All quantum expectation values as classical statistical mean values

Given a physical quantum system described by a Hilbert H, for any bounded quantum observable (a bounded self-adjoint operator) T it is possible to define several "hidden observable" functions f:H->R associated to T and for any quantum mixed state (a density matrix) D it is possible to define several "hidden mixed states" (probability measures) m on H associated to D in such a way that the following equality is verified: Trace[ b(T). D] =integral[b(f(psi)).dm(psi) whatever is the continuous function b:R->R. This formula gives a general way to express any expectation value computable in a quantum theory as a classical statistical mean value.