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

 

Mutual existence of product integrals


Author: Jon C. Helton
Journal: Proc. Amer. Math. Soc. 42 (1974), 96-103
MSC: Primary 26A39
MathSciNet review: 0349925
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Abstract: Definitions and integrals are of the subdivision-refinement type, and functions are from $ R \times R$ to R, where R represents the real numbers. Let $ O{M^ \circ }$ be the class of functions G such that $ _x\prod {^y} (1 + G)$ exists for $ a \leqq x < y \leqq b$ and $ \smallint _a^b\vert 1 + G - \prod {(1 + G)\vert = 0} $. Let $ O{P^ \circ }$ be the class of functions G such that $ \vert\prod\nolimits_{q = i}^j {(1 + {G_q})\vert} $ is bounded for refinements $ \{ {x_q}\} _{q = 0}^n$ of a suitable subdivision of [a, b]. If F and G are functions from $ R \times R$ to R such that $ F \in O{P^ \circ }$ on [a, b], $ {\lim _{x,y \to {p^ + }}}F(x,y)$ and $ {\lim _{x,y \to {p^ - }}}F(x,y)$ exist and are zero for $ p \in [a,b]$, each of $ {\lim _{x \to {p^ + }}}F(p,x),{\lim _{x \to {p^ - }}}F(x,p),{\lim _{x \to {p^ + }}}G(p,x)$ and $ {\lim _{x \to {p^ - }}}G(x,p)$ exist for $ p \in [a,b]$, and G has bounded variation on [a, b], then any two of the following statements imply the other: (1) $ F + G \in O{M^ \circ }$ on [a, b], (2) $ F \in O{M^ \circ }$ on [a, b], and (3) $ G \in O{M^\circ }$ on [a, b].


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0349925-0
PII: S 0002-9939(1974)0349925-0
Keywords: Product integral, sum integral, subdivision-refinement integral, interval function
Article copyright: © Copyright 1974 American Mathematical Society