Uncountable sample spaces are way too large

[summary: 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 $\Omega$. ]

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 uniform distribution] over the interval $[0,2]$, which intuitively should assign equal probability to everywhere in the interval.