This chapter deals with probabilistic choice when the number of options is infinite. The choice space is a compact set S⊆R k   and we model choice over S  as a limit of choices over triangular sequences {x n1 ,…,x nn }⊆S  as n→∞  . We employ the theory of random sup measures and show that in the limit when n→∞  , people behave as though they are maximising over a random sup measure. Thus, our results complement Resnick and Roy’s [18] theory of probabilistic choice over infinite sets. They define choice as a maximisation over a stochastic process on S  with upper semi-continuous (usc) paths. This connects to our model as their random usc function can be defined as a sup-derivative of a random sup measure, and their maximisation problem can be transformed into a maximisation problem over this random sup measure. One difference remains though: with our model the limiting random sup measures are independently scattered, without usc paths. A benefit of our model is that we provide a way of connecting the stochastic process in their model with finite case distributional assumptions, which are easier to interpret. In particular, when choices are valued additively with one deterministic and one random part, we explore the importance of the tail behaviour of the random part, and show that the exponential distribution is an important boundary case between heavy-tailed and light-tailed distributions.