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:
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)