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 {Bα} 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