In measure theory, a branch of mathematics, a continuity set of a measure μ is any Borel set B such that
\( {\displaystyle \mu (\partial B)=0\,,} \)
where \( \partial B \) is the boundary of B. For signed measures, one asks that
\( {\displaystyle |\mu |(\partial B)=0\,.} \)
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
\( {\displaystyle \Pr[X\in \partial B]=0.} \)
Continuity set of a function
The continuity set C(f) of a function f is the set of points where f is continuous.
References
Cuppens, R. (1975) Decomposition of multivariate probability. Academic Press, New York.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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