  We give a characterization of commutative semispectral measures by means of
Feller and Strong Feller Markov kernels. In particular:
  {itemize} we show that a semispectral measure $F$ is commutative if and only
if there exist a self-adjoint operator $A$ and a Markov kernel
$\mu_{(\cdot)}(\cdot):\Gamma\times\mathcal{B}(\mathbb{R})\to[0,1]$,
$\Gamma\subset\sigma(A)$, $E(\Gamma)=\mathbf{1}$, such that
$$F(\Delta)=\int_{\Gamma}\mu_{\Delta}(\lambda)\,dE_{\lambda},$$ \noindent and
$\mu_{(\Delta)}$ is continuous for each $\Delta\in R$ where,
$R\subset\mathcal{B}(\mathbb{R})$ is a ring which generates the Borel
$\sigma$-algebra of the reals $\mathcal{B}(\mathbb{R})$. Moreover,
$\mu_{(\cdot)}(\cdot)$ is a Feller Markov kernel and separates the points of
$\Gamma$. we prove that $F$ admits a strong Feller Markov kernel
$\mu_{(\cdot)}(\cdot)$, if and only if $F$ is uniformly continuous. Finally, we
prove that if $F$ is absolutely continuous with respect to a regular finite
measure $\nu$ then, it admits a strong Feller Markov kernel. {itemize} The
mathematical and physical relevance of the results is discussed giving a
particular emphasis to the connections between $\mu$ and the imprecision of the
measurement apparatus.
