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  defined the integral for in terms of point functions.
|||E. Hellinger, "Neue Begründung der Theorie quadratischer Formen von unendlichvielen Veränderlichen" J. Reine Angew. Math. , 136 (1909) pp. 210–271|
|||V.I. Smirnov, "A course of higher mathematics" , 5 , Addison-Wesley (1964) (Translated from Russian)|
Hellinger integral. I.A. Vinogradova (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hellinger_integral&oldid=18237