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In the framework of the generalized measure theory the decomposableprobabilistic-valued set functions are introduced with triangle functions$\tau$ in an appropriate probabilistic metric space as natural candidates forthe "addition", leading to the concept of $\tau$-decomposable measures. Severalset functions, among them the classical (sub)measures, previously defined$\tau_T$-submeasures, $\tau_{L,A}$-submeasures as well as recently introducedShen's (sub)measures are described and investigated in a unified way. Basicproperties and characterizations of $\tau$-decomposable (sub)measures are alsostudied and numerous extensions of results from the above mentioned papers areprovided.

Probabilistic-valued decomposable set functions with respect to triangle functions