A geometrical characterization of commutative positive operator valued measures

We show that a POV measure F on the Borel σ-algebra of the reals B(R) is commutative if and only if there exists a PV measure E on B(R) and, for every λ in the spectrum of E, a probability measure γ(·)(λ) on B(R) such that the effect F(Δ) coincides with γΔ(A), where A is the self-adjoint operator associated to E. The relevance of this result to the theory of the sharp reconstruction is analyzed