Continuity set

In <a href="/facts/Measure_theory/yCq7zlI4">measure theory</a>, a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a continuity set of a <a href="/facts/Measure_(mathematics)/yCq7zlI4">measure</a> μ is any <a href="/facts/Borel_set/jvSsfpkP">Borel set</a> B such that

μ
        (
        ∂
        B
        )
        =
        0
        ,
      
    
    {\displaystyle \mu (\partial B)=0,}

where 
 
 
 
 ∂
 B
 
 
 {\displaystyle \partial B}
 
 is the (topological) <a href="/facts/Boundary_(topology)/WgdH18kp">boundary</a> of B. For <a href="/facts/Signed_measure/vukVf84A">signed measures</a>, one instead asks that

|
        
        μ
        
          |
        
        (
        ∂
        B
        )
        =
        0.
      
    
    {\displaystyle |\mu |(\partial B)=0.}

The collection of all continuity sets for a given measure μ forms a <a href="/facts/Ring_of_sets/l3vKYcHu">ring of sets</a>.
Similarly, for a <a href="/facts/Random_variable/TwTBXnLT">random variable</a> X, a set B is called a continuity set of X if

Pr
        [
        X
        ∈
        ∂
        B
        ]
        =
        0.
      
    
    {\displaystyle \Pr[X\in \partial B]=0.}

Continuity set open-in-new

Continuity set