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
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and the Hellinger integral of with respect to
is defined as
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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:
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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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hellinger_integral&oldid=47207
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