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