  Submodular set functions are undoubtedly among the most important building
blocks of combinatorial optimization. Somewhat surprisingly, continuous
counterparts of such functions have also appeared in an analytic line of
research where they found applications in the theory of finitely additive
measures, nonlinear integrals, and electric capacities. Recently, a number of
connections between these two branches have been established, and the aim of
this paper is to generalize further results on submodular set functions on
finite sets to the analytic setting.
  We first extend the notion of duality of matroids to submodular set
functions, and characterize the uniquely determined decomposition of a
submodular set function into the sum of a nonnegaive charge and an increasing
submodular set function in which the charge is maximal. Then, we describe basic
properties of infinite-alternating set functions, a subclass of submodular set
functions that serves as an analytic counterpart of coverage functions. By
relaxing the monotonicity assumption in the definition, we introduce a new
class of submodular functions with distinguished structural properties that
includes, among others, weighted cut functions of graphs. We prove that, unlike
general submodular set functions over an infinite domain, any
infinite-alternating set function can be written as the sum of an increasing
and a decreasing submodular function or as the difference of two increasing
submodular functions, thus giving extension of results on monotonic
decompositions in the finite case. Finally, motivated by its connections to
graph parameters such as the maximum size of a cut and the maximum size of a
fractional triangle packing, we study the structure of such decompositions for
weighted cut functions of undirected graphs.
