# Definition:Pre-Measure of Finite Stieltjes Function

## Definition

Let $\mathcal J_{ho}$ 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 $\mu_f: \mathcal J_{ho} \to \overline \R_{\ge 0}$ defined by:

$\mu_f \left({ \left[{a \,.\,.\, b}\right) \, }\right) := \begin{cases} f \left({b}\right) - f\left({a}\right) & \text{if } b \ge a \\ 0 & \text{otherwise} \end{cases}$

where $\overline \R_{\ge 0}$ denotes the set of positive extended real numbers.