  We introduce a multivariate Markov transform which generalizes the well-known
one-dimensional Stieltjes transform from the Moment problem and Spectral
theory. Our main result states that two measures {\mu} and {\nu} with bounded
support contained in the zero set of a polynomial P(x) are equal if they
coincide on the subspace of all polynomials of polyharmonic degree N_{P} where
the natural number N_{P} is explictly computed by the properties of the
polynomial P(x). The method of proof depends on a definition of a multivariate
Markov transform which another major objective of the present paper. The
classical notion of orthogonal polynomial of second kind is generalized to the
multivariate setting: it is a polyharmonic function which has similar features
as in the one-dimensional case.
