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| | In [[measure theory]], a '''continuity set''' of a [[measure (mathematics)|measure]] ''μ'' is any [[Borel set]] ''B'' such that |
| | : <math> |
| | \mu(\partial B) = 0\,. |
| | </math> |
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| |
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| | where <math>\partial B</math> is the [[boundary set]] of ''B''. For [[signed measure]]s, one asks that |
| | : <math> |
| | |\mu|(\partial B) = 0\,. |
| | </math> |
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| | The class of all continuity sets for given measure ''μ'' forms a [[Ring_of_sets|ring]].<ref>Cuppens, R. (1975) Decomposition of multivariate probability. Academic Press, New York.</ref> |
| | |
| | Similarly, for a random variable ''X'' a set ''B'' is called '''continuity set''' if |
| | : <math> |
| | \Pr[X \in \partial B] = 0, |
| | </math> |
| | otherwise ''B'' is called the ''discontinuity set''. The collection of all discontinuity sets is [[sparse set|sparse]]. In particular, given any collection of sets {''B<sub>α</sub>''} with pairwise disjoint boundaries, all but at most countably many of them will be the continuity sets.<ref>van der Vaart (1998) Asymptotic statistics. Cambridge University Press. ISBN 978-0-521-78450-4. Page 7</ref> |
| | |
| | The '''continuity set''' ''C''(''f'') of a function ''f'' is the set of points where ''f'' is continuous. |
| | |
| | == References == |
| | {{reflist}} |
| | |
| | [[Category:Measure theory]] |
| | |
| | {{mathanalysis-stub}} |
Revision as of 04:26, 22 January 2014
In measure theory, a continuity set of a measure μ is any Borel set B such that
where 
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.[1]
Similarly, for a random variable X a set B is called continuity set if
![{\displaystyle \Pr[X\in \partial B]=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c88f5cb972612431282df571f55d0cf92a9684e)
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
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Template:Mathanalysis-stub
- ↑ Cuppens, R. (1975) Decomposition of multivariate probability. Academic Press, New York.
- ↑ van der Vaart (1998) Asymptotic statistics. Cambridge University Press. ISBN 978-0-521-78450-4. Page 7