Probabilistic-valued decomposable set functions with respect to triangle functions

Real world applications often require dealing with the situations in which the exact numerical values of the (sub)measure of a set may not be provided, but at least some probabilistic assignment still could be done. Also, several concepts in uncertainty processing are linked to the processing of distribution functions. In the framework of generalized measure theory we introduce the probabilistic-valued decomposable set functions which are related to triangle functions as natural candidates for the “addition” in an appropriate probabilistic metric space. Several set functions, among them the classical (sub)measures, previously defined τTτT-submeasures, τL,AτL,A-submeasures as well as recently introduced Shen’s (sub)measures are described and investigated in a unified way. Basic properties and characterizations of τ-decomposable (sub)measures are also studied and numerous extensions of results from the above mentioned papers are provided.