The Radon-Nikodym Theorem

  • Jean Jacod
  • Philip Protter
Part of the Universitext book series (UTX)


Let (ω, F, P) be a probability space. Suppose a random variable X ≥ 0 a.s. has the property E{X| = 1. Then if we define a set function Q on F by
$$ Q\left( \wedge \right) = E\left\{ {1_ \wedge X} \right\} $$
then it is easy to see that Q defines a new probability (see Exercise 9.5). Indeed
$$ Q\left( \Omega \right) = E\left\{ {1_\Omega X} \right\} = E\left\{ X \right\} = 1 $$
and A1, A2, A3, … are disjoint in F then
$$ \begin{gathered} Q\left( {\bigcup\limits_{i = 1}^\infty {A_i } } \right) = E\{ 1_{ \cup _{i = 1}^\infty A_i } X\} \hfill \\ = E\left\{ {\sum\limits_{i = 1}^\infty {1_{A_i } X} } \right\} \hfill \\ = \sum\limits_{i = 1}^\infty {E\{ 1_{A_i } ,X\} } \hfill \\ = \sum\limits_{i = 1}^\infty {Q(A_i )} \hfill \\ \end{gathered} $$
and we have countable additivity. The interchange of the expectation and the summation is justified by the Monotone Convergence Theorem (Theorem 9.1(d)).


Conditional Expectation Positive Random Variable Monotone Convergence Theorem Countable Additivity Hilbert Space Theory 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Jean Jacod
    • 1
  • Philip Protter
    • 2
  1. 1.Laboratoire de ProbabilitésUniversité de ParisParis Cedex 05France
  2. 2.School of Operations Research and Industrial EngineeringCornell UniversityIthacaUSA

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