P. Williams, Probability with Martingales. New York, NY: Cambridge Press, 1991. Home/Search Document Not in Database Summary Related Articles Check |
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....the conditional expectation by integration over B. Thus instead of a number we get a # measurable function called the conditional expectation given # and is written E p ( #) It is not at all obvious that such a function should exist. It is a fundamental result of Kolmogorov (see for instance [15], p.84) that such conditional expectations exist. Theorem 1 (Kolmogorov) Let (S, #, p) be a probability triple, X be in and # be a sub # algebra of #, then there exists a Y (S, #, p) such that #B # #. Y dp. 1) Not only does the conditional expectation exist, but it has a lot of ....
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