This paper presents a model of random matching between individuals chosen from large populations. We assume that the populations and the set of encounters are infinite but countable and that the encounters are i.i.d. random variables. Furthermore, the probability distribution on individuals according to which they are chosen for each encounter is "uniform", which also implies that it is only finitely additive. Although the probability measure which governs the whole matching process also fails to be (fully) sigma-additive, it still retains enough continuity properties to allow for the use of the law of large numbers. This, in turn, guarantees that the aggregate process will (almost surely) behave "nicely", i.e., that there will be no aggregate uncertainty.