Probabilistic Measures and Integrals: How to Aggregate Imprecise Data

This paper develops the theory of probabilistic-valued measures and integrals as a suitable aggregation tool for dealing with certain types of imprecise information. The motivation comes from Moore's interval mathematics, where the use of intervals in data processing is due to measurement inaccuracy errors. In case of rounding, the intervals can be considered in distribution function form linked to random variables uniformly distributed over the relevant intervals. We demonstrate how the convolution of distribution functions is taken into account, and integration with respect to probabilistic-valued measures is converted into convolving certain distribution functions. We also improve some existing features of the integral and investigate its convergence properties.