The aim of this master thesis was to provide the motivation for the introduction of a random closed set. Since a random closed set is a natural generalization of the concept of a random vector, at the beginning we provided an overview of some well known results. After defining the concept of a random closed set, we presented some simple examples. We have introduced the distribution and capacity functions and demonstrated their basic properties. We have presented the Choquet’s theorem which gives the necessary and sufficient conditions for a function to be a capacity function of a random closed set. We have defined the Choquet integral which allows the integration of the nonadditive functions. Capacity functionals are nonadditive so we use Choquet integral to define the three modes of convergence of a series of random closed sets. Lastly, we observed what happens with the convergence of the ME plots, that is, the mean excess function plots. ME plots are used in the analysis and modeling of extreme value observations. On the simulated data from the generalized Pareto distribution and the F distribution and on the real–world data, we illustrated the theorem of the convergence of ME plots to a linear function.