Continuity set

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

$\mu(\partial B) = 0\,.$

where $\partial B$ is the boundary set of B. For signed measures, one asks that

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

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

$\Pr[X \in \partial B] = 0,$

otherwise B is called the discontinuity set. The collection of all discontinuity sets is sparse. In particular, given any collection of sets {B_α} with pairwise disjoint boundaries, all but at most countably many of them will be the continuity sets.^[2]

The continuity set C(f) of a function f is the set of points where f is continuous.

References

1. Cuppens, R. (1975) Decomposition of multivariate probability. Academic Press, New York.
2. van der Vaart (1998) Asymptotic statistics. Cambridge University Press. ISBN 978-0-521-78450-4. Page 7