Let \({\mathcal U}_{+}\) be a family of upper semi-continuous real-valued positive functions on \(R^{d}\). For any \(f\in {\mathcal U}_{+}\) the capacity integral is defined as \(E_{A}f=\int_{0}^{\infty}T(\{u\in R^{d}: f(u)\geq t\})dt\), where \(A\) is a random closed set, \(T(K)=P\{A\cap K\neq \emptyset\}\) is a capacity functional, \(K\) is running the class of compact subsets of \(R^{d}\). The author proposes the following definition of the expectation of a random closed set: for the family \(F\) of numerical functions the set \(E_{F}A=\{x:f(x)\leq E_{A}f,\forall f\in F\}\) is said to be \(F\)-expectation of \(A\). If \(F\) coincides with the class of all linear functions, then \(E_{F}A\) coincides with the Aumann expectation of \(A\). The corresponding law of large numbers which generalizes the Arstein-Vitale law of large numbers for the Minkowski addition is proved.
For the entire collection see [Zbl 0896.00028].