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The theory of random sets is demonstrated to prove useful for the theory ofrandom operators. A random operator is here defined by requiring the graph tobe a random set. It is proved that the spectrum and the set of eigenvalues ofrandom operators are random sets. These results seem to be a novelty even inthe case of random bounded operators. The main technical tools are given by themeasurable selection theorem, the measurable projection theorem, and acharacterisation of the spectrum by approximate eigenvalues of the operator andthe adjoint operator. A discussion of some of the existing definitions of theconcept of a random operator is included at the end of the paper.  Keywords: Random operators; Set-valued functions; General topics in linearspectral theory; Random operators and equations; Stochastic integrals;Disordered systems

The spectrum of a random operator is a random set