TITLE: In probability, how can a sigma-algebra represent the total information?
QUESTION [5 upvotes]: Why does a sigma-algebra represent the information available at a given time?
I understand the idea of filtration and stopping-time, given that each sigma-algebra represent the info we have at a specific time, but why is that?
For instance in a game of dice rolls (or anyone you want), what would be total universe and the available information in forms of sigma-algebra, at the n-th turn?
thanks

REPLY [2 votes]: The Doob-Dynkin lemma relates them in an intuitive way for most of the standard applications in probability theory.
Suppose you have a probability space $(\Omega,\Sigma,\mu)$ and two random variables $f:\Omega\to\mathbb{R}$ and $g:\Omega\to\mathbb{R}$. That $g$ only depends on $f$ can be interpreted as saying that you know the value of $g$ whenever you know the value of $f$. This means that you can find a function $h:\mathbb{R}\to\mathbb{R}$ such that $g(\omega)=h(f(\omega))$. In other words, $g=h\circ f$. Now the Doob-Dynkin emma says that the following are equivalent:

There is a measurable function $h:\mathbb{R}\to\mathbb{R}$ such that $g=h\circ f$.
The random variable $g$ is measurable with respect to the $\sigma$-algebra generated by $f$, that is the $\sigma$-algebra $\{f^{-1}(B):B\textrm{ is a Borel set}\}$. 

Most naturally occuring $\sigma$-algebras are of the form $\{f^{-1}(B):B\textrm{ is a Borel set}\}$ for some random variable $f$. This is equivalent to the $\sigma$-algebra being countably generated.