|
|
| Line 1: |
Line 1: |
| − | A generalization of the [[Perron integral|Perron integral]] for a function of one real variable. A finite function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p0723801.png" /> is said to be integrable in the sense of Perron–Stieltjes with respect to a finite function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p0723802.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p0723803.png" /> if on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p0723804.png" /> there exists a major function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p0723805.png" /> and a minor function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p0723806.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p0723807.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p0723808.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p0723809.png" /> having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238010.png" /> and such that at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238011.png" />, | + | {{TEX|done}} |
| | + | A generalization of the [[Perron integral|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 $x\in[a,b]$, |
| | | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238012.png" /></td> </tr></table>
| + | $$M(x+\beta)-M(x-\alpha)\geq f(x)(G(x+\beta)-G(x-\alpha))$$ |
| | | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238013.png" /></td> </tr></table>
| + | $$n(x+\beta)-m(x-\alpha)\leq f(x)(G(x+\beta)-G(x-\alpha))$$ |
| | | | |
| − | for all sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238015.png" />, while the greatest lower bound of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238017.png" /> is any such major function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238018.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238019.png" />, and the least upper bound of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238021.png" /> is any such minor function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238022.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238023.png" />, coincide. Their common value is called the Perron–Stieltjes integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238024.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238025.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238026.png" /> and is denoted by | + | 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 |
| | | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238027.png" /></td> </tr></table>
| + | $$(P-S)\int_a^bf(x)dG(x).$$ |
| | | | |
| | This generalization of the Perron integral was introduced by A.J. Ward [[#References|[1]]]. | | This generalization of the Perron integral was introduced by A.J. Ward [[#References|[1]]]. |
| Line 17: |
Line 18: |
| | | | |
| | ====Comments==== | | ====Comments==== |
| − | A major function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238028.png" /> of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238029.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238030.png" /> with respect to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238031.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238032.png" /> is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238033.png" /> such that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238034.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238035.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238036.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238037.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238038.png" />. A minor function is defined similarly, but with the inequality sign reversed. Thus, a suitable lower derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238039.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238040.png" /> majorizes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238041.png" />. More generally one considers additive interval functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238043.png" /> satisfying the above property, cf. [[#References|[2]]] for details. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238044.png" /> is an ordinary function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238045.png" />, then its associated additive interval function, denoted by the same letter, is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238046.png" />. A major function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238047.png" />, without any specification of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238048.png" />, is one with respect to the identify function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072380/p07238050.png" />. | + | 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. [[#References|[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 $x\mapsto x$, $x\in[a,b]$. |
Revision as of 08:18, 23 July 2014
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
$$(P-S)\int_a^bf(x)dG(x).$$
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 $x\mapsto x$, $x\in[a,b]$.
How to Cite This Entry:
Perron-Stieltjes integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Perron-Stieltjes_integral&oldid=22895
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article