Real Analysis Exchange

A characterization theorem for the existence of a Hellinger-type integral

William D. L. Appling

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Suppose that \(a \lt b\) and each of \(h\) and \(m\) is a real-valued function defined on \([a;b]\) with \(m\) nondecreasing such that if \([p;q]\subseteq [a;b]\) and \(m\vert_p^q = 0\), then \(h\vert_p^q = 0\). There are developed, among other things, necessary and sufficient conditions in order that for each real-valued function \(f\) defined and quasi-continuous on \([a;b]\), the Hellinger-type integral \[\int_{[a;b]}{{dfdh}\over {dm}}\] exists. As is well known, this integral has arisen in connection with, among other things, representation theorems for certain classes of continuous linear functionals.

Article information

Real Anal. Exchange, Volume 22, Number 1 (1996), 236-264.

First available in Project Euclid: 1 June 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A25: Integration with respect to measures and other set functions

integral existence characterization bounded difference quotient variation convex decomposition


Appling, William D. L. A characterization theorem for the existence of a Hellinger-type integral. Real Anal. Exchange 22 (1996), no. 1, 236--264.

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