  A Positive Operator Valued Measure (POVM) is a map
$F:\mathcal{B}(X)\to\mathcal{L}_s^+(\mathcal{H})$ from the Borel
$\sigma$-algebra of a topological space $X$ to the space of positive
self-adjoint operators on a Hilbert space $\mathcal{H}$. We assume $X$ to be
Hausdorff, locally compact and second countable and prove that a POVM $F$ is
commutative if and only if it is the smearing of a spectral measure $E$ by
means of a Feller Markov kernel. Moreover, we prove that the smearing can be
realized by means of a strong Feller Markov kernel if and only if $F$ is
uniformly continuous. Finally, we prove that a POVM which is norm bounded by a
finite measure $\nu$ admits a strong Feller Markov kernel.
  That provides a characterization of the smearing which connects a commutative
POVM $F$ to a spectral measure $E$ and is relevant both from the mathematical
and the physical viewpoint since smearings of spectral measures form a large
and very relevant subclass of POVMs: they are paradigmatic for the modeling of
certain standard forms of noise in quantum measurements, they provide optimal
approximators as marginals in joint measurements of incompatible observables
\cite{Busch}, they are important for a range of quantum information processing
protocols, where classical post-processing plays a role \cite{Heinosaari}.
  The mathematical and physical relevance of the results is discussed and
particular emphasis is given to the connections between the Markov kernel and
the imprecision of the measurement process.
