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Transactions of the American Mathematical Society

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Existence of sum and product integrals


Author: Jon C. Helton
Journal: Trans. Amer. Math. Soc. 182 (1973), 165-174
MSC: Primary 26A39
DOI: https://doi.org/10.1090/S0002-9947-1973-0352368-0
MathSciNet review: 0352368
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Abstract: Functions are from $ R \times R$ to R, where R represents the set of real numbers. If c is a number and either (1) $ \smallint _a^b{G^2}$ exists and $ \smallint _a^bG$ exists, (2) $ \smallint _a^bG$ exists and $ _a{{\mathbf{\Pi }}^b}(1 + G)$ exists and is not zero or (3) each of $ _a{{\mathbf{\Pi }}^b}(1 + G)$ and $ _a{\Pi ^b}(1 - G)$ exists and is not zero, then $ \smallint _a^bcG$ exists, $ \smallint _a^b\vert cG - \smallint cG\vert = 0{,_x}{{\mathbf{\Pi }}^y}(1 + cG)$ exists for $ a \leq x < y \leq b$ and $ \smallint _a^b\vert 1 + cG - {\mathbf{\Pi }}(1 + cG)\vert = 0$. Furthermore, if H is a function such that $ {\lim _{x \to {p^ - }}}H(x,p),{\lim _{x \to {p^ + }}}H(p,x),{\lim _{x,y \to {p^ - }}}H(x,y)$ and $ {\lim _{x,y \to {p^ + }}}H(x,y)$ exist for each $ p \in [a,b],n \geq 2$ is an integer, and G satisfies either (1), (2) or (3) of the above, then $ \smallint _a^bH{G^n}$ exists, $ \smallint _a^b\vert H{G^n} - \smallint H{G^n}\vert = 0{,_x}{{\mathbf{\Pi }}^y}(1 + H{G^n})$ exists for $ a \leq x < y \leq b$ and $ \smallint _a^b\vert 1 + H{G^n} - {\mathbf{\Pi }}(1 + H{G^n})\vert = 0$.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0352368-0
Keywords: Sum integral, product integral, subdivision-refinement integral, existence, interdependency, interval function
Article copyright: © Copyright 1973 American Mathematical Society

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