Measure Preserving Transformations
نویسنده
چکیده
This gives us a new probability measure on (Ω,F), so we may define expectations with respect to this conditioned probability measure. Thus for F measurable Y : Ω → R we define the conditional expectation E[Y | X = x] by taking the expectation of Y with respect to the measure (1.1). Consider now how to generalize the idea of conditional probability to the case when P (X = x) = 0. We wish to do this in a way which is consistent with Bayes formula. Thus we want for all bounded Borel measurable functions f : R→ R the identity
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