# [ilmath]p[/ilmath]-system

From Maths

## Contents

## Definition

A *[ilmath]p[/ilmath]-system* or *product-system*^{[1]}^{[Note 1]} is the name given to a system of subsets of [ilmath]X[/ilmath], [ilmath]P\subseteq\mathcal{P}(X)[/ilmath] where it is closed under finite intersections^{[1]}, that is to say:

- Given [ilmath]A,B\in P[/ilmath] that [ilmath]A\cap B\in P[/ilmath]

## See also

- [ilmath]d[/ilmath]-system (Dynkin system)
- A collection of subsets of [ilmath]X[/ilmath], [ilmath]\mathcal{A} [/ilmath] is a [ilmath]\sigma[/ilmath]-algebra if and only if it is both a [ilmath]p[/ilmath]-system and a [ilmath]d[/ilmath]-system

## Notes

- ↑ Product sometimes means 'intersection' according to
*Probability and Stochastics - Erhan Cinlar*- this is Dynkin's own naming system

## References

- ↑
^{1.0}^{1.1}Probability and Stochastics - Erhan Cinlar