Continuity set

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

where ∂ B is the boundary set of B. For signed measures, one asks that

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

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

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.

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