A collection of subsets is a $\sigma$-algebra $\iff$ it is both a $p$-system and a $d$-system

Terminology note:

• A collection of subsets of $X$, $\mathcal{A}$, is a $\sigma$-algebra if and only if it is a $d$-system (another name for a Dynkin system) and $\cap$-closed (which is sometimes called a $p$-system).

Dynkin himself used the $p$-system/$d$-system terminology using it we get the much more concise statement.

Statement

• A collection of subsets of $X$, $\mathcal{A}$ is a $\sigma$-algebra if and only if is is both a $p$-system and a $d$-system.

Proof

A collection of subsets is a sigma-algebra iff it is a Dynkin system and closed under finite intersections/Proof

---

TODO: Page 33 in and like page 3 in

---