# Definition:Stieltjes Function of Measure on Real Numbers

## Definition

Let $\mu$ be a measure on $\R$ with the Borel $\sigma$-algebra $\mathcal B \left({\R}\right)$.

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

$F_\mu \left({x}\right) := \begin{cases} \mu \left({ \left[{0 \,.\,.\, x}\right) \, }\right) & \text{if } x > 0\\ 0 & \text{if } x = 0\\ - \mu \left({ \left[{x \,.\,.\, 0}\right) \, }\right) & \text{if } x < 0 \end{cases}$

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