Random sets play an essential role in modelling several phenomena in biology, medicine and material science. The first part of the paper, which refers to the theoretical results, a random closed set in \(\mathbb{R}^{d}\) is defined as a function over the probability space and illustrative examples are given. Then the concept of capacity functional was defined and its properties were investigated. In the central chapter of the paper Choquet’s theorem is proved. It tells us what are the necessary and sufficient conditions for a functional to be a capacity functional of a unique random closed set, and thus to uniquely determine the distribution of the corresponding random closed set. In the final part of the paper a simulation study was made the goal of which was to investigate how well the capacity functional describes the realization of a random set and whether the similarity measure based on the capacity functional can be used for clustering the realizations of different models. The results of the research showed that the capacity functional distinguishes the corresponding realizations very well, so it can be used as a measure of similarity for clustering.