Difference between revisions of "Dynkin system generated by"
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==Definition== | ==Definition== | ||
Given a set {{M|X}} and another set {{M|\mathcal{G}\subseteq\mathcal{P}(X)}} which we shall call the ''generator'' then we can define ''the [[Dynkin system]] generated by {{M|\mathcal{G} }}'' as<ref name="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref>: | Given a set {{M|X}} and another set {{M|\mathcal{G}\subseteq\mathcal{P}(X)}} which we shall call the ''generator'' then we can define ''the [[Dynkin system]] generated by {{M|\mathcal{G} }}'' as<ref name="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref>: | ||
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Contents
[hide]Definition
Given a set X and another set G⊆P(X) which we shall call the generator then we can define the Dynkin system generated by G as[1]:
- The smallest Dynkin system that contains G
And we denote this as: δ(G). This is to say that:
- δ(G) is the smallest Dynkin system such that G⊆δ(G)
(Claim 1) This is the same as:
- δ(G):=⋂D is a Dynkin systemand G⊆DD
Proof of claims
[Expand]
Claim 1: δ(G):=⋂D is a Dynkin systemand G⊆DD is the smallest Dynkin system containing G
See also
- Types of set algebras
- Sigma-algebra
- Conditions for a Dynkin system to be a sigma-algebra
- Conditions for a generated Dynkin system to be a sigma-algebra
References
- ↑ Jump up to: 1.0 1.1 Measures, Integrals and Martingales - Rene L. Schilling
