On Two Generalizations for k-Additivity.

There are two generalizations for k-additive set functions: constructive k-additivity and formulaic k-additivity. We study some properties around these concepts and their relations. A constructively k-additive set function is always formulaic k-additive. For a distorted measure, these two concepts are equivalent. Under certain conditions of “bounded variation” and “continuity at the \(\emptyset \),” we prove the constructive k-additivity for a formulaic k-additive set function.