smallest $\sigma$-algebra

smallest $\sigma$-algebra

Intersections of $\sigma$-algebras

For a set X and set $\mathcal{A}$ of subsets of X. The intersection of 20231205121417-sigma_algebras containing $\mathcal{A}$ is a $\sigma$-algebra.

Proof:

Let ${\mathcal{S}}_0$ be the intersection of all the $\sigma$-algebras containing $\mathcal{A}$

1. $\varnothing \in {\mathcal{S}}_0$ Since $\varnothing$ is in each $\sigma$-algebra by definition it must be in the intersection.
2. $E$ \in ${\mathcal{S}}_0⟹E^C\in {\mathcal{S}}_0$ Since E is in each $sigma$-algebra, then $E^C$ is in each - thus it is in the intersection.
3. Countable union by the same logic…

The smallest $\sigma$-algebra

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