What is the proper definition of cylinder sets? in class we defined the terminal $\sigma$-algebra for a sequence of random variables $(X_i)$ with $X_i:\Omega \rightarrow \mathbb{R}$ as $G_{\infty}:=\bigcap_i G_i$, with $G_i:=\sigma(X_i,X_{i+1},...)$. The question I asked myself was what the proper definition of $\sigma(X_i,X_{i+1},...)$ is? I know what it is, if we are only dealing with finitely many functions(random variables). My lecturer told me that the proper definition involves cylinder sets, where you would only have to look at a finite set of these random variables. Actually these cylinder sets are supposed to 'create' this sigma-algebra. Unfortunately, I could not find a proper definition of cylinder set in this context. I found only one that involves projections. 
Could anybody here help me making sense out of this hint?

The idea is about the same as for finite collections. We want to define $\sigma(X_{i},X_{i+1},\dots)$ so that it has just enough sets for each of the variables $X_i$, $X_{i+1}$, ... to be measurable. What do we need for this? For every open set $A\subset \mathbb R$ and every $j\ge i$ the set  $X_{j}^{-1}(A)\phantom{}$ must be in our $\sigma$-algebra. These are the basic cylinders. Of course, we can't have just these sets: we need a $\sigma$-algebra. The standard way is to get one is to take the intersection of all $\sigma$-algebras that contain all cylinder sets.
