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.