Mutual existence of product integrals in normed rings
Author:
Jon C. Helton
Journal:
Trans. Amer. Math. Soc. 211 (1975), 353363
MSC:
Primary 28A45; Secondary 46G99
MathSciNet review:
0387536
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Abstract: Definitions and integrals are of the subdivisionrefinement type, and functions are from to N, where R denotes the set of real numbers and N denotes a ring which has a multiplicative identity element represented by 1 and a norm with respect to which N is complete and . If G is a function from to N, then on [a, b] only if (i) exists for and (ii) if , then there exists a subdivision D of [a, b] such that, if is a refinement of D and , then and on [a, b] only if (i) exists for and (ii) the integral exists and is zero. Further, on [a, b] only if there exist asubdivision D of [a, b] and a number B such that, if is a refinement of D and , then . If F and G are functions from to N, on [a, b], each of and exists and is zero for , each of and exists for , and G has bounded variation on [a, b], then any two of the following statements imply the other: (1) on [a, b], (2) on [a, b], and (3) on [a, b]. In addition, with the same restrictions on F and G, any two of the following statements imply the other: (1) on [a, b], (2) on [a, b], and (3) on [a, b]. The results in this paper generalize a theorem contained in a previous paper by the author [Proc. Amer. Math. Soc. 42 (1974), 96103]. Additional background on product integration can be obtained from a paper by B. W. Helton [Pacific J. Math. 16 (1966), 297322].
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197503875367
PII:
S 00029947(1975)03875367
Keywords:
Sum integral,
product integral,
subdivisionrefinement integral,
existence,
interval function,
normed complete ring
Article copyright:
© Copyright 1975
American Mathematical Society
