The Radon-Nikodym Theorem

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

Abstract

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
$$ Q\left( \wedge \right) = E\left\{ {1_ \wedge X} \right\} $$
(28.1)
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)).

Keywords

Radon 

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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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