Abstract This note generalizes Gul and Pesendorfer’s random expected utility theory, a stochastic reformulation of von Neumann–Morgenstern expected utility theory for lotteries overaﬁnitesetofprizes,tothecircumstanceswithacontinuumofprizes.Let[0, M]denote thiscontinuumofprizes;assumethateachutilityfunctioniscontinuous,letC0[0, M]bethe set of all utility functions which vanish at the origin, and deﬁne a random utility function to be a ﬁnitely additive probability measure on C0[0, M] (associated with an appropriate algebra).Itisshownherethatarandomchoiceruleismixturecontinuous,monotone,linear, and extreme if, and only if, the random choice rule maximizes some regular random utility function. To obtain countable additivity of the random utility function, we further restrict ourconsiderationtothoseutilityfunctionsthatarecontinuouslydifferentiableon[0, M]and vanish at zero. With this restriction, it is shown that a random choice rule is continuous, monotone, linear, and extreme if, and only if, it maximizes some regular, countably additive random utility function. This generalization enables us to make a discussion of risk aversion in the framework of random expected utility theory. Keywords Expected utility·Random utility·Random choice·Independence axiom·Risk aversion