Let $\mu$ be a positive borel measure on $\mathbb{R}$ with $\int_{\mathbb{R}} x^n d\mu (x) = s_n, n \in N_0$. Find a hilbert space $\tilde{\mathcal{H}}$ and a self adjoint
extension $A$ of $S$ in $\tilde{\mathcal{H}}$ such that $\mu = \mu_A$, where
$S_0$ is the symmetric densly defined and closable operator $p \mapsto zp(z)$ for $p \in \mathbb{C}[z]$ and $S = \overline{S_0}$ is its closure on $\mathcal{C}(\mathcal{H})$ where $\mathcal{H}$ is the hilbert space completion of $\mathbb{C}[z]$.
I tried to define the hilbert space $\tilde{\mathcal{H}}$ through $p \in \tilde{\mathcal{H}}$ if there exists a $p_n \in \mathcal{H}$ with $p_n \to p$ for $n \to \infty$ with respect to the norm of $\mathcal{H}$ and
\begin{equation}
(S(p_n - p_m), p_n - p_m) \to 0 \mbox{ for } n \to \infty.
\end{equation}
Now I defined $A$ through a quadratic form as follows
\begin{equation}
(Ap, p)_{\tilde{\mathcal{H}}} := \lim_{n \to \infty}(Sp_n, p_n )_{\mathcal{H}}.
\end{equation}
This operator is self adjoint and non negative with $\mbox{dom } A = \mbox{dom } A^*$ and therefore there exists a spectral measure $E_A$ such that I can define
\begin{equation}
\mu_A := (E_A(\Delta) 1, 1)
\end{equation}
for borel sets $\Delta$ of $\mathbb{R}$ and get
\begin{equation}
s_n = ((S)^n 1, 1) = ((A)^n1, 1) = \int_{\mathbb{R}}{x^n d \mu_A(x)}
\end{equation}
where I suppressed the product notation. Now
\begin{equation}
\mu_A(\mathbb{R}) = \int_{\mathbb{R}}{x^0 d \mu_A(x)} = s_0 = \int_{\mathbb{R}}{x^0 d \mu(x)} = \mu(\mathbb{R})
\end{equation}
I'am not sure if this is a way to construct such a operator $A$.
I also thought about using $\mu$ to define $A$ but struggled to figure out how the Operator $A$ with spectral measure $E_A$ and $\mu_A(\Delta) = (E_A(\Delta) 1, 1) = \mu(\Delta)$ for borel sets $\Delta$ would be a self adjoint extension of $S$.