Discrete measure

In mathematics, more precisely in measure theory, a measure on the real line is called a discrete measure (in respect to the Lebesgue measure) if it is concentrated on an at most countable set. Note that the support need not be a discrete set. Geometrically, a discrete measure (on the real line, with respect to Lebesgue measure) is a collection of point masses.

A measure ${\displaystyle \mu }$ defined on the Lebesgue measurable sets of the real line with values in ${\displaystyle [0,\infty ]}$ 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 ${\displaystyle \delta .}$ One has ${\displaystyle \delta (\mathbb {R} \backslash \{0\})=0}$ and ${\displaystyle \delta (\{0\})=1.}$

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