  We consider questions related to a quantization scheme in which a classical
variable f:\Omega\to R on a phase space \Omega is associated with a
semispectral measure E^f, such that the moment operators of E^f are required to
be of the form \Gamma(f^k), with \Gamma a suitable mapping from the set of
classical variables to the set of (not necessarily bounded) operators in some
Hilbert space. In particular, we investigate the situation where the map \Gamma
is implemented by the operator integral with respect to some fixed positive
operator measure. The phase space \Omega is first taken to be an abstract
measurable space, then a locally compact unimodular group, and finally R^2,
where we determine explicitly the relevant operators \Gamma(f^k) for certain
variables f, in the case where the quantization map \Gamma is implemented by a
translation covariant positive operator measure. In addition, we consider the
question under what conditions a positive operator measure is projection
valued.
