Editing Field of sets (section)

== Definitions ==

A field of sets is a pair <math>( X, \mathcal{F} )</math> consisting of a [[Set (mathematics)|set]] <math>X</math> and a [[Family of sets|family]] <math>\mathcal{F}</math> of [[subset]]s of <math>X,</math> called an '''algebra over <math>X,</math>''' that has the following properties:

<ol>
<li>{{em|Closed under [[Complement (set theory)|complementation]] in <math>X</math>}}: <math display="block">X \setminus F \in \mathcal{F} \text{ for all } F \in \mathcal{F}.</math></li>
<li>{{em|Contains the [[empty set]] (or contains <math>X</math>)}} as an element: <math>\varnothing \in \mathcal{F}.</math>
* Assuming that (1) holds, this condition (2) is equivalent to: <math>X \in \mathcal{F}.</math>
</li>
<li>Any/all of the following equivalent<ref group=note>The listed statements are equivalent if (1) and (2) hold. The equivalence of statements (a) and (b) follows from [[De Morgan's laws]]. This is also true of the equivalence of statements (c) and (d).</ref> conditions hold:
<ol style="list-style-type: lower-latin;">
<li>{{em|Closed under binary [[Union (set theory)|unions]]}}: 
<math display="block">F \cup G \in \mathcal{F} \text{ for all } F, G \in \mathcal{F}.</math></li>
<li>{{em|Closed under binary [[Intersection (set theory)|intersections]]}}: 
<math display="block">F \cap G \in \mathcal{F} \text{ for all } F, G \in \mathcal{F}.</math></li>
<li>{{em|Closed under finite unions}}: 
<math display="block">F_1 \cup \cdots \cup F_n \in \mathcal{F} \text{ for all integers } n \geq 1 \text{ and all } F_1, \ldots, F_n \in \mathcal{F}.</math></li>
<li>{{em|Closed under finite intersections}}: 
<math display="block">F_1 \cap \cdots \cap F_n \in \mathcal{F} \text{ for all integers } n \geq 1 \text{ and all } F_1, \ldots, F_n \in \mathcal{F}.</math></li>
</ol>
</li>
</ol>

In other words, <math>\mathcal{F}</math> forms a [[subalgebra]] of the power set [[Boolean algebra (structure)|Boolean algebra]] of <math>X </math> (with the same identity element <math>X \in \mathcal{F}</math>).
Many authors refer to <math>\mathcal{F}</math> itself as a field of sets. 
Elements of <math>X</math> are called '''points''' while elements of <math>\mathcal{F}</math> are called '''complexes''' and are said to be the '''admissible sets''' of <math>X.</math>

A field of sets <math>( X, \mathcal{F} )</math> is called a '''σ-field of sets''' and the algebra <math>\mathcal{F}</math> is called a '''[[σ-algebra]]''' if the following additional condition (4) is satisfied: 

<ol start=4>
<li>Any/both of the following equivalent conditions hold:
<ol style="list-style-type: lower-latin;">
<li>{{em|Closed under [[Countable set|countable]] unions}}: 
<math display="block">\bigcup_{i=1}^{\infty} F_i := F_1 \cup F_2 \cup \cdots \in \mathcal{F}</math> for all <math>F_1, F_2, \ldots \in \mathcal{F}.</math></li>
<li>{{em|Closed under countable intersections}}: 
<math display="block">\bigcap_{i=1}^{\infty} F_i := F_1 \cap F_2 \cap \cdots \in \mathcal{F}</math> for all <math>F_1, F_2, \ldots \in \mathcal{F}.</math></li>
</ol>
</li>
</ol>