subsets of <math>X</math> is <math display=block>\limsup_{n\to\infty} A_n = \bigcap_{n=1}^\infty \bigcup_{m=n}^\infty A_m = \bigcap_{n=1}^\infty A_n \cup A_{n+1} \cup \cdots.</math> It consists of all points <math>x</math> that are in infinitely many of these sets (or equivalently, that are in [[Cofinal (mathematics)|{{em|cofinally}} many]] of them). That is, <math>x \in \limsup_{n\to\infty} A_n</math> if and only if there exists an infinite [[subsequence]] <math> A_{n_1}, A_{n_2}, \ldots</math> (where <math>n_1 < n_2 < \cdots</math>) of sets that all contain <math>x;</math> that is, such that <math>x \in A_{n_1} \cap A_{n_2} \cap \cdots.</math>
* The {{em|[[limit infimum]]}} or {{em|inner limit}} of a sequence <math>A_1, A_2, A_3, \ldots</math> of subsets of <math>X</math> is <math display=block>\liminf_{n\to\infty} A_n = \bigcup_{n=1}^\infty \bigcap_{m=n}^\infty A_m = \bigcup_{n=1}^\infty A_n \cap A_{n+1} \cap \cdots.</math> It consists of all points that are in all but finitely many of these sets (or equivalently, that are {{em|eventually}} in all of them). That is, <math>x \in \liminf_{n\to\infty} A_n</math> if and only if there exists an index <math>N \in \N</math> such that <math>A_N, A_{N+1}, \ldots</math> all contain <math>x;</math> that is, such that <math>x \in A_N \cap A_{N+1} \cap \cdots.</math>

The inner limit is always a subset of the outer limit: <math display=block>\liminf_{n\to\infty} A_n ~\subseteq~ \limsup_{n\to\infty} A_n.</math> 
If these two sets are equal then their limit <math>\lim_{n\to\infty} A_n</math> exists and is equal to this common set: 
<math display=block>\lim_{n\to\infty} A_n := \liminf_{n\to\infty} A_n = \limsup_{n\to\infty} A_n.</math>