Algebra of sets

A non-empty collection of subsets of some set that is closed under the set-theoretic operations (of union, intersection, taking complements), carried out a finite number of times. In order for a class of subsets of a set to be an algebra of sets, it is necessary (and sufficient) for it to be closed under finite unions and taking the complement. An algebra of sets that is closed under countable unions is known as a -algebra of sets. Any -algebra of sets is closed under the set-theoretic operations carried out a countable number of times.

Examples.

1) The collection of finite subsets of an arbitrary set and their complements is an algebra of sets; the collection consisting of the at most countable subsets of and their complements is a -algebra of sets.

2) The collection of finite unions of intervals of the type

forms an algebra of sets.

3) let be a topological space; the -algebra of sets generated by the open subsets of (in other words, the smallest -algebra of sets containing all open subsets of ) is known as the Borel -algebra of subsets of , while the sets belonging to are known as Borel sets.

4) let , where is an arbitrary set (i.e. is the set of all real functions on ); the class of sets of the type

where is a Borel subset of , is an algebra of sets; in the theory of random processes a probability measure is often originally defined only on an algebra of this type, and is their subsequently extended to a wider class of sets (to the -algebra generated by ).

5) The collection of Lebesgue-measurable subsets of is a -algebra of sets.

Algebras (respectively, -algebras) are the natural domain of definition of finitely-additive (respectively, -additive) measures. According to the theorem of extension of measures, any -finite, -additive measure, defined on an algebra , can be uniquely extended to a -additive measure defined on the -algebra generated by .

References

[1]	N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)
[2]	P.R. Halmos, "Measure theory" , v. Nostrand (1950)
[3]	J. Neveu, "Bases mathématiques du calcul des probabilités" , Masson (1970)