  In his influential work Choquet systematically studied capacities on Boolean
algebras in a topological space, and gave a probabilistic interpretation for
completely monotone (and completely alternating) capacities. Beyond complete
monotonicity we can view a capacity as a marginal condition for probability
distribution over the distributive lattice of dual order ideals. In this paper
we discuss a combinatorial approach when capacities are defined over a finite
lattice, and investigate Fr\'{e}chet bounds given the marginal condition,
probabilistic interpretation of difference operators, and stochastic
inequalities with completely monotone capacities.
