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 $μ_{(\cdot)}(\cdot):Γ\times\mathcal{B}(\mathbb{R})\to[0,1]$, $Γ\subsetσ(A)$, $E(Γ)=\mathbf{1}$, such that $$F(Δ)=\int_Γμ_Δ(λ)\,dE_λ,$$ \noindent and $μ_{(Δ)}$ is continuous for each $Δ\in R$ where, $R\subset\mathcal{B}(\mathbb{R})$ is a ring which generates the Borel $σ$-algebra of the reals $\mathcal{B}(\mathbb{R})$. Moreover, $μ_{(\cdot)}(\cdot)$ is a Feller Markov kernel and separates the points of $Γ$. we prove that $F$ admits a strong Feller Markov kernel $μ_{(\cdot)}(\cdot)$, if and only if $F$ is uniformly continuous. Finally, we prove that if $F$ is absolutely continuous with respect to a regular finite measure $ν$ 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 $μ$ and the imprecision of the measurement apparatus.