Weakly measurable function

<h2 id="definition">Definition</h2>
<p>If 
  
    
      
        (
        X
        ,
        Σ
        )
      
    
    {\displaystyle (X,\Sigma )}
  
 is a <a href="/facts/Measurable_space/AJ98lhT4">measurable space</a> and 
  
    
      
        B
      
    
    {\displaystyle B}
  
 is a Banach space over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> 
  
    
      
        
          K
        
      
    
    {\displaystyle \mathbb {K} }
  
 (which is the <a href="/facts/Real_number/R02gw5Pb">real numbers</a> 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 or <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbb {C} }
  
), then 
  
    
      
        f
        :
        X
        →
        B
      
    
    {\displaystyle f:X\to B}
  
 is said to be weakly measurable if, for every <a href="/facts/Continuous_linear_functional/KcqjDBrJ">continuous linear functional</a> 
  
    
      
        g
        :
        B
        →
        
          K
        
        ,
      
    
    {\displaystyle g:B\to \mathbb {K} ,}
  
 the function

g
        ∘
        f
        :
        X
        →
        
          K
        
        
        
           defined by 
        
        
        x
        ↦
        g
        (
        f
        (
        x
        )
        )
      
    
    {\displaystyle g\circ f\colon X\to \mathbb {K} \quad {\text{ defined by }}\quad x\mapsto g(f(x))}

is a measurable function with respect to 
  
    
      
        Σ
      
    
    {\displaystyle \Sigma }
  
 and the usual <a href="/facts/Borel_sigma_algebra/jvSsfpkP">Borel 
  
    
      
        σ
      
    
    {\displaystyle \sigma }
  
-algebra</a> on 
  
    
      
        
          K
        
        .
      
    
    {\displaystyle \mathbb {K} .}

</p><p>A measurable function on a <a href="/facts/Probability_space/dEcv2VrU">probability space</a> is usually referred to as a <a href="/facts/Random_variable/TwTBXnLT">random variable</a> (or <a href="/facts/Random_vector/qMfooyVf">random vector</a> if it takes values in a vector space such as the Banach space 
  
    
      
        B
      
    
    {\displaystyle B}
  
).
Thus, as a special case of the above definition, if 
  
    
      
        (
        Ω
        ,
        
          
            P
          
        
        )
      
    
    {\displaystyle (\Omega ,{\mathcal {P}})}
  
 is a probability space, then a function 
  
    
      
        Z
        :
        Ω
        →
        B
      
    
    {\displaystyle Z:\Omega \to B}
  
 is called a (
  
    
      
        B
      
    
    {\displaystyle B}
  
-valued) weak random variable (or weak random vector) if, for every continuous linear functional 
  
    
      
        g
        :
        B
        →
        
          K
        
        ,
      
    
    {\displaystyle g:B\to \mathbb {K} ,}
  
 the function

g
        ∘
        Z
        :
        Ω
        →
        
          K
        
        
        
           defined by 
        
        
        ω
        ↦
        g
        (
        Z
        (
        ω
        )
        )
      
    
    {\displaystyle g\circ Z\colon \Omega \to \mathbb {K} \quad {\text{ defined by }}\quad \omega \mapsto g(Z(\omega ))}

is a 
  
    
      
        
          K
        
      
    
    {\displaystyle \mathbb {K} }
  
-valued random variable (i.e. measurable function) in the usual sense, with respect to 
  
    
      
        Σ
      
    
    {\displaystyle \Sigma }
  
 and the usual Borel 
  
    
      
        σ
      
    
    {\displaystyle \sigma }
  
-algebra on 
  
    
      
        
          K
        
        .
      
    
    {\displaystyle \mathbb {K} .}

</p>
<h2 id="properties">Properties</h2>
<p>The relationship between measurability and weak measurability is given by the following result, known as <a href="/facts/B._J._Pettis/pG9mrUKr">Pettis</a>' theorem or Pettis measurability theorem.
</p><p>A function 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is said to be <a href="/facts/Almost_surely/fWaxzpBA">almost surely</a> separably valued (or essentially separably valued) if there exists a subset 
  
    
      
        N
        ⊆
        X
      
    
    {\displaystyle N\subseteq X}
  
 with 
  
    
      
        μ
        (
        N
        )
        =
        0
      
    
    {\displaystyle \mu (N)=0}
  
 such that 
  
    
      
        f
        (
        X
        ∖
        N
        )
        ⊆
        B
      
    
    {\displaystyle f(X\setminus N)\subseteq B}
  
 is separable.
</p>

<p>Theorem (Pettis, 1938)—A function 
  
    
      
        f
        :
        X
        →
        B
      
    
    {\displaystyle f:X\to B}
  
 defined on a <a href="/facts/Measure_space/w9Dx4KY1">measure space</a> 
  
    
      
        (
        X
        ,
        Σ
        ,
        μ
        )
      
    
    {\displaystyle (X,\Sigma ,\mu )}
  
 and taking values in a Banach space 
  
    
      
        B
      
    
    {\displaystyle B}
  
 is (strongly) measurable (that equals a.e. the limit of a sequence of measurable countably-valued functions) <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> it is both weakly measurable and almost surely separably valued.
</p>

<p>In the case that 
  
    
      
        B
      
    
    {\displaystyle B}
  
 is separable, since any subset of a separable Banach space is itself separable, one can take 
  
    
      
        N
      
    
    {\displaystyle N}
  
 above to be empty, and it follows that the notions of weak and strong measurability agree when 
  
    
      
        B
      
    
    {\displaystyle B}
  
 is separable.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Bochner_measurable_function/HxXFeAgt">Bochner measurable function</a></li>
<li><a href="/facts/Bochner_integral/RpzcTu9n">Bochner integral</a> – Concept in mathematics</li>
<li><a href="/facts/Bochner_space/tSuPabsJ">Bochner space</a> – Type of topological space</li>
<li><a href="/facts/Pettis_integral/NLPpIDsf">Pettis integral</a></li>
<li><a href="/facts/Vector_measure/pQXUI3mv">Vector measure</a></li></ul>

<ul><li><a href="/facts/Billy_James_Pettis/pG9mrUKr">Pettis, B. J.</a> (1938). <a href="https://doi.org/10.2307%2F1989973">"On integration in vector spaces"</a>. <i>Trans. Amer. Math. Soc</i>. 44 (2): 277–304. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1989973">10.2307/1989973</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-9947">0002-9947</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1501970">1501970</a>.</li>
<li>Showalter, Ralph E. (1997). "Theorem III.1.1". <a href="https://archive.org/details/monotoneoperatio00show"><i>Monotone operators in Banach space and nonlinear partial differential equations</i></a>. Mathematical Surveys and Monographs 49. Providence, RI: American Mathematical Society. p. <a href="https://archive.org/details/monotoneoperatio00show/page/n109">103</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-0500-2. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1422252">1422252</a>.</li></ul>

Weakly measurable function open-in-new

Weakly measurable function