# 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.

Parents:

• Sample space

The set of possible things that could happen in a part of the world that you are uncertain about.