[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.