AbstractA special class of “analytic measures” in the totality of measures orthogonal to the algebra of rational functions on a compact set X⊂C is introduced. It is proved that there always exist nontrivial (i.e., nonzero) analytic measures provided that R(X) ≠ C(X). We also give sufficient conditions in order to have the linear span of analytic measures be weak (∗) dense in the whole annihilator of the algebra R(X)