In measure theory, a branch of mathematics, a continuity set of a measure μ is any Borel set B such that

${\displaystyle \mu (\partial B)=0\,,}$

where ${\displaystyle \partial B}$ is the (topological) boundary of B. For signed measures, one asks that

${\displaystyle |\mu |(\partial B)=0\,.}$

The class of all continuity sets for given measure μ forms a ring.^[1]

Similarly, for a random variable X, a set B is called continuity set if

${\displaystyle \Pr[X\in \partial B]=0.}$