I have a definition:
Given $C \supset 2^{\Omega}$, the $\sigma$– algebra generated by $C$ written, $\sigma(C)$ is the "smallest" $\sigma$– algebra containing $C$
I understand what this means but I just don't understand what "generated by $C$" means.
Similarly, I am given an example of the Borel $\sigma$– algebra as $\sigma(T)$ where $T = ${open sets of $\mathbb{R}$}
So a Borel $\sigma$– algebra is equal to a $\sigma$– algebra generated by all the open sets of $\mathbb{R}$
Can someone please explain what the word "generated by" means?
I know a $\sigma$– algebra is a collection of subsets of the power set $2^{\Omega}$ where $\Omega$ is any set. So does this mean that if a $\sigma$– algebra is generated by something else, that something else is just the set $\Omega$?
Sorry for such a basic question, just looking for some clarification.
Best Answer
You can give meaning to the "smallest" $\sigma$-algebra containing $C\subset2^\Omega$ by first noting that any intersection of $\sigma$-algebras on $\Omega$ is again a $\sigma$-algebra on $\Omega$ and then setting
$$ \sigma(C):=\bigcap_{\mathcal A \in \mathbb A}\mathcal A, \quad \text{where $\mathbb A:=\{\mathcal A\subset 2^\Omega : \text{$\mathcal A$ is a $\sigma$-algebra on $\Omega$ and $C\subset\mathcal A$}\}$.} $$