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 ${\mu }_{(\cdot )}(\cdot ):\Gamma \times \mathcal{B}(\mathbb{R})\to [0,1]$, $\Gamma \subset \sigma (A)$, $E(\Gamma )=1$, such that $$F(\Delta )={\int }_{\Gamma }{\mu }_{\Delta }(\lambda )d{E}_{\lambda },$$ \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.

27 pages

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