The infinite monkey theorem states something like:
If a random process generates a countably infinite ordered set of characters C, then any finite ordered set of characters S of length L is almost surely an ordered subset of C.
Since every string S of length L is almost surely going to appear arbitrarily many times in C, then isn't the sample space generated by strings of length L uncountable, creating an uncountable value of almost sure outcomes in a countable sample space C?
I understand that every finite string is a substring of countably infinitely many finite strings, and it's quite trivial that any single character is almost sure to appear. The proof where the probability of an arbitrarily chosen string not appearing approaches zero also seems clear. I understand that removing S and all characters before S in C yields a sample space C*=C\X (X representing the set from index 0 to the index at the end of S), where C* and C have the same cardinality.
My issue is when you collect all of the arbitrary strings of arbitrary length, we seem to have more outcomes as almost sure then there are outcomes available. How could it be true that uncountably many finite strings are almost surely substrings of a countable string? Isn't that a measurable space of almost sure outcomes in a sample space of measure 0?
A simpler question for intuition is how can S almost always be almost sure to appear in C\S (removing the first time it appears), if the operation is repeated arbitrarily many times?
Nothing about the sample space is uncountable. A string of length $L$ is made from a language with $k$ different characters, i.e. simply an element of $[L]^k$. The point is that a given string always has a finite length. The space of all possible infinite-length strings is no longer countable, since we can inject it into the real numbers $abcde... \mapsto a.bcde...$.