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  text: '[summary: \nIf the sample space $\\Omega$ is [2w0 uncountable], then in general we can't even define a probability distribution over $\\Omega$ in the same way we defined\n[3tb 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 $\\Omega$. ]\n\n\n\nIf the sample space $\\Omega$ is [2w0 uncountable], then in general we can't even define a probability distribution over $\\Omega$ in the same way we defined\n[3tb 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\n[uniform_distribution uniform distribution] over the interval $[0,2]$, which intuitively should assign equal probability to everywhere in the interval. \n',
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