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
Famous quotes containing the words definition and/or properties:
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)