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:

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

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

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

We are given that:

$\FF \subseteq \GG$.

Hence:

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

$\blacksquare$


Sources

  • 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $3.5 \ \text{(iii)}$
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.