In Apostol's Calculus II, he splits the calculus of probability into two chapters:
- Finite and Countable sets
- Uncountable sets
He only remarks that explaining why different tools are needed for uncountable sets "would take us too far afield."
I have an intuition for why a probability measure defined over a finite set would have different requirements than one defined over an infinite set, but I'm not really sure why uncountable sets are so different.
I suspect it may have something to do with the first theorem of the chapter (if $S$ is an uncountable set, there are an most a countable number of $s\in S$ where $P(s)\not=0$), but I'm not sure what exactly.
A search online doesn't seem to turn up much, so maybe this is just a decision the author made to separate them that others don't agree with? More likely I just don't know the terms to search for.