Sigma-Algebra is Dynkin System

Theorem

Let $X$ be a set, and let $\Sigma$ be a $\sigma$-algebra on $X$.


Then $\Sigma$ is a Dynkin system on $X$.


Proof

The axioms $(1)$ and $(2)$ for both $\sigma$-algebras and Dynkin systems are identical.

Dynkin system axiom $(3)$ is seen to be a specification of $\sigma$-algebra axiom $(3)$ to pairwise disjoint sequences.


Hence $\Sigma$ is trivially a Dynkin system on $X$.

$\blacksquare$


Sources

  • 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $5.2$, $\S 5$: Problem $1$
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