A generalization of the concept of the Riemann integral, realizing the notion of integrating a function with respect to another function . Let two functions and be defined and bounded on and let . A sum of the form
where , , is called a Stieltjes integral sum. A number is called the limit of the integral sums (1) when if for each there is a such that if , the inequality holds. If the limit exists when and is finite, then the function is said to be integrable with respect to the function over , and the limit is called the Stieltjes integral (or the Riemann–Stieltjes integral) of with respect to , and is denoted by
the function is said to be the integrating function. Th.J. Stieltjes  hit upon the idea of such an integral when studying the positive "distribution of masses" on a straight line defined by an increasing function , the points of discontinuity of which correspond to masses that are "concentrated at one point" .
The Riemann integral is a particular case of the Stieltjes integral, when a function , where , is taken as the integrating function .
When the integrating function increases monotonically, the upper and lower Darboux–Stieltjes sums are studied:
where and are the greatest lower and least upper bounds of on .
For a Stieltjes integral to exists, it is sufficient for one of the following conditions to be fulfilled:
1) the function is continuous on , while the function is of bounded variation on ;
2) the function is Riemann integrable on , while the function satisfies a Lipschitz condition on , i.e. , where , for any and from ;
3) the function is Riemann integrable on , while the function can be represented on as an integral with a variable upper bound,
where is absolutely integrable over .
When condition 3) is fulfilled, the integral (2) reduces to a Lebesgue integral by the formula
(The right-hand side is a Riemann integral if is Riemann integrable.) In particular, (4) holds if has a bounded and Riemann-integrable derivative on ; in this case .
If is integrable with respect to over , then is also integrable with respect to over . This statement leads to a number of further conditions on the existence of Stieltjes integrals.
The Stieltjes integral has the property of linearity relative to both the integrand and the integrating function (given the condition that every one of the Stieltjes integrals on the right-hand side exists):
Generally speaking, Stieltjes integrals do not possess the property of additivity: The existence of does not follow from the existence of both the integrals and (the converse is, instead, true if ).
If is bounded on , , and increases monotonically on , then there exists a satisfying the inequality , such that the mean-value formula
holds for a Stieltjes integral. In particular, if is continuous on , then there exists a point such that .
A Stieltjes integral , where is of bounded variation, provides the general form of a continuous linear functional on the space of continuous functions on (Riesz' theorem).
When the function is of bounded variation, the value of the Stieltjes integral coincides with the value of the corresponding Lebesgue–Stieltjes integral.
|||Th.J. Stieltjes, "Recherches sur les fractions continues" C.R. Acad. Sci. Paris , 118 (1894) pp. 1401–1403|
|||V.I. Smirnov, "A course of higher mathematics" , 5 , Addison-Wesley (1964) (Translated from Russian)|
|||V.I. Glivenko, "The Stieltjes integral" , Moscow-Leningrad (1936) (In Russian)|
|[a1]||K.A. Ross, "Elementary analysis: The theory of calculus" , Springer (1980)|
|[a2]||W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 75–78|
|[a3]||T.M. Apostol, "Mathematical analysis" , Addison-Wesley (1974)|
Stieltjes integral. V.A. Il'in (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Stieltjes_integral&oldid=17367