Semispectral Measures and Feller markov Kernels

Abstract

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.