it might be a stupid question but I was discussing with a colleague when the 3rd axiom of probability (sigma additivity) is really needed. I argue that in the case of a discrete distribution, say a single die, the first two axioms are sufficient as this distribution has a finite number of events. Is this right?
And then I am stretching it a bit by arguing that for simple/well behaved continuous distributions, such as the uniform distribution, the first two axioms are sufficient to ensure a proper probability distribution. Is that right?
I always had the idea that the 3rd axiom was rather there to ensure that more cumbersome distributions or convolutions of distributions would still be proper probability distributions (although I don't have an example at hand). Or is axiom 3 much more vital than I understand?
Thanks for any input! Best, Stefan