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

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Terminology note:

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

Statement

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. 1.0 1.1 Measures, Integrals and Martingales
  2. 2.0 2.1 2.2 2.3 2.4 Probability and Stochastics - Erhan Cinlar