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
,
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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
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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
,
.
Perron-Stieltjes integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Perron-Stieltjes_integral&oldid=22895