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I can see that a filtration on a probability space $\{\Omega, \mathcal{F}, P\}$is defined thus: An increasing collection of $\sigma$-fields, $\mathcal{F_0} \subseteq \mathcal{F_1} \subseteq \mathcal{F_n} ... \subseteq \mathcal{F}$

This means that it is a collection of $\sigma$-fields that tops out at $\mathbb{P}(\Omega)$, which is the largest element of this collection.

The text(s) then usually define what it means for a sequence of random variables to be adapted to a filtration, and suggest that this is a way to model knowledge at some time $n$.

This is what I find confusing; for a set of finite outcomes, there is clearly a limit on the largest possible $\sigma$-algebra. How can the filtration model a potentially infinite time series?

For instance, if I have two outcomes- $H$ and $T$ - the largest $\sigma$ alegebra I can possibly have is $\mathcal{F} = \{\emptyset, \{H\},\{T\},\{H,T\}\}$

If the information I am trying to represent is $HHHTTTHHH$, how would my filtration represent it differently to $TTTHHHTTT$? My understanding is that the filtration at the end of either of these runs of data will sequence will remain a collection of sub-$\sigma$-algebras of $\mathcal{F}$, which provides me no information about how to distinguish them, or indeed represent any data stream in a nontrivial way that actually captures the order of arrival.

I've had a look at other questions, particularly this one: Filtration and measure change. However, I think I'm at a more basic level right now.

I've asked a previous question on this, but it was very, very vague- I will delete it.

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    Indeed you are at a more basic level. First, if $\mathcal{F} = \{\emptyset, \{H\},\{T\},\{H,T\}\}$ is a sigma-algebra, it is so on the set $\Omega=\{H,T\}$ which is not suited to model $n$ throws of a coin (except if $n=1$), in particular HHHTTTHHH and TTTHHHTTT are not "informations" one can "represent" in this probability space. – 2017-01-16
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    I suppose then a set that does model $n$ throws of a coin must contain all possible runs of arbitrary length- $\{H, T, HH, TT, HT, TH, HHH, ... \}$ If that is correct, I think you resolved my confusion – 2017-01-16
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    To model $n$ throws of a coin, a common choice is $\Omega=\{H,T\}^n$. To model an infinite sequence of throws of a coin, a common choice is $\Omega=\{H,T\}^\mathbb N$ the set of sequences of $H$ or $T$. Of course the latter can be used to model the former, for every $n$. – 2017-01-16

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