Conditioning and Martingales

  • Vivek S. Borkar
Part of the Universitext book series (UTX)


Let (Ω,F, P) be a probability space and A, BF events such that P(B) > 0. In this section we seek to formalize the intuitive content of the phrase “probability of A given (or, equivalently, “conditioned on”) B”. The words “given B ” imply prior knowledge of the fact that the sample point ω lies in B. It is then natural to consider the reduced probability space (Ω, Fb, Pb) with ΩB = B,FB = A∩B AF (the “trace σ-field”), PB(C) = P(C)/P(B) for CTB, and define the probability of A given B, denoted by P(A/B), as PB(A ∩ B) = P(AB)/P(B). The reader may convince himself of the reasonableness of this procedure by evaluating, say, the probability of two consecutive heads in four tosses of a fair coin given the knowledge that the number of heads was even, using the principle of insufficient reason.


Conditional Expectation Dominate Convergence Theorem Polish Space Martingale Theory Real Random Variable 
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Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • Vivek S. Borkar
    • 1
  1. 1.Department of Electrical EngineeringIndian Institute of ScienceBangaloreIndia

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