From Durrett's textbook on probability:
P. 232 A filtration is an increasing sequence of $\sigma$-fields.
i.e. $\mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}$ where $\mathcal{F}$ is the $\sigma$ algebra associated with the underlying probability space $(\Omega, \mathcal{F}, P)$.
My question is what is the interpretation of the underlying probability space and the filtration $\{\mathcal{F}_k\}$ defined on it? I am asking because I have not seen a concrete example of these sets.
Let's consider the simple random walk example on the same page:
Let $\xi_n = +1$ if the $n$th toss is head, $-1$ otherwise and define $X_n = \xi_1 + \ldots + \xi_n$, $X_0 = 0$ and $\mathcal{F}_0 = \{\varnothing, \Omega\}$. It was shown that $X_n$ is a martingale adapted to $\mathcal{F}_n$.
So in this simple example,
Is it correct to say that the sample space for this example is $\Omega$ = all possible infinite sequences of binary (i.e., $\pm 1$) numbers? Or is it perhaps all possible sequence of binary numbers of length $n$? I am not sure if this set simply contains all possible values generated by $\xi_n$ as $n$ becomes large, that is, is $\Omega$ finite, countable or uncountable?
How would you construct $\mathcal{F}_1$? Since I am not sure what $\Omega$ contains, therefore I am not sure how I should go about constructing $\mathcal{F}_1 = \{\varnothing, \text{something }, \Omega\}$.
Following from the above question, for this example, how would you define the "limiting" $\sigma$ algebra $\mathcal{F}$? i.e., what sets should it contain?
Typically, one doesn't care about an underlying probability space (unless one works in continuous time). It suffices to know that such a space exists (e.g., using Kolmogorov's extension theorem). The simplest filtration in your example is one generated by the sequence $\{\xi_n\}$, that is, $\mathcal{F}_n=\sigma(\xi_1,\ldots,\xi_n)$, which makes $\{\xi_n\}$ adapted to $\{\mathcal{F}_n\}$. However, this filtration can be larger, depending on the context.