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 \(\sum_{\omega \in \Omega} f(\omega) = 1\) 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.