Continuity set

In measure theory, a branch of mathematics, a '''continuity set''' of a measure {{mvar|μ}} is any Borel set {{mvar|B}} such that <math display=block>\mu(\partial B) = 0,</math> where <math>\partial B</math> is the (topological) boundary of {{mvar|B}}. For signed measures, one instead asks that <math display=block>|\mu|(\partial B) = 0.</math>

The collection of all continuity sets for a given measure {{mvar|μ}} forms a ring of sets.<ref>Cuppens, R. (1975) Decomposition of multivariate probability. Academic Press, New York.</ref>

Similarly, for a random variable {{mvar|X}}, a set {{mvar|B}} is called a '''continuity set''' of {{mvar|X}} if <math display=block>\Pr[X \in \partial B] = 0.</math>

==Continuity set of a function== The '''continuity set''' {{math|''C''(''f'')}} of a function {{mvar|f}} is the set of points where {{mvar|f}} is continuous.{{citation needed|date=February 2025}}

== References == {{reflist}}

Category:Measure theory

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