Classical and nonclassical randomness in quantum measurements

The space of positive operator-valued measures on the Borel sets of a compact (or even locally compact) Hausdorff space with values in the algebra of linear operators acting on a d-dimensional Hilbert space is studied from the perspectives of classical and non-classical convexity through a transform $\Gamma$ that associates any positive operator-valued measure with a certain completely positive linear map of the homogeneous C*-algebra $C(X)\otimes B(H)$ into $B(H)$. This association is achieved by using an operator-valued integral in which non-classical random variables (that is, operator-valued functions) are integrated with respect to positive operator-valued measures and which has the feature that the integral of a random quantum effect is itself a quantum effect. A left inverse $\Omega$ for $\Gamma$ yields an integral representation, along the lines of the classical Riesz Representation Theorem for certain linear functionals on $C(X)$, of certain (but not all) unital completely positive linear maps $\phi:C(X)\otimes B(H) \rightarrow B(H)$. The extremal and C*-extremal points of the space of POVMS are determined.