Are nonmeasurable sets significant for epistemology?

Probabilism holds that rational credence functions are probability functions defined over some probability space $(\Omega, \F, P)$. According to some recent philosophical arguments, in some situations, rational credence function must be \textit{total}, i.e. $\F=2^\Omega$, a view which I call \textit{credence totalism}. Arguments for credence totalism are based on the premise that non-Lebesgue measurable subsets of $\mathbb{R}$ are epistemically significant, in the sense that an agent has reasons to assign probability to these sets. This paper argues that nonmeasurable sets are not epistemically significant in this sense. Consequently, the arguments for credence totalism are not successful. My argument is based on a careful consideration of the role of the Axiom of Choice in probabilistic practice. I also discuss some topics considered closely related, viz. the existence of total chance functions and the truth value of the Continuum Hypothesis. I argue that the role of nonmeasurability in epistemology does not shed light on these issues.