Difference between revisions of "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 [1].

References

 [1] A.J. Ward, "The Perron–Stieltjes integral" Math. Z. , 41 (1936) pp. 578–604 [2] S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) [3] I.A. Vinogradova, V.A. Skvortsov, "Generalized integrals and Fourier series" Itogi Nauk. Mat. Anal. 1970 (1971) pp. 65–107 (In Russian)

A major function $U$ of a function $f$ on $[a,b]$ with respect to a function $G$ on $[a,b]$ is a function $U$ such that for each $x\in[a,b]$ there is an $\epsilon>0$ such that $U(d)-U(c)\geq f(x)(G(d)-G(c))$ for all $c\leq x\leq d$ with $|d-c|<\epsilon$. A minor function is defined similarly, but with the inequality sign reversed. Thus, a suitable lower derivative of $U$ with respect to $G$ majorizes $f$. More generally one considers additive interval functions $U$ and $G$ satisfying the above property, cf. [2] for details. If $G$ is an ordinary function on $[a,b]$, then its associated additive interval function, denoted by the same letter, is $G([c,d])=G(d)-G(c)$. A major function of $f$, without any specification of a $G$, is one with respect to the identify function $x\mapsto x$, $x\in[a,b]$.