An integral with respect to probabilistic-valued decomposable measures

• We examine probabilistic-valued decomposable (sub)measures.
• We introduce an integral of non-negative real-valued functions with respect to them.
• Additivity and linearity of the integral depend on the underlying triangle function.
• The integral brings a new tool in approximate reasoning and uncertainty processing.

Several concepts of approximate reasoning in uncertainty processing are linked to the processing of distribution functions. In this paper we make use of probabilistic framework of approximate reasoning by proposing a Lebesgue-type approach to integration of non-negative real-valued functions with respect to probabilistic-valued decomposable (sub)measures. Basic properties of the corresponding probabilistic integral are investigated in detail. It is shown that certain properties, among them linearity and additivity, depend on the properties of the underlying triangle function providing (sub)additivity condition of the considered (sub)measure. It is demonstrated that the introduced integral brings a new tool in approximate reasoning and uncertainty processing with possible applications in several areas.