  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.
