The concept of Fuzzy-random-variable was introduced as an analogous notion to randomvariable in order to extend statistical analysis to situations when the outcomes of some random experiment are fuzzy sets. But in contrary to the classical statistical methods no unique definition has been established yet. In this paper a set-theoretical concept of fuzzy-random-variable will be presented. This notion provides a useful framework to compare different concepts of fuzzy-random-variables, using methods of general topology and drawing on results from topological measure theory and the theory of analytic spaces. As the main result, it will be shown that the introduced concept of fuzzy-random-variable is a unification of the already known ones.