Definition:Pre-Measure of Finite Stieltjes Function

Definition

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.

Also see

• Pre-Measure of Finite Stieltjes Function is Pre-Measure
• Pre-Measure of Finite Stieltjes Function Extends to Unique Measure
• Definition:Measure of Finite Stieltjes Function

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 7$: Problem $9 \ \text{(ii)}$