Stieltjes Function of Measure is Stieltjes Function

Theorem
Let $\mu$ be a measure on $\R$ with the Borel $\sigma$-algebra $\mathcal B \left({\R}\right)$.

Then $F_\mu: \R \to \overline{\R}$, the Stieltjes function of $\mu$, is a Stieltjes function.