# Perron-Stieltjes integral

A generalization of the Perron integral for a function of one real variable. A finite function is said to be integrable in the sense of Perron–Stieltjes with respect to a finite function on if on there exists a major function and a minor function for with respect to on having and such that at each point ,

for all sufficiently small and , while the greatest lower bound of the numbers , where is any such major function of with respect to , and the least upper bound of the numbers , where is any such minor function of with respect to , coincide. Their common value is called the Perron–Stieltjes integral of with respect to on and is denoted by

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) |

#### Comments

A major function of a function on with respect to a function on is a function such that for each there is an such that for all with . A minor function is defined similarly, but with the inequality sign reversed. Thus, a suitable lower derivative of with respect to majorizes . More generally one considers additive interval functions and satisfying the above property, cf. [2] for details. If is an ordinary function on , then its associated additive interval function, denoted by the same letter, is . A major function of , without any specification of a , is one with respect to the identify function , .

**How to Cite This Entry:**

Perron-Stieltjes integral.

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