Mutual existence of product integrals

Author:
Jon C. Helton

Journal:
Proc. Amer. Math. Soc. **42** (1974), 96-103

MSC:
Primary 26A39

DOI:
https://doi.org/10.1090/S0002-9939-1974-0349925-0

MathSciNet review:
0349925

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Abstract | References | Similar Articles | Additional Information

Abstract: Definitions and integrals are of the subdivision-refinement type, and functions are from to *R*, where *R* represents the real numbers. Let be the class of functions *G* such that exists for and . Let be the class of functions *G* such that is bounded for refinements of a suitable subdivision of [*a, b*]. If *F* and *G* are functions from to *R* such that on [*a, b*], and exist and are zero for , each of and exist 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*].

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1974-0349925-0

Keywords:
Product integral,
sum integral,
subdivision-refinement integral,
interval function

Article copyright:
© Copyright 1974
American Mathematical Society