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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Mutual existence of product integrals in normed rings


Author: Jon C. Helton
Journal: Trans. Amer. Math. Soc. 211 (1975), 353-363
MSC: Primary 28A45; Secondary 46G99
MathSciNet review: 0387536
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Abstract: Definitions and integrals are of the subdivision-refinement type, and functions are from $ R \times R$ 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 $ \vert \cdot \vert$ with respect to which N is complete and $ \vert 1\vert = 1$. If G is a function from $ R \times R$ to N, then $ G \in O{M^\ast}$ on [a, b] only if (i) $ _x{\Pi ^y}(1 + G)$ exists for $ a \leq x < y \leq b$ and (ii) if $ \varepsilon > 0$, then there exists a subdivision D of [a, b] such that, if $ \{ {x_i}\} _{i = 0}^n$ is a refinement of D and $ 0 \leq p < q \leq n$, then

$\displaystyle \left\vert{}_{x_{p}}\prod ^{x_q} (1 + G) - \prod\limits_{i = p + 1}^q {(1 + {G_i})} \right\vert < \varepsilon ;$

and $ G \in O{M^ \circ }$ on [a, b] only if (i) $ _x{\Pi ^y}(1 + G)$ exists for $ a \leq x < y \leq b$ and (ii) the integral $ \smallint _a^b\vert 1 + G - \Pi (1 + G)\vert$ exists and is zero. Further, $ G \in O{P^ \circ }$ on [a, b] only if there exist a-subdivision D of [a, b] and a number B such that, if $ \{ {x_i}\} _{i = 0}^n$ is a refinement of D and $ 0 < p \leq q \leq n$, then $ \vert\Pi _{i = p}^q(1 + {G_i})\vert < B$.

If F and G are functions from $ R \times R$ to N, $ F \in O{P^ \circ }$ on [a, b], each of $ {\lim _{x,y \to {p^ + }}}F(x,y)$ and $ {\lim _{x,y \to {p^ - }}}F(x,y)$ exists and is 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)$ exists 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 OM^\ast$ on [a, b], (2) $ F \in OM^\ast$ on [a, b], and (3) $ G \in OM^\ast$ on [a, b].

In addition, with the same restrictions on F and G, any two of the following statements imply the other:

(1) $ F + G \in OM^\circ$ on [a, b], (2) $ F \in OM^\circ $ on [a, b], and (3) $ G \in OM^\circ$ 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), 96-103]. Additional background on product integration can be obtained from a paper by B. W. Helton [Pacific J. Math. 16 (1966), 297-322].


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0387536-7
PII: S 0002-9947(1975)0387536-7
Keywords: Sum integral, product integral, subdivision-refinement integral, existence, interval function, normed complete ring
Article copyright: © Copyright 1975 American Mathematical Society