I am reading David Williams' "Probability with Martingales" and in this book he supposes that you are given the following information about some 'outcome' $\omega \in \Omega$: $$X_n(\omega) \ \text{s.t.} \ n\in \mathbb{N}$$ for a series of random variables $X_n:\Omega \rightarrow \mathbb{R}$. He then says, based on this information, $\sigma(X_n : n \in \mathbb{N})$ consists of 'events' $F$ such that for each point $\omega$, you can tell whether the event $F$ has occurred or not (that $\omega \in F$).
However, no matter how hard I try to wrap my head around this vague idea, I can't seem to understand it. I don't understand how the 'information' provided helps to understand whether some event has occurred or not and what this means in terms of a sigma algebra generated by random variables (which also confuses me). Is there any easier way to understand what a sigma algebra generated by these functions actually is?
