Intuitive idea of adapted processes with respect to "information" My text says the following,
"If the process $X_{n}$ is adapted to the filtration $\mathcal{F}_{n}$ the value $X_{n}(\omega)$ is known to us at time $n$."
This sounds wierd to me since it is all still random stuff, nothing is said about a realisation of $X_{n}$ up until $n$ for instance. 
At time $n$ we ONLY KNOW that a certain amout of sets are in $\mathcal{F}_{n}$ . What he must mean is that given that we know which sets  in $\mathcal{F}_{n}$ that $\omega$  "belongs" and "not belongs" to for all $\omega$, then we know $X_{n}(\omega)$, is this correct?
I also get confused by his statement regarding filtration in general i.e
"For any $\omega$ the information we have about it given the filtaration at time $n$ is exactly the values of any $\mathcal{F_{n}}$ measureable function"
This again must be dependant of the information which sets in $\mathcal{F}_{n}$ that $\omega$ belongs to and does not belong to, for all $\omega$. I think my second problem is the same as the one above in term of thinking about these matters. I know this is kind of a stupid question since it all seams reasonable but it would be nice if anyone would care to confirm this.
Update
Part of the misconseption was that I regarding the realisation wrong. As far I understand now the sigma algebra evolves with the index of the process and so does the $\omega$ in some sense aswell, the "whole" $\omega$ is not a factor until one considers the whole process i.e all $n$ are realized. Until then sets in each $\mathcal{F}_{n}$ are such that one can distingusih between paths up to time $n$. The remaining times are still unknown and are still within the same event.
 A: "If the process $X_{n}$ is adapted to the filtration $\mathcal{F}_{n}$ the value $X_{n}(\omega)$ is known to us at time $n$."
This means that the value of process will be known to you not before the time $n$. If you know $X_1,\dots,X_{n-1}$, i.e., $\mathcal F_{n-1}$, you cannot predict $X_n$. Look at it from conditional expectation point of view:
$$
E(X_n\mid \mathcal F_{n-1})\neq X_{n}.
$$
This is the intuition behind the adapted process. Basically only $\mathcal F_n$ (or future sigma-algebras) contains what we can know about $X_n$. 

"For any $\omega$ the information we have about it given the filtaration at time $n$ is exactly the values of any $\mathcal{F_{n}}$ measureable function"
This is another way of saying that any information you can have about $\omega$ at time $n$ is simply included in $\mathcal{F}_n$. Knowing $\mathcal F_n$ and $f$ a $\mathcal F_n$-measurable function, $f(\omega)$ is a random variable which gives information about $\omega$ knowing $\mathcal F_n$.
Let's have an example. Suppose that we are interested in knowing  if $\omega$ belongs to $A\subset \Omega$. If $A\in\mathcal F_n$, this information can be represented as $1(\omega\in A)$ which is a $\mathcal F_n$-measurable function. 
