On a problem of Radko Mesiar: General form of IF-probabilities

Recently, axiomatic definition has been given of probability on the family of all IF-events. Any IF-event consists of two fuzzy events and the probability of the event is a compact interval on the real line, hence IF-probability is a mapping from the family of IF-events to the family of all compact intervals. Prof. Mesiar asked to construct such a probability measure which is nondivisible into two components. In this paper, we describe the form of all IF-probability measures. As a consequence we obtain an existence of a nondivisible probability, hence the positive answer to the Mesiar question.