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.
Contents
Statement
- A collection of subsets of [ilmath]X[/ilmath], [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
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
