Generated Sigma-Algebra Contains Generated Dynkin System

Theorem

Let $X$ be a set.

Let $\GG \subseteq \powerset X$ be a collection of subsets of $X$.

Then $\map \delta \GG \subseteq \map \sigma \GG$.

Here $\delta$ denotes generated Dynkin system, and $\sigma$ denotes generated $\sigma$-algebra.

Proof

By Sigma-Algebra is Dynkin System, $\map \sigma \GG$ is a Dynkin system.

The definition of $\map \delta \GG$ now ensures that $\map \delta \GG \subseteq \map \sigma \GG$.

$\blacksquare$

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $5.3$