smallest $\sigma$-algebra

smallest -algebra

Intersections of -algebras

For a set X and set of subsets of X. The intersection of 20231205121417-sigma_algebras containing is a -algebra.

Proof:

Let be the intersection of all the -algebras containing

1. Since is in each -algebra by definition it must be in the intersection.
2. \in Since E is in each -algebra, then is in each - thus it is in the intersection.
3. Countable union by the same logic…

The smallest -algebra

The intersection of all -algebras containing a set A is called the smallest -algebra containing A.