Let \((\Omega ,{\mathcal A},P)\) be a probability space, \(E\) a Hausdorff, locally compact and second countable topological space, and \(\mathcal U\) the family of all upper semicontinuous (u.s.c., for short) functions \(f:E\to [0,1]\). Using hypographs, \(\mathcal U\) can be embedded into the space \({\mathcal F}(E\times [0,1])\) of all closed subsets of the product \(E\times [0,1]\). The space \({\mathcal F}(E\times [0,1])\) is endowed with the hit-or-miss topology (called also Fell's topology); it is Hausdorff, compact and second countable. The authors prove that the space \({\mathcal U}^-\) of hypographs of elements of \(\mathcal U\) is a closed subset of \({\mathcal F}(E\times [0,1])\). Any measurable map \(X:\Omega\to {\mathcal F}(E\times [0,1])\) is called a random closed set on \(E\times [0,1]\). Probability laws of such random sets are characterized by capacity functionals on the family of compact subsets of \(E\times [0,1]\), via the Choquet theorem. Random u.s.c. functions are random closed sets on \(E\times [0,1]\) which take values in \({\mathcal U}^-\). The authors study special properties of capacity functionals corresponding to random u.s.c. functions. Recently related problems were studied by [\textit{Y. Ogura}, Soft methods for integrated uncertainty modelling. Proceedings of the 2006 international workshop on soft methods in probability and statistics (SMPS 2006), Bristol, UK, September 5--7, 2006. Berlin: Springer. Advances in Soft Computing, 145--151 (2006; Zbl 1110.54011)].