Given a nonnegative monotone set function and a nonnegative measurable
function on a measurable space, the Choquet integral determines a new
nonnegative monotone set function that is absolutely continuous with
respect to the original one (in a generalized sense for monotone set f
unctions). This new set function preserves almost all desirable struct
ural characteristics of the original monotone set function, such as co
ntinuity, subadditivity, superadditivity, null-additivity, converse-nu
ll-additivity, autocontinuity, converse-autocontinuity, uniform autoco
ntinuity, uniform converse-autocontinuity, and fuzzy multiplicativity.
Such a construction is a useful method to define sound fuzzy measures
in various applications.