Definition:Stieltjes Function of Measure on Real Numbers

Definition

Let $\mu$ be a measure on $\R$ with the Borel $\sigma$-algebra $\map \BB \R$.

The Stieltjes function of $\mu$ is the mapping $F_\mu: \R \to \overline \R$ defined by:

$\map {F_\mu} x := \begin{cases}

\map \mu {\hointr 0 x} & \text{if } x > 0\\ 0 & \text{if } x = 0\\ - \map \mu {\hointr x 0} & \text{if } x < 0 \end{cases}$

where $\overline \R$ denotes the extended real numbers.

Also see

• Stieltjes Function of Measure is Stieltjes Function

Sources

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