 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 to a cottage industry of exploring whether regularity can be preserved by allowing the ranges of probability functions to be richer fields than the real numbers. For example, Bernstein and Wattenberg (1969) show that there exists a regular hyperreal-valued probability function for the dart throw at [0, 1] that we imagined earlier. Each landing point receives infinitesimal probability. Williamson (2007) argues that an infinite sequence of tosses of a fair coin all landing heads must receive probability 0 rather than some infinitesimal probability; Howson (2019) challenges the argument. The debate for and against preserving regularity continues, with Easwaran (2014) and Pruss (2012, 2013b, 2014) against, Benci, Horsten and Wenmackers (2012, 2016) for—offering NAP as a way of doing so, again assigning infinitesimal probabilities where Kolmogorov’s theory would assign 0’s.