# Uncountable sample spaces are way too large

https://arbital.com/p/uncountable_sample_spaces_are_too_large

by Tsvi BT May 25 2016 updated Jun 16 2016

We can't define probability distributions over uncountable sample spaces by just assigning numbers to each point in the sample space.

[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.