In measure theory, a branch of mathematics, a continuity set of a measure μ is any Borel set B such that μ ( ∂ B ) = 0 , {\displaystyle \mu (\partial B)=0,} where ∂ B {\displaystyle \partial B} is the (topological) boundary of B. For signed measures, one instead asks that | μ | ( ∂ B ) = 0. {\displaystyle |\mu |(\partial B)=0.}

The collection of all continuity sets for a given measure μ forms a ring of sets.

Similarly, for a random variable X, a set B is called a continuity set of X if Pr [ X ∈ ∂ B ] = 0. {\displaystyle \Pr[X\in \partial B]=0.}