Probability Theory pp 37-63 | Cite as

# Conditioning and Martingales

## Abstract

Let (Ω,*F, P*) be a probability space and *A*, *B* ∈*F* 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, *F*_{b}, *P*_{b}) with Ω_{B} = *B,F*_{B} = A∩B *A* ∈ *F* (the “trace σ-field”), *P*_{B}(C) = P(C)/P(B) for *C* ∈ *T*_{B}, and define the probability of *A* given *B*, denoted by *P(A/B), as P*_{B}(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.

## Keywords

Conditional Expectation Dominate Convergence Theorem Polish Space Martingale Theory Real Random Variable## Preview

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