The author generalizes the classical notions of weak convergence and strong convergence in measure theory. This is done by taking a set E together with two partial orders such that the first order satisfies the countable Dedekind condition (that is, every nonempty countable subset of E which is bounded above has a supremum), and the second order is also subject to certain conditions. Now take the set of all positive-valued functions on E which are increasing with respect to the first order. The usual concepts of measure theory, such as upper and lower envelopes of a function, weak convergence, etc., are adapted to this general setting. The results so developed are then applied to capacities and to certain special classes of capacities.
