## Real Analysis Exchange

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

William D. L. Appling

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 1 June 2012

https://projecteuclid.org/euclid.rae/1338515218

Mathematical Reviews number (MathSciNet)
MR1433611

Zentralblatt MATH identifier
0879.28007

#### Citation

Appling, William D. L. A characterization theorem for the existence of a Hellinger-type integral. Real Anal. Exchange 22 (1996), no. 1, 236--264. https://projecteuclid.org/euclid.rae/1338515218

#### References

• W. D. L. Appling, Fields of sets, set functions, set function integrals, and finite additivity, Internat. J. Math. & Math. Sci., 7(2) (1984), 209–233, (expository paper).
• E. Hellinger, Die Orthogonalivarianten Quadratischer Formen von Unendlich Vielen Variablen, Diss, Gottingen, 1907.
• H. S. Kaltenborn, Linear functional operations on functions having discontinuities of the first kind, Bull. Amer. Math. Soc., 40 (1934), 702–708.
• A. N. Kolmogoroff, Untersuchungen über den integralbegriff, Math. Ann., 103 (1930), 654–696.
• R. E. Lane, The integral of a function with respect to a function, Proc. Amer. Math. Soc., 5 (1954), 59–66.
• J. A. Reneke, Linear functionals on the space of quasi-continuous functions, Bull. Amer. Math. Soc., 72 (1966), 1023–1025.
• J. R. Webb, A Hellinger integral representation for bounded linear functionals, Doctoral Dissertation, University of Texas, Austin, 1960.