At first the author proves the existence of a unique measurable solution for the following random variational inequality: Find for every elementary event
is a closed convex non-empty subset of a separable Hilbert space and for every
being a linear form, respectively, bilinear form. It is assumed that
is nonnegative. Then the author considers in detail a more specialized inequality, where the data decompose in deterministic data and given real-valued random variables. Existence and uniqueness results are given, various approximation procedures and its convergence are studied. The general theory is applied to a Helmholtz-like elliptic equation with Signorini boundary conditions.