Suppose that C is a class of subsets of $\Omega$ and $B\in \sigma(C)$. Show that there exists a countable class $C_B\subset C$ such that $B\in \sigma(C_B)$.
Sigma algebra related
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probability-theory
measure-theory
1 Answers
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I'm guessing that the somewhat cryptic notation $\sigma(\mathcal C)$ denotes the $\sigma$-algebra generated by $\mathcal C.$ If I guessed right, then the assertion you want to prove is that $\sigma(\mathcal C)=\mathcal S$ where $\mathcal S=\bigcup\{\sigma(\mathcal D):\mathcal D\subseteq\mathcal C,\ \mathcal D\text{ countable}\}.$ For that you just have to show that $\mathcal C\subseteq\mathcal S\subseteq\sigma(\mathcal C)$ (which is pretty obvious) and that $\mathcal S$ is a $\sigma$-algebra. The trickiest part is showing that $\mathcal S$ is closed under countable union. I guess you know that a countable union of countable sets is countable?