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 to *R*, where *R* represents the real numbers. Let be the class of functions *G* such that exists for and . Let be the class of functions *G* such that is bounded for refinements of a suitable subdivision of [*a, b*]. If *F* and *G* are functions from to *R* such that on [*a, b*], and exist and are zero for , each of and exist 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*].

**[1]**William D. L. Appling,*Interval functions and real Hilbert spaces*, Rend. Circ. Mat. Palermo (2)**11**(1962), 154–156. MR**0154081****[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**, 10.1090/S0002-9939-1969-0248310-8**[4]**J. C. Helton,*Product integrals, bounds and inverses*, Texas J. Sci. (to appear).**[5]**Jon C. Helton,*Bounds for products of interval functions*, Pacific J. Math.**49**(1973), 377–389. MR**0360969****[6]**A. Kolmogoroff,*Untersuchungen über denIntegralbegriff*, Math. Ann.**103**(1930), no. 1, 654–696 (German). MR**1512641**, 10.1007/BF01455714**[7]**J. S. MacNerney,*Integral equations and semigroups*, Illinois J. Math.**7**(1963), 148–173. MR**0144179**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1974-0349925-0

Keywords:
Product integral,
sum integral,
subdivision-refinement integral,
interval function

Article copyright:
© Copyright 1974
American Mathematical Society