Hellinger integral

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An integral of Riemann type of a set function . If is a space with a finite, non-negative, non-singular measure; if , , is a totally-additive function with for ; and if is a partition of , then

and the Hellinger integral of with respect to is defined as

provided that this supremum is finite. Hellinger's integral can also be regarded as the limit over a directed set of partitions: if is a subdivision of .

If is a summable function such that is the Lebesgue integral , then the Hellinger integral can be expressed in terms of the Lebesgue integral:

E. Hellinger in [1] defined the integral for in terms of point functions.


[1] E. Hellinger, "Neue Begründung der Theorie quadratischer Formen von unendlichvielen Veränderlichen" J. Reine Angew. Math. , 136 (1909) pp. 210–271
[2] V.I. Smirnov, "A course of higher mathematics" , 5 , Addison-Wesley (1964) (Translated from Russian)
How to Cite This Entry:
Hellinger integral. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by I.A. Vinogradova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article