Understanding Random Walks and Related Theorems/Definitions I'm reading Durret's book, chapter 4 on Random Walks.  Rather than spending time dwelling in my sorrow and frustration over how bad I think this book is, I think I'll take to Stack Exchange to help me understand some things.  
I'm going to ask several questions over three posts so that it's more likely for people to respond.
Also, I did not know anything about Random Walks before reading this book.(Maybe that's relevant?)
If $(S,F,\mu)$ is a measure sapce, Durret tells us that for this section our probability space is 
$\Omega = S^{\mathbb{N}}$ $($i.e sequences of elements of $S)$ where $F^{\mathbb{N}}$ is the corresponding $\sigma$ algebra, and $P = \mu \times \mu ...$ is our measure.  Lastly, $X_n(\omega) = \omega$
Given a finite permutation, $\pi$, of $\mathbb{N}$, Durret refers to an event $A \in F^\mathbb{N}$ being permutable for any finite permutation $\pi$ provided $\pi^{-1}(A) = A$.  The collection of such events is called the exchangeable $\sigma$ field. Ok cool.
According to Durret, if $S = \mathbb{R}$ and $S_n(\omega) = X_1(\omega) + X_2(\omega) + ... X_n(\omega) = \omega_1 + \omega_2 + ...$
$1:\{\omega: S_n(\omega) \in B \ \ i.o \}$ is permutable;however, it's not a tail event. As expected, Durret does not say what $B$ is. I assume its just some subset of $S$ though. I don't understand why it's not a tail event. Please explain this if you can.
$2$: In his proof of the Hewitt-Savage $0-1$ law, Durret says that we can choose $A_n \in \sigma(X_1,...X_n)$ such that $P(A_n \Delta A) \to 0$ I don't understand what the sets $A_n \Delta A$ looks like and why $A_n$ can be chosen so that $P(A_n \Delta A) \to 0$
I'll stop here and ask more questions on another post.
 A: I'll work from Edition 4.1 of Durrett's text, available here:
1) The set $B$ is intended to be any Borel subset of the state space. As for why the event $\{\omega : S_n(\omega) \in B \text{ i.o.} \}$ is not a tail event: note that the event $\{\omega : S_n(\omega) \in B\}$ is contained in the $\sigma$-field $\mathcal F_n = \sigma(X_1, \dots, X_n)$. The tail field is in some sense the opposite of this; it's $\cap_n \mathcal F_n'$, where $\mathcal F_n' = \sigma(X_n, X_{n+1}, \dots)$. In short: permutable events are ones that do not depend on the order of the first $n$ terms, but tail events are ones that do not depend on those terms at all.
Here's an instructive example, although it's admittedly not quite the subject of the chapter: let $X_0, X_1, \dots$ be such that $X_0 = \pm 10$ and for $n \geq 1$, $X_n = \pm 1/2^n$, all independent and with each possibility having a $50\%$ chance. The sum $\sum_{n=1}^{\infty} X_n(\omega)$ will converge absolutely for each $\omega$ to a value at most $1$. Thus, for any $n$ the sum $S_n$ will be at most $1$ away from either $10$ or $-10$, depending on the outcome of $S_0$. Or, put another way: the event $\{S_n \in [9, 11] \text{ i.o.}\}$ will have a probability of $50\%$. This automatically tells us that it's not a tail event, because we know from Kolmogorov's 0-1 Law (Thm 2.5.1) that tail events can have probability only $0$ or $1$. Yet, permuting the first $m$ coordinates of the random walk will not change the value of $S_n$ so long as $n > m$, so this is indeed an element of the exchangeable $\sigma$-field.
I mentioned that this example was not the topic of the chapter, and that's precisely because it's not a random walk; the novelty of a random walk is that the sum must be constructed out of $X_i$ terms that are not only independent, as those ones were, but also identically distributed. That extra condition changes the game significantly. It's also the nature of Durrett's comment right before the Hewitt-Savage 0-1 law: 

The next result shows that for an i.i.d. sequence there is no difference between $\mathcal E$ and $\mathcal T$. They are both trivial.

2) This gets into the weeds of the Caretheodory extension theorem a bit, but the proof of this result is found in the appendix (and Durrett cites this in his proof of the Hewitt-Savage Law). I won't reproduce the proof here, because it's easy to find in the text for yourself, but I'll try to give some intuition for it: since $A \in \sigma(X_1, X_2, \dots)$, there is a sequence of events $A_n \in \sigma(X_1, \dots, X_n)$ that should approximate $A$ in the manner described (i.e. $A_n$ and $A$ are not that different, and their difference diminishes as $n$ increases). Roughly, this comes from constructing the $\sigma$-field $\sigma(X_1, \dots )$ in a limiting process from $\sigma(X_1, \dots, X_n)$.
