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 below:

Statement

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

Proof

$\sigma$-algebra $\implies$ both $p$-system and $d$-system

It needs to be shown that:

• A sigma-algebra is itself a Dynkin system

Then it is EVEN more trivial that a sigma-algebra is $\cap$-closed

$p$-system and $d$-system $\implies$ a $\sigma$-algebra

• 

  Proof done on paper for like the third time

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TODO: Page 33 in and like page 3 in

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