# A collection of subsets is a [ilmath]\sigma[/ilmath]-algebra [ilmath]\iff[/ilmath] it is both a [ilmath]p[/ilmath]-system and a [ilmath]d[/ilmath]-system

From Maths

**Terminology note:**

- A collection of subsets of [ilmath]X[/ilmath], [ilmath]\mathcal{A} [/ilmath], is a [ilmath]\sigma[/ilmath]-algebra
*if and only if*^{[1]}^{[2]}it is a [ilmath]d[/ilmath]-system (another name for a Dynkin system) and [ilmath]\cap[/ilmath]-closed (which is sometimes called a [ilmath]p[/ilmath]-system^{[2]}).

Dynkin himself used the [ilmath]p[/ilmath]-system/[ilmath]d[/ilmath]-system terminology^{[2]} using it we get the much more concise statement below:

## Contents

## Statement

- A collection of subsets of a set [ilmath]X[/ilmath], say [ilmath]\mathcal{A} [/ilmath], is a [ilmath]\sigma[/ilmath]-algebra
*if and only if*is is both a [ilmath]p[/ilmath]-system and a [ilmath]d[/ilmath]-system^{[2]}.

## Proof

### [ilmath]\sigma[/ilmath]-algebra [ilmath]\implies[/ilmath] both [ilmath]p[/ilmath]-system and [ilmath]d[/ilmath]-system

It needs to be shown that:

Then it is EVEN more trivial that a sigma-algebra is [ilmath]\cap[/ilmath]-closed

### [ilmath]p[/ilmath]-system and [ilmath]d[/ilmath]-system [ilmath]\implies[/ilmath] a [ilmath]\sigma[/ilmath]-algebra

TODO: Page 33 in^{[1]} and like page 3 in^{[2]}

## References

- ↑
^{1.0}^{1.1}Measures, Integrals and Martingales - ↑
^{2.0}^{2.1}^{2.2}^{2.3}^{2.4}Probability and Stochastics - Erhan Cinlar