Pre-Measure of Finite Stieltjes Function is Pre-Measure

Theorem
Let $\mathcal{J}_{ho}$ denote the collection of half-open intervals in $\R$.

Let $f: \R \to \R$ be a finite Stieltjes function.

Then the pre-measure of $f$, $\mu_f: \mathcal{J}_{ho} \to \overline{\R}_{\ge0}$ is a pre-measure.

Here, $\overline{\R}_{\ge 0}$ denotes the set of positive extended real numbers.