infinite monkey problem - probability of an infinite sequence containing an infinite sequence Note: This question is specifically about when the infinite monkey theorem is extended to reproducing an infinite sequence (as oppose to a finite one)
I was browsing wikipedia, and came across the infinite monkey problem. I understand that if Random keys are selected for an infinite period of time, eventually any finite combination of characters will eventually be produced. For example, a complete copy of Hamlet. However, I am confused about what happens when this is extended to infinity. According to wikipedia:

If the monkey's allotted length of text is infinite, the chance of typing only the digits of pi is $0$, which is just as possible as typing nothing but Gs (also probability $0$).

http://en.wikipedia.org/wiki/Infinite_monkey_theorem#Almost_surely
However, this suggests that the probability of typing any combination of characters (that are infinitely long) is also $0$. However, the probability that a combination is typed is 1. However the sum of all possible events appears to be $0$, not $1$ (as the probability of any combination of infinite length being typed such as pi is $0$ as $0+0+0+0+...+0=0$ 
I'm not sure if I have just misunderstood this problem or if I have made a false assumption. It might have something to do with concept of "Almost certainly", but I was a bit intimidated by the wikipedia article.

Here is a similar example: let $x$ be a number chosen uniformly from $[0,1]$. For every $y \in [0,1]$, the probability that $x = y$ is $0$; and yet $x$ must be equal to something! What goes wrong here is that there are uncountably many numbers in $[0,1]$, while the uniform measure over $[0,1]$ is only countably additive. Measures with stronger additivity properties are studied in measure theory (in the set-theoretic parts), but the usual measures one studies in probability theory are only countably additive (or, as it is more commonly known, $\sigma$-additive).
