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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On existence and a dominated convergence theorem for weighted $ g$-summability


Authors: Fred M. Wright and Melvin L. Klasi
Journal: Proc. Amer. Math. Soc. 34 (1972), 479-488
MSC: Primary 26A39; Secondary 28A25
MathSciNet review: 0296223
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Abstract: Let $ ({w_1},{w_2},{w_3})$ be an ordered triple of real numbers such that $ {w_1} + {w_2} + {w_3} = 1$. 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 $ ({w_1},{w_2},{w_3})$ g-summable over [a, b], and we define the integral

$\displaystyle [F,({w_1},{w_2},{w_3})]s\int_a^b {f(x)dg(x)} $

when f has this property. We show that this integral extends the weighted refinement integral $ [F,({w_1},{w_2},{w_3})]\int_a^b {f(x)dg(x)} $ 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.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0296223-8
PII: S 0002-9939(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, $ ({w_1},{w_2},{w_3})$ g-summable, Lebesgue Dominated Convergence Theorem, iterated limits, Banach space
Article copyright: © Copyright 1972 American Mathematical Society