## Sigma Algebra

Let be a Set. Then a -algebra is a nonempty collection of Subsets of such that the following hold:

1. The Empty Set is in .

2. If is in , then so is the complement of .

3. If is a Sequence of elements of , then the Union of the s is in .

If is any collection of subsets of , then we can always find a -algebra containing , namely the Power Set of . By taking the Intersection of all -algebras containing , we obtain the smallest such -algebra. We call the smallest -algebra containing the -algebra generated by .

See also Borel Sigma Algebra, Borel Space, Measurable Set, Measurable Space, Measure Algebra, Standard Space

© 1996-9 Eric W. Weisstein
1999-05-26