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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-1975-0387536-7
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].


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

  • [1] W. D. L. Appling, Interval functions and real Hilbert spaces, Rend. Circ. Mat. Palermo (2) 11 (1962), 154-156. MR 27 #4040. MR 0154081 (27:4040)
  • [2] B. W. Helton, Integral equations and product integrals, Pacific J. Math. 16 (1966), 297-322. MR 32 #6167. MR 0188731 (32:6167)
  • [3] -, A product integral representation for a Gronwall inequality, Proc. Amer. Math. Soc. 23 (1969), 493-500. MR 40 #1562. MR 0248310 (40:1562)
  • [4] J. C. Helton, An existence theorem for sum and product integrals, Proc. Amer. Math. Soc. 39 (1973), 149-154. MR 0317048 (47:5596)
  • [5] -, Bounds for products of interval functions, Pacific J. Math. 49 (1973), 377-389. MR 0360969 (50:13416)
  • [6] -, Mutual existence of product integrals, Proc. Amer. Math. Soc. 42 (1974), 96-103. MR 0349925 (50:2418)
  • [7] -, Mutual existence of sum and product integrals, Pacific J. Math. 56 (1975). MR 0405098 (53:8894)
  • [8] A. Kolmogoroff, Untersuchungen über den Integralbegriff, Math. Ann. 103 (1930), 654-696. MR 1512641
  • [9] J. S. MacNerney, Integral equations and semigroups, Illinois J. Math. 7 (1963), 148-173. MR 26 #1726. MR 0144179 (26:1726)
  • [10] P. R. Masani, Multiplicative Riemann integration in normed rings, Trans. Amer. Math. Soc. 61 (1947), 147-192. MR 8, 321. MR 0018719 (8:321c)

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Additional Information

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

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