Commutative POV-measures: form the Choquet representation to the Markov kernel and back

We study two relevant characterizations of a commutative positive operator valued measure (POVM) F. The first one is a Choquet type of an integral representation. It introduces a measure ν on the space of the projection valued measures (PVMs) and describes F as an integral over this space. The second one represents a commutative POVM F as the randomization of a single PVM E by means of a Markov kernel μ. We show that one can be derived from the other. We also elaborate upon some previous results on Choquet’s representation of Markov kernels and find a functional relationship between ν and μ. Finally, we analyze some relevant particular cases and provide some physically relevant examples which include the unsharp position observables.