[ilmath]p[/ilmath]-system
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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
