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

Abstract

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.