Generated Sigma-Algebra Preserves Subset

Theorem

Let $X$ be a set.

Let $\FF, \GG \subseteq \powerset X$ be collections of subsets of $X$.

Suppose that $\FF \subseteq \GG$.

Then $\map \sigma \FF \subseteq \map \sigma \GG$, where $\map \sigma \GG$ denotes the $\sigma$-algebra generated by $\GG$

Proof

By definition of $\sigma$-algebra generated by $\GG$:

$\GG \subseteq \map \sigma \GG$

It follows that also:

$\FF \subseteq \map \sigma \GG$

Hence, by definition of $\sigma$-algebra generated by $\FF$:

$\map \sigma \FF \subseteq \map \sigma \GG$

$\blacksquare$

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $3.5 \ \text{(iii)}$