Tail events and exchangeable events In this problem I have $X_1, X_2, \cdots$ independent identically distributed RVs taking values $\pm1$ with the equal probability of $1/2$ and my trajectory is defined by $S_n=\sum_i^n X_i$ (so pretty much like a coin flipping experiment).
I am trying to show that certain events are tail events, exchangeable events or both, but I have some questions about these events.
The definition of the tail sigma algebra is $T=\bigcap_{n\geq 1}\bigcup_{k=n}^\infty \mathcal{F_k}$  and $\mathcal{F_n}$ are a sequence of $\sigma$-algebras on $\Omega$. I think that $\Omega$ here would be the union of all $n$-sized sequences of $\pm 1$? i.e. $\Omega=\{1\},\{0\},\{11\},\{10\},\cdots$
So how exactly are the $\mathcal{F}_n$ $\sigma$-algebras indexed? I thought that it could be related to $n$ in $S_n$ and that $\mathcal{F}_n$ is the $\sigma$-algebra generated by all possible trajectories of the $n$-step trajectory, i.e. $\mathcal{F_1}=\{\varnothing,\{1\},\{0\},\{1,0\}\}$
I'm not sure this makes sense, because there is a distinction between the set $\Omega$ and the set, for example in $\mathcal{F_1}$, $\{0,1\}$
 A: Try $\Omega=\{0,1\}^{\mathbb N}$,  $\mathcal F_n=\{A\times\{0,1\}^{\{n+1,n+2,\ldots\}}\mid A\subseteq\{0,1\}^n\}$ and $X_n(\omega)=\omega_n$ for every $n\geqslant0$.
Thus, $\Omega$ is the set of all infinite $\{0,1\}$-valued sequences, not the union on $n\geqslant0$ of the sets of $\{0,1\}$-valued sequences of length $n$.
Edit: An alternative characterization of $\mathcal F_n$ is that $B\subseteq\Omega$ is in $\mathcal F_n$ if and only if property $P_n(B)$ holds:

$P_n(B):$ Let $\omega=(\omega_k)_{k\in\mathbb N}$ and $\omega'=(\omega'_k)_{k\in\mathbb N}$ denote two elements of $\Omega$. If $\omega$ is in $B$ and if $\omega'_k=\omega_k$ for every $1\leqslant k\leqslant n$, then $\omega'$ is in $B$.

Note that each $\mathcal F_n$ contains $\varnothing$ since $\varnothing=\varnothing\times\{0,1\}^{\{n+1,n+2,\ldots\}}$ and $\varnothing\subseteq\{0,1\}^n$, and that each $\mathcal F_n$ contains $\Omega$ since $\Omega=\{0,1\}^n\times\{0,1\}^{\{n+1,n+2,\ldots\}}$ and $\{0,1\}^n\subseteq\{0,1\}^n$. Furthermore, for each $n$, $B_n=\{0\}^n\times\{0,1\}^{\{n+1,n+2,\ldots\}}$ is in $\mathcal F_k$ if and only if $n\leqslant k$. An equivalent formulation of the set $B_n$ is that $B_n=\bigcap\limits_{i=1}^n[X_i=0]$.
Edit-edit: The above corresponds to the description of the sequence $(\mathcal F_n)_n$ in the post. However, in the context of tail sigma-algebras, a more natural choice would be $\mathcal F_n$ the sigma-alebra generated by $X_n$ alone (not by every $X_k$ with $k\leqslant n$ as above). Then $B\subseteq\Omega$ is in $\mathcal F_n$ if and only if property $Q_n(B)$ holds:

$Q_n(B):$ Let $\omega=(\omega_k)_{k\in\mathbb N}$ and $\omega'=(\omega'_k)_{k\in\mathbb N}$ denote two elements of $\Omega$. If $\omega$ is in $B$ and if $\omega'_n=\omega_n$, then $\omega'$ is in $B$.

Then the tail sigma-algebra $\mathcal T=\bigcap\limits_n\bigcup\limits_{k\geqslant n}\mathcal F_k$ contains events such as $[X_n=1\ \text{for infinitely many}\ n]$ or $[S_n/n\to c]$ for every $c$ (the proofs are not that difficult but this should be proven), and a powerful result in this domain is the so-called Kolmogorov zero-one law, which states that, if the sequence $(\mathcal F_n)$ is independent then $\mathcal T$ contains events of probability $0$ or $1$ only.
