A Geometrical Characterization of Commutative Positive Operator Valued Measures

We show that a POV measure $F$ on the Borel $\sigma$-algebra of the reals $\mathcal{B}(\mathbb{R})$ is commutative if and only if there exists a PV measure $E$ on $\mathcal{B}(\mathbb{R})$ and, for every $\lambda$ in the spectrum of $E$, a probability measure $\gamma_{(\cdot)}(\lambda)$ on $\mathcal{B}(\mathbb{R})$ such that the effect $F(\Delta)$ coincides with $\gamma_{\Delta}(A)$, where $A$ is the self-adjoint operator associated to $E$. The relevance of this result to the theory of the sharp reconstruction is analysed.