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.

Proof
By definition, $F_\mu$ is a Stieltjes function iff it is increasing and left-continuous.

$F_\mu$ is increasing
Let $x, y \in \R$ such that $x \le y$.

It is apparent that for all $z \in \R$ that:


 * $z \le 0 \implies F_\mu \left({z}\right) \le 0$
 * $z \ge 0 \implies F_\mu \left({z}\right) \ge 0$

Therefore, only the cases $x \le y \le 0$ and $0 \le x \le y$ remain.

In the first of these cases, note that:


 * $\left[{y \,.\,.\, 0}\right) \subseteq \left[{x \,.\,. 0}\right)$

Hence from Measure is Monotone, it follows that:


 * $F_\mu \left({x}\right) = -\mu \left({\left[{x \,.\,.\, 0}\right)}\right) \le -\mu \left({\left[{y \,.\,.\, 0}\right)}\right) = F_\mu \left({y}\right)$

The remaining case is analogous, using the observation that:


 * $\left[{0 \,.\,.\, x}\right) \subseteq \left[{0 \,.\,.\, y}\right)$

Hence $F_\mu$ is increasing.