We present a general formalism for giving a measure space paired with a separable Hilbert space a quantum version based on a normalized positive operator-valued measure. The latter are built from families of density operators labeled by points of the measure space. We especially focus on various probabilistic aspects of these constructions. Simple ormore elaborate examples illustrate the procedure: circle, two-sphere, plane and half-plane. Links with Positive-Operator Valued Measure (POVM) quantum measurement and quantum statistical inference are sketched.