Doob–Dynkin lemma

<p>In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, the Doob–Dynkin lemma, named after <a href="/facts/Joseph_L._Doob/HLcsS24G">Joseph L. Doob</a> and <a href="/facts/Eugene_Dynkin/RMIBmG9q">Eugene Dynkin</a> (also known as the factorization lemma), characterizes the situation when one <a href="/facts/Random_variable/TwTBXnLT">random variable</a> is a function of another by the <a href="/facts/Subset/SJGERmbA">inclusion</a> of the <a href="/facts/Sigma_algebra/8LJINLNc">
  
    
      
        σ
      
    
    {\displaystyle \sigma }
  
-algebras</a> generated by the random variables. The usual statement of the lemma is formulated in terms of one random variable being <a href="/facts/Measurable_function/NgIHnb1b">measurable</a> with respect to the 
  
    
      
        σ
      
    
    {\displaystyle \sigma }
  
-algebra generated by the other.
</p><p>The lemma plays an important role in the <a href="/facts/Conditional_expectation/3GxG1Q3x">conditional expectation</a> in probability theory, where it allows replacement of the conditioning on a <a href="/facts/Random_variable/TwTBXnLT">random variable</a> by conditioning on the <a href="/facts/Sigma-algebra/8LJINLNc">
  
    
      
        σ
      
    
    {\displaystyle \sigma }
  
-algebra</a> that is <a href="/facts/Sigma_algebra/8LJINLNc">generated</a> by the random variable.
</p>

Doob–Dynkin lemma open-in-new

Doob–Dynkin lemma