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An integral of Riemann type of a [[Set function|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|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 [[#References|[1]]] defined the integral for  $  X = [ a, b] $
in terms of point functions.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Hellinger,   "Neue Begründung der Theorie quadratischer Formen von unendlichvielen Veränderlichen"  ''J. Reine Angew. Math.'' , '''136'''  (1909)  pp. 210–271</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Smirnov,   "A course of higher mathematics" , '''5''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR></table>