In mathematics, two positive (or signed or complex) measures ${\displaystyle \mu }$ and ${\displaystyle \nu }$ defined on a measurable space ${\displaystyle (\Omega ,\Sigma )}$ are called singular if there exist two disjoint measurable sets ${\displaystyle A,B\in \Sigma }$ whose union is ${\displaystyle \Omega }$ such that ${\displaystyle \mu }$ is zero on all measurable subsets of ${\displaystyle B}$ while ${\displaystyle \nu }$ is zero on all measurable subsets of ${\displaystyle A.}$ This is denoted by ${\displaystyle \mu \perp \nu .}$

A refined form of Lebesgue's decomposition theorem decomposes a singular measure into a singular continuous measure and a discrete measure. See below for examples.