# 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.