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
This article was adapted from an original article by I.A. Vinogradova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article