Real-valued probability functions cannot assign positive probability to every point in an uncountable sample space while remaining countably additive and summing to 1

We have seen a striking violation of regularity in de Finetti’s lottery: his assignment of 0 to each ticket. Regularity may be preserved here by countably additive probabilities, but at the expense of a uniform distribution—for example, \(\frac{1}{2}\) to ticket 1, \(\frac{1}{4}\), to ticket 2, \(\frac{1}{8}\) to ticket 3, and so on. It may be shown that if \(F\) is uncountable, a Kolmogorovian (real-valued) probability distribution must violate regularity. (See e.g. Hájek 2003b.) This has led