Sub-probability measure

<h2 id="definition">Definition</h2>
<p>Let 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 be a <a href="/facts/Measure_(mathematics)/yCq7zlI4">measure</a> on the <a href="/facts/Measurable_space/AJ98lhT4">measurable space</a>  
  
    
      
        (
        X
        ,
        
          
            A
          
        
        )
      
    
    {\displaystyle (X,{\mathcal {A}})}
  
.
</p><p>Then 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 is called a sub-probability measure if 
  
    
      
        μ
        (
        X
        )
        ≤
        1
      
    
    {\displaystyle \mu (X)\leq 1}
  
.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<h2 id="properties">Properties</h2>
<p>In measure theory, the following implications hold between measures:

probability
        
        
        ⟹
        
        
          sub-probability
        
        
        ⟹
        
        
          finite
        
        
        ⟹
        
        σ
        
          -finite
        
      
    
    {\displaystyle {\text{probability}}\implies {\text{sub-probability}}\implies {\text{finite}}\implies \sigma {\text{-finite}}}

</p><p>So every probability measure is a sub-probability measure, but the converse is not true. Also every sub-probability measure is a <a href="/facts/Finite_measure/GrjjuSXt">finite measure</a> and a <a href="/facts/%25CE%25A3-finite_measure/5PXwoAX7">σ-finite measure</a>, but the converse is again not true.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Helly%2527s_selection_theorem/TYBsKs5p">Helly's selection theorem</a></li>
<li><a href="/facts/Helly%25E2%2580%2593Bray_theorem/xtuSccx4">Helly–Bray theorem</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 247. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
 <a href="978-1-84800-047-6" target="_blank">978-1-84800-047-6</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 30. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
 <a href="978-3-319-41596-3" target="_blank">978-3-319-41596-3</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Sub-probability measure open-in-new

Sub-probability measure