I know this question was already answered on here, but I feel I am missing something fundamental in the reasoning in this problem. We state it as follows:
"A fair coin is tossed repeatedly. Show that the probability that a heads eventually turns up is one."
From what I understand about probability so far, to calculate the probability we take the set of all events where a heads shows up, and divide it by the size of the outcome space. Here I would think the sample space is the set of all finite sequences of heads and tails (which is clearly infinite).
Now in the solution listed here: Probability that a head eventually turns up (From Grimmett and Stirzaker)
they assume the sample space is $\Omega = \{H, TH, TTH, TTTH, ...\}$ and then they define $A_i$ to be the event that $H$ turns up on the $i$-th toss.
I have two confusions with this setup:
(1) Why do they assume the sample space is just the set of all coin tosses wherein a head appears?
(2) If our sample space is infinite, how can we calculate $\mathbb{P}(A_i) = \frac{1}{2^i}$? Why can we just restrict the probability measure to finite subset of the $\sigma$-algebra when making calculations?
Thank you for your time
