Discrete Measure - Definition and Properties

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

\delta_{s_i}(X) = \begin{cases} 1 & \mbox { if } s_i \in X\\ 0 & \mbox { if } s_i \not\in X\\ \end{cases}

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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