I am trying to solve the exercise at the end of this page, the framework is that of measure theory where we are tossing a coin infinitely often so we are working with a probability triple $( \Omega, F, P)$, where $w$ is a point of $\Omega$, $\Omega =[H,T]^N, \qquad w =(w_1, w_2, \dots ), \quad w_n \in \{H,T \},$ and
$$F = \sigma(\{ w \in \Omega : w_n = W \}: n \in N, W \in \{H,T\})$$
The formal definition (2.3,b) that is cited in the text below is
$$F^* = \left \{ w : \frac{\text{number of } (k \le n : w_k = H)}{n} \rightarrow \frac{1}{2} \right \} $$
I think that for any $w$ an $\alpha$ s.t. $w \notin F_\alpha$ is $\alpha(k) = 1$, correct?
Moreover I am missing the importance of this discussion, why is the fact that the intersection is the empty set important?
This is going to be a little fuzzy. It's a fuzzy question on a fuzzy topic... the whole point is why axiomatic measure-theoretic probability theory was needed to unfuzzify things.
A person might attempt to define probability in terms of those limits: By "definition", saying that $P(H)=1/2$ means that that limit is $1/2$.
Then when someone else points out that the sequence $HHH\dots$ is possible the reply is that that's not a "random" sequence. Hard to argue with that since we haven't defined random...
So. We've attempted to define $P=1/2$ by saying that it means that we get asymptotically half heads in any "random" sequence of tosses. But now wait, we have a problem. If we make a "random" sequence of tosses then the results from the odd-numbered tosses should certainly still count as "random", so half the odd-numbered tosses must also be heads. And the same for any subsequence of tosses.
The exercise shows this can't work, by showing that it simply can't happen that half of any subsequence of tosses is heads. No matter what sequence we toss, there's some subsequence that violates this. So our definition leads to a contradiction. Bad.