Capacities with Values in Compact Hausdorff Lattices

For a compact Hausdorff topological lattice L the set M _L X of L-valued normed regular capacities on a compact Hausdorff topological space X is investigated. It is shown that the set M _L X carries a compact Hausdorff topology, and M _L extends to a weakly τ-normal functor in the category of compacta. If L is an upper and lower Lawson semilattice, then M _L is the functorial part of two semimonads. These semimonads coincide and are a monad if and only if L is distributive, i.e., is a Lawson lattice. The obtained results have a natural interpretation if capacities are regarded as subjective estimates of likelihood of realization of events in conditions of uncertainty.