We begin by recalling the definition of a conditional expectation relative to a given field. Given the probability triple (Ω, , P), a random variable ζ with E(ζ) < ∞ and an augmented Borel subfield of , any ω-function χ (·) which is measurable and such that
$$\int\limits_{M}{\chi (\omega )}P(d\omega )=\int\limits_{M}{\xi (\omega )}P(d\omega )$$
for every M ∈ , is called a version of the conditional expectation of ζ relative ℊ, and denoted collectively by E(ζ|). Thus \(\tilde{\chi }\) is another version of the conditional probability if and only if \(\tilde{\chi }\) is measurable and \(\chi =\tilde{\chi }\) with probability one.


Conditional Probability Conditional Expectation Markov Property Continuous Parameter Sample Function 
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Copyright information

© Springer-Verlag OHG. Berlin · Göttingen · Heidelberg 1960

Authors and Affiliations

  • Kai Lai Chung
    • 1
  1. 1.Syracuse UniversityUSA

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