If the sample space is uncountable, then in general we can't even define a probability distribution over $Ω$ in the same way we defined probability distributions over countable sample spaces, i.e. by just assigning numbers to each point in the sample space. Any function $f : Ω \to [0, 1]$ with $\sum_{ω \in Ω} f (ω) = 1$ can only assign positive values to at most countably many elements of $Ω$ . But this means we can't, for example, talk about a uniform distribution over the interval $[0, 2]$ , which intuitively should assign equal probability to everywhere in the interval.