A note on the coincidence of decomposition integrals and superdecomposition integrals

Abstract

It is known that the decomposition integral IH and the superdecomposition integral IH related to the system H of all finite chains from A⧹{∅}, where A is a σ-algebra of the subsets of a nonvoid set X, coincide with each other for each monotone measure and for each nonnegative measurable function, and they are equal to the Choquet integral. In this note, we show that the converse is also true. That is, for a complete system H of finite set systems from A⧹{∅}, if IH=IH holds for each monotone measure and for each nonnegative measurable function, then the system H and the system of all finite chains from A⧹{∅} are equivalent in some sense and the integrals involved are the Choquet integral.