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Abstract

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.

Keywords

Conditional Probability Conditional Expectation Markov Property Continuous Parameter Sample Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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