Infinite sequences of linear functionals, positive operator valued measures and Naimark extension theorem

Let $F:\mathcal{B}(\mathbb{R})\to\mathcal{F(H)}$ be a commutative positive operator valued measure from the Borel $\sigma$-algebra of the reals to the space of bounded positive operators on the Hilbert space $\mathcal{H}$. It is well known that there exist a self-adjoint operator $A$ and a Markov kernel $\mu_{(\cdot)}(\lambda)$ such that $F(\Delta)=\mu_{\Delta}(A)$, $\Delta\in\mathcal{B}(\mathbb{R})$. We prove that if, for any $\Delta\in\mathcal{B}(\mathbb{R})$, $F(\Delta)$ is a discrete operator then, $A$ is a linear combination of the moments of $F$. This result allows us to characterize the quantum observable represented by $A$ by means of the Naimark dilation of $F$. The result is connected with the following properties of the sequences of linear functionals. Consider an infinite sequence of linear functionals $\{T_i\}_{i\in\mathbb{N}}$, $T_if=\int f(t)\,d\mu_t(i)$, corresponding to an infinite sequence of probability measures $\{\mu_{(\cdot)}(i)\}_{i\in\mathbb{N}}$, on the Borel $\sigma$-algebra $\mathcal{B}([0,1])$ such that, $\mu_{(\cdot)}(i)\neq\mu_{(\cdot)}(j)$, $i,j\in\mathbb{N},\,\,i\neq j$. There exists a real, bounded, one-to-one continuous function $f$ such that $$T_if=\int f(t)\,d\mu_t(i)\neq \int f(t)\,d\mu_t(j)=T_jf, \quad i,j\in\mathbb{N},\,i\neq j.$$