Often, in the course of my (self-)study of statistics, I've met the terminology "$\sigma$-algebra generated by a random variable". I don't understand the definition on Wikipedia, but most importantly I don't get the intuition behind it. Why/when do we need $\sigma-$algebras generated by random variables? What is their meaning? I know the following:
- a $\sigma$-algebra on a set $\Omega$ is a nonempty collection of subsets of $\Omega$ which contains $\Omega$, is closed under complement and under countable union.
- we introduce $\sigma$-algebras to build probability spaces on infinite sample spaces. In particular, if $\Omega$ is uncountably infinite, we know there can exist unmeasurable subsets (sets for which we cannot define a probability). Thus, we can't just use the power set of $\Omega$ $\mathcal{P}(\Omega)$ as our set of events $\mathcal{F}$. We need a smaller set, which is still large enough so that we can define the probability of interesting events, and we can talk about convergence of a sequence of random variables.
In short, I think I have a fair intuitive understanding of $\sigma-$algebras. I would like to have a similar understanding for the $\sigma-$algebras generated by random variables: definition, why we need them, intuition, an example...