Fuzzy Observables: from Weak Markov Kernels to Markov Kernels

Abstract

We provide a proof based on transfinite induction that every weak Markov kernel is equivalent to a Markov kernel. We only assume the space where the weak Markov kernel is defined to be second countable and metrizable. That generalizes some previous results where the kernel is required to be defined on a standard Borel space (which is second countable and completely metrizable) and the framework is the theory of stochastic operators. This property of weak Markov kernels is at the root of the characterization of a commutative POVM as the fuzzification of a spectral measure through a Markov kernel. As a result, the characterization of commutative POVMs is also generalized. We then revisit the relationships between weak Markov kernels, Markov kernels, commutative POVMs and fuzzy observables.