It's similar, but with a finite set it wouldn't show up unless you had some outcomes have probability 0 (in which case, why not just delete them from the outcom

It's similar, but with a finite set it wouldn't show up unless you had some outcomes have probability 0 (in which case, why not just delete them from the outcome space?). Where you need the idea is in continuous spaces - like a uniform [0,1] random variable. With probability 1 it's not a rational number, etc.