On existence and a dominated convergence theorem for weighted -summability

Authors:
Fred M. Wright and Melvin L. Klasi

Journal:
Proc. Amer. Math. Soc. **34** (1972), 479-488

MSC:
Primary 26A39; Secondary 28A25

DOI:
https://doi.org/10.1090/S0002-9939-1972-0296223-8

MathSciNet review:
0296223

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an ordered triple of real numbers such that . Let *g* be a real-valued function on the entire real axis which is of bounded variation on every closed interval. For *f* a real-valued function on the entire real axis which is bounded on a closed interval [*a, b*], we use the F. Riesz step function approach to define the concept of *f* being *g*-summable over [*a, b*], and we define the integral

*f*has this property. We show that this integral extends the weighted refinement integral for

*f*'s as above. This paper generalizes the method of Pasquale Porcelli for the Stieltjes mean sigma integral. We present an existence theorem for the integral defined here involving saltus and continuous parts of

*g*. We establish a convergence theorem for this integral which is analogous to the Lebesgue Dominated Convergence Theorem for the Lebesgue-Stieltjes integral.

**[1]**F. M. Wright and J. D. Baker,*On integration-by-parts for weighted integrals*, Proc. Amer. Math. Soc.**22**(1969), 42-52. MR**39**#7056. MR**0245750 (39:7056)****[2]**P. Porcelli,*Concerning integrals*, Proc. Amer. Math. Soc.**5**(1954), 395-400. MR**15**, 944; 1140. MR**0062204 (15:944a)****[3]**F. M. Wright, M. L. Klasi and D. R. Kennebeck,*The Gronwall inequality for weighted integrals*, Proc. Amer. Math. Soc.**30**(1971), 504-510. MR**0283147 (44:380)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0296223-8

Keywords:
Step function,
bounded variation,
Stieltjes mean sigma integral,
F. Riesz step function approach,
saltus function,
continuous function,
weighted refinement integral,
Lebesgue-Stieltjes integral,
*g*-summable,
Lebesgue Dominated Convergence Theorem,
iterated limits,
Banach space

Article copyright:
© Copyright 1972
American Mathematical Society