The distribution function of a probability measure on a Polish ultrametric space

In applied sciences, the scientific community uses simultaneously different kinds of information coming from several sources in order to infer a conclusion or working decision. In the literature there are many techniques for merging the information and providing, hence, a meaningful fused data. In mostpractical cases such fusion methods are based on aggregation operators on somenumerical values, i.e. the aim of the fusion process is to obtain arepresentative number from a finite sequence of numerical data. In the aforementioned cases, the input data presents some kind of imprecision and for thisreason it is represented as fuzzy sets. Moreover, in such problems the comparisons between the numerical values that represent the information described by the fuzzy sets become necessary. The aforementioned comparisons are made by means of a distance defined on fuzzy sets. Thus, the numerical operators aggregating distances between fuzzy sets as incoming data play a central role in applied problems. Recently, J.J. Nieto and A. Torres gave some applications of the aggregation of distances on fuzzy sets to the study of real medical data in /cite{Nieto}. These applications are based on the notion of segment joining two given fuzzy sets and on the notion of set of midpoints between fuzzy sets. A few results obtained by Nieto and Torres have been generalized in turn by Casasnovas and Rossell/'{o} in /cite{Casas,Casas2}. Nowadays, quasi-metrics provide efficient tools in some fields of computer science and in bioinformatics. Motivated by the exposed facts, a study of segments joining two fuzzy sets and of midpoints between fuzzy sets when the measure, used for comparisons, is a quasi-metric has been made in /cite{Casas3, SebVal2013,TiradoValero}. (Author)