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 $1$