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.

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