Positive and negative sets

<h2 id="properties">Properties</h2>
<p>It follows from the definition that every measurable subset of a positive or negative set is also positive or negative. Also, the union of a sequence of positive or negative sets is also positive or negative; more formally, if 
  
    
      
        
          A
          
            1
          
        
        ,
        
          A
          
            2
          
        
        ,
        …
      
    
    {\displaystyle A_{1},A_{2},\ldots }
  
 is a sequence of positive sets, then

⋃
          
            n
            =
            1
          
          
            ∞
          
        
        
          A
          
            n
          
        
      
    
    {\displaystyle \bigcup _{n=1}^{\infty }A_{n}}

is also a positive set; the same is true if the word "positive" is replaced by "negative".
</p><p>A set which is both positive and negative is a 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
-<a href="/facts/Null_set/kMudL8Mn">null set</a>, for if 
  
    
      
        E
      
    
    {\displaystyle E}
  
 is a measurable subset of a positive and negative set 
  
    
      
        A
        ,
      
    
    {\displaystyle A,}
  
 then both 
  
    
      
        μ
        (
        E
        )
        ≥
        0
      
    
    {\displaystyle \mu (E)\geq 0}
  
 and 
  
    
      
        μ
        (
        E
        )
        ≤
        0
      
    
    {\displaystyle \mu (E)\leq 0}
  
 must hold, and therefore, 
  
    
      
        μ
        (
        E
        )
        =
        0.
      
    
    {\displaystyle \mu (E)=0.}

</p>
<h2 id="hahn-decomposition">Hahn decomposition</h2>
<p>The <a href="/facts/Hahn_decomposition_theorem/FW5oGvJe">Hahn decomposition theorem</a> states that for every measurable space 
  
    
      
        (
        X
        ,
        Σ
        )
      
    
    {\displaystyle (X,\Sigma )}
  
 with a signed measure 
  
    
      
        μ
        ,
      
    
    {\displaystyle \mu ,}
  
 there is a <a href="/facts/Partition_of_a_set/VeP9g8CA">partition</a> of 
  
    
      
        X
      
    
    {\displaystyle X}
  
 into a positive and a negative set; such a partition 
  
    
      
        (
        P
        ,
        N
        )
      
    
    {\displaystyle (P,N)}
  
 is unique <a href="/facts/Up_to/ydz4ztzX">up to</a> 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
-null sets, and is called a <i><a href="/facts/Hahn_decomposition/FW5oGvJe">Hahn decomposition</a></i> of the signed measure 
  
    
      
        μ
        .
      
    
    {\displaystyle \mu .}

</p><p>Given a Hahn decomposition 
  
    
      
        (
        P
        ,
        N
        )
      
    
    {\displaystyle (P,N)}
  
 of 
  
    
      
        X
        ,
      
    
    {\displaystyle X,}
  
 it is easy to show that 
  
    
      
        A
        ⊆
        X
      
    
    {\displaystyle A\subseteq X}
  
 is a positive set if and only if 
  
    
      
        A
      
    
    {\displaystyle A}
  
 differs from a subset of 
  
    
      
        P
      
    
    {\displaystyle P}
  
 by a 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
-null set; equivalently, if 
  
    
      
        A
        ∖
        P
      
    
    {\displaystyle A\setminus P}
  
 is 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
-null. The same is true for negative sets, if 
  
    
      
        N
      
    
    {\displaystyle N}
  
 is used instead of 
  
    
      
        P
        .
      
    
    {\displaystyle P.}

</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Set_function/Dy46B9EJ">Set function</a> – Function from sets to numbers</li></ul>

Positive and negative sets open-in-new

Positive and negative sets