# Hellinger integral

From Encyclopedia of Mathematics

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.

#### References

[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. I.A. Vinogradova (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Hellinger_integral&oldid=18237

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098