\begin{definition}[Definition:Pre-Measure of Finite Stieltjes Function]
Let $\JJ_{ho}$ denote the collection of half-open intervals in $\R$.
Let $f: \R \to \R$ be a finite Stieltjes function.
The '''pre-measure of $f$''' is the mapping $\mu_f: \JJ_{ho} \to \overline \R_{\ge 0}$ defined by:
:$\map {\mu_f} {\hointr a b} := \begin{cases}
\map f b - \map f a & \text{if } b \ge a \\
0 & \text{otherwise}
\end{cases}$
where $\overline \R_{\ge 0}$ denotes the set of positive extended real numbers.
\end{definition}