# Perron-Stieltjes integral

A generalization of the Perron integral for a function of one real variable. A finite function $f$ is said to be integrable in the sense of Perron–Stieltjes with respect to a finite function $G$ on $[a,b]$ if on $[a,b]$ there exists a major function $M$ and a minor function $m$ for $f$ with respect to $G$ on $[a,b]$ having $M(a)=m(a)=0$ and such that at each point $x\in[a,b]$,

$$M(x+\beta)-M(x-\alpha)\geq f(x)(G(x+\beta)-G(x-\alpha))$$

$$n(x+\beta)-m(x-\alpha)\leq f(x)(G(x+\beta)-G(x-\alpha))$$

for all sufficiently small $\alpha\geq0$ and $\beta\geq0$, while the greatest lower bound of the numbers $M(b)$, where $M$ is any such major function of $f$ with respect to $G$, and the least upper bound of the numbers $m(b)$, where $m$ is any such minor function of $f$ with respect to $G$, coincide. Their common value is called the Perron–Stieltjes integral of $f$ with respect to $G$ on $[a,b]$ and is denoted by

$$(P-S)\int\limits_a^bf(x)dG(x).$$

This generalization of the Perron integral was introduced by A.J. Ward .

How to Cite This Entry:
Perron-Stieltjes integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Perron-Stieltjes_integral&oldid=32524
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article