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 | References | Similar Articles | Additional Information

Abstract: Definitions and integrals are of the subdivision-refinement type, and functions are from 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 with respect to which *N* is complete and . If *G* is a function from to *N*, then on [*a, b*] only if (i) exists for and (ii) if , then there exists a subdivision *D* of [*a, b*] such that, if is a refinement of *D* and , then

*a, b*] only if (i) exists for and (ii) the integral exists and is zero. Further, on [

*a, b*] only if there exist a-subdivision

*D*of [

*a, b*] and a number

*B*such that, if is a refinement of

*D*and , then .

If *F* and *G* are functions from to *N*, on [*a, b*], each of and exists and is zero for , each of and exists for , and *G* has bounded variation on [*a, b*], then any two of the following statements imply the other:

(1) on [*a, b*], (2) on [*a, b*], and (3) on [*a, b*].

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

(1) on [*a, b*], (2) on [*a, b*], and (3) 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].

**[1]**William D. L. Appling,*Interval functions and real Hilbert spaces*, Rend. Circ. Mat. Palermo (2)**11**(1962), 154–156. MR**0154081**, https://doi.org/10.1007/BF02843951**[2]**Burrell W. Helton,*Integral equations and product integrals*, Pacific J. Math.**16**(1966), 297–322. MR**0188731****[3]**Burrell W. Helton,*A product integral representation for a Gronwall inequality*, Proc. Amer. Math. Soc.**23**(1969), 493–500. MR**0248310**, https://doi.org/10.1090/S0002-9939-1969-0248310-8**[4]**Jon C. Helton,*An existence theorem for sum and product integrals*, Proc. Amer. Math. Soc.**39**(1973), 149–154. MR**0317048**, https://doi.org/10.1090/S0002-9939-1973-0317048-1**[5]**Jon C. Helton,*Bounds for products of interval functions*, Pacific J. Math.**49**(1973), 377–389. MR**0360969****[6]**Jon C. Helton,*Mutual existence of product integrals*, Proc. Amer. Math. Soc.**42**(1974), 96–103. MR**0349925**, https://doi.org/10.1090/S0002-9939-1974-0349925-0**[7]**Jon C. Helton,*Mutual existence of sum and product integrals*, Pacific J. Math.**56**(1975), no. 2, 495–516. MR**0405098****[8]**A. Kolmogoroff,*Untersuchungen über denIntegralbegriff*, Math. Ann.**103**(1930), no. 1, 654–696 (German). MR**1512641**, https://doi.org/10.1007/BF01455714**[9]**J. S. MacNerney,*Integral equations and semigroups*, Illinois J. Math.**7**(1963), 148–173. MR**0144179****[10]**P. R. Masani,*Multiplicative Riemann integration in normed rings*, Trans. Amer. Math. Soc.**61**(1947), 147–192. MR**0018719**, https://doi.org/10.1090/S0002-9947-1947-0018719-6

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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