Uncountable sample spaces are way too large

If the sample space $\Omega$ is uncountable, then in general we can't even define a probability distribution over $\Omega$ 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 $f:\Omega \to [0,1]$ with ${\sum }_{\omega \in \Omega }f\left(\omega \right)=1$ can only assign positive values to at most countably many elements of $\Omega$. 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.