Sigma-Algebra

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

1.  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 .

REFERENCES:

Jech, T. J. Set Theory, 2nd ed. Berlin: Springer-Verlag, p. 494, 1997.