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

An existence theorem for sum and product integrals


Author: Jon C. Helton
Journal: Proc. Amer. Math. Soc. 39 (1973), 149-154
MSC: Primary 46G99; Secondary 26A45
MathSciNet review: 0317048
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Abstract: Functions are from $ S \times S$ to $ N$, where $ S$ and $ N$ denote a linearly ordered set and a normed ring, respectively.

Theorem. If $ \{ a,b\} \in S \times S,G$ has bounded variation on $ \{ a,b\} $, $ \{ {F_n}\} _1^\infty $ converges uniformly to a bounded function $ F$ on $ \{ a,b\} $ and either

$\displaystyle (1){\quad _x}{\prod ^y}(1 + {F_n}G)\quad and\quad \int_a^b {\vert 1 + {F_n}G - \prod {(1 + {F_n}G)} \vert} $

exist for $ n = 1,2, \cdots $ and each subdivision $ \{ a,x,y,b\} $ of $ \{ a,b\} $ and $ \{ \smallint _a^b\vert 1 + {F_n}G - \prod ( 1 + {F_n}G)\vert\} _1^\infty $ converges to zero or

$\displaystyle (2)\quad \int_a^b {{F_n}G\quad and \quad \int_a^b {\vert{F_n}G - \int {{F_n}G\vert} } } $

exist for $ n = 1,2, \cdots $ and $ \{ \smallint _a^b\vert{F_n}G - \smallint {F_n}G\vert\} _1^\infty $ converges to zero, then (conclusion) $ _a\prod {^b} (1 + FG){\;_a}\prod {^b} (1 + \vert FG\vert),\smallint _a^bFG$ and $ \smallint _a^b\vert FG\vert$ exist. Further, $ \smallint _a^b\vert 1 + FG - \prod {(1 + FG)\vert,\smallint _a^b\vert 1 + \ver... ... \prod {(1 + \vert FG\vert)\vert,\smallint _a^b\vert FG - \smallint FG\vert} } $ and $ \smallint _a^b\vert\vert FG\vert - \smallint \vert FG\vert\vert$ exist and are zero.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1973-0317048-1
PII: S 0002-9939(1973)0317048-1
Keywords: Sum integral, product integral, subdivision-refinement integral, existence, sequence, limit, normed ring, interval function
Article copyright: © Copyright 1973 American Mathematical Society