Free moment measures and laws

In [3], it was shown that convex, almost everywhere continuous functions coordinatize a broad class of probability measures on Rn by the map U↦(∇U)#e−Udx. We consider whether there is similar coordinatization non-commutative spaces, with Gibbs measure e−Udx replaced corresponding free law. call laws parameterized in this way moment laws. first case single (and thus commutative) random variable and then regime n variables which are perturbations freely independent semi-circular variables. prove exist little restriction for one dimensional case, small even semi-circle general case.