  We present a general formalism for giving a measure space paired with a
separable Hilbert space a quantum version based on normalized positive
operator-valued measure. The latter are built from families of density
operators labelled by points of the measure space. We specially focus on
various probabilistic aspects of these constructions. Simple or more elaborate
examples illustrate the procedure: circle, 2-sphere, plane, half-plane. Links
with POVM quantum measurement and quantum statistical inference are sketched.
