Counting measure on a set 
is the measure on the power
set of 
that counts the elements of each subset.
For 
,
it is defined by
 |
(1)
|
Thus, every singleton set has counting measure 1, and every infinite set has counting measure 
, regardless of its cardinal
number.
Counting measure is countably additive. For nonnegative functions on a countably infinite set, the integral with respect to counting measure is the corresponding sum. This identifies Lp-spaces for counting measure with the corresponding spaces of sequences.
In point process theory, the term counting measure is also used for a measure that counts a specified collection of points in an ambient space, possibly with multiplicity. Under this usage, a singleton set need not have measure 1 (Westcott 1972, pp. 448-450).
