Continuity set explained

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

\mu(\partial B) = 0 ,

where

\partial B is the (topological) boundary 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 .

Continuity set of a function

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

Notes and References

1. Cuppens, R. (1975) Decomposition of multivariate probability. Academic Press, New York.