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), 479488
MSC:
Primary 26A39; Secondary 28A25
MathSciNet review:
0296223
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Abstract: Let be an ordered triple of real numbers such that . Let g be a realvalued function on the entire real axis which is of bounded variation on every closed interval. For f a realvalued 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 gsummable over [a, b], and we define the integral when 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 LebesgueStieltjes integral.
 [1]
Fred
M. Wright and James
D. Baker, On integrationbyparts for weighted
integrals, Proc. Amer. Math. Soc. 22 (1969), 42–52. MR 0245750
(39 #7056), http://dx.doi.org/10.1090/S00029939196902457508
 [2]
Pasquale
Porcelli, Concerning integrals, Proc. Amer. Math. Soc. 5 (1954), 395–400. MR 0062204
(15,944a), http://dx.doi.org/10.1090/S00029939195400622041
 [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), http://dx.doi.org/10.1090/S00029939197102831474
 [1]
 F. M. Wright and J. D. Baker, On integrationbyparts for weighted integrals, Proc. Amer. Math. Soc. 22 (1969), 4252. MR 39 #7056. MR 0245750 (39:7056)
 [2]
 P. Porcelli, Concerning integrals, Proc. Amer. Math. Soc. 5 (1954), 395400. 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), 504510. MR 0283147 (44:380)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197202962238
PII:
S 00029939(1972)02962238
Keywords:
Step function,
bounded variation,
Stieltjes mean sigma integral,
F. Riesz step function approach,
saltus function,
continuous function,
weighted refinement integral,
LebesgueStieltjes integral,
gsummable,
Lebesgue Dominated Convergence Theorem,
iterated limits,
Banach space
Article copyright:
© Copyright 1972 American Mathematical Society
