Definition and Properties
A measure defined on the Lebesgue measurable sets of the real line with values in is said to be discrete if there exists a (possibly finite) sequence of numbers
such that
The simplest example of a discrete measure on the real line is the Dirac delta function One has and
More generally, if is a (possibly finite) sequence of real numbers, is a sequence of numbers in of the same length, one can consider the Dirac measures defined by
for any Lebesgue measurable set Then, the measure
is a discrete measure. In fact, one may prove that any discrete measure on the real line has this form for appropriately chosen sequences and
Read more about this topic: Discrete Measure
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