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Refinement-Unbounded Interval Functions and Absolute Continuity

Published online by Cambridge University Press:  20 November 2018

William D. L. Appling
William D. L. Appling*
Affiliation:
North Texas State University, Denton, Texas
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In this paper we prove the following characterization theorem (Section 3) :

Theorem 1. If each of g and m is a real-valued non-decreasing function on the number interval [a, b], then the following two statements are equivalent: (1) If R is a real-valued, refinement-unbounded (Section 3) function of subintervals of [a, b], then the integral (Section 2)

exists and is equal to g (b) — g (a), and

(2) g is absolutely continuous with respect to m.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 17 , 1965 , pp. 257 - 260
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Hellinger, E., Die Orthogonalinvarianten quadratischer Formen von unendlichvielen Variablen, Diss. (Göttingen, 1907).Google Scholar