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. 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.

Given two (positive) σ-finite measures ${\displaystyle \mu }$ and ${\displaystyle \nu }$ on a measurable space ${\displaystyle (X,\Sigma )}$. Then ${\displaystyle \mu }$ is said to be discrete with respect to ${\displaystyle \nu }$ if there exists an at most countable subset ${\displaystyle S\subset X}$ in ${\displaystyle \Sigma }$ such that

A measure ${\displaystyle \mu }$ on ${\displaystyle (X,\Sigma )}$ is discrete (with respect to ${\displaystyle \nu }$) if and only if ${\displaystyle \mu }$ has the form

with ${\displaystyle a_{i}\in \mathbb {R} _{>0}}$ and Dirac measures ${\displaystyle \delta _{s_{i}}}$ on the set ${\displaystyle S=\{s_{i}\}_{i\in \mathbb {N} }}$ defined as

for all ${\displaystyle i\in \mathbb {N} }$.

One can also define the concept of discreteness for signed measures. Then, instead of conditions 2 and 3 above one should ask that ${\displaystyle \nu }$ be zero on all measurable subsets of ${\displaystyle S}$ and ${\displaystyle \mu }$ be zero on measurable subsets of ${\displaystyle X\backslash S.}$^{[clarification needed]}

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