# Hellinger integral

An integral of Riemann type of a set function $ f $.
If $ ( X, \mu ) $
is a space with a finite, non-negative, non-singular measure; if $ f ( E) $,
$ E \subset X $,
is a totally-additive function with $ f ( E) = 0 $
for $ \mu E = 0 $;
and if $ \delta = \{ E _ {n} \} _ {1} ^ {N} $
is a partition of $ X $,
then

$$ S _ \delta = \ \sum _ {n = 1 } ^ { N } \frac{f ^ { 2 } ( E _ {n} ) }{\mu E _ {n} } $$

and the Hellinger integral of $ f ( E) $ with respect to $ X $ is defined as

$$ \int\limits _ { X } \frac{f ^ { 2 } ( dE) }{d \mu } = \ \sup _ \delta \ S _ \delta , $$

provided that this supremum is finite. Hellinger's integral can also be regarded as the limit over a directed set of partitions: $ \delta _ {1} < \delta _ {2} $ if $ \delta _ {2} $ is a subdivision of $ \delta _ {1} $.

If $ \phi : X \rightarrow \mathbf R $ is a summable function such that $ f ( E) $ is the Lebesgue integral $ \int _ {E} \phi d \mu $, then the Hellinger integral can be expressed in terms of the Lebesgue integral:

$$ \int\limits _ { X } \frac{f ^ { 2 } ( dE) }{d \mu } = \ \int\limits _ { X } \phi ^ {2} d \mu . $$

E. Hellinger in [1] defined the integral for $ X = [ a, b] $ 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.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Hellinger_integral&oldid=47207