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An expectation of a random variable with respect to a regular conditional probability is equal to its conditional expectation.
=== Interpretation of conditioning on a Sigma Field ===
Consider the probability space <math>(\Omega, \mathcal{F}, \mathbb{P})</math>
and a sub-sigma field <math>\mathcal{A} \subset \mathcal{F}</math>.
The sub-sigma field <math>\mathcal{A}</math> can be loosely interpreted as containing a subset of the information in <math>\mathcal{F}</math>. For example, we might think of <math>\mathbb{P}(B|\mathcal{A})</math> as the probability of the event <math>B</math> given the information in <math>\mathcal{A}</math>.
Also recall that an event <math>B</math> is independent of a sub-sigma field <math>\mathcal{A}</math> if <math>\mathbb{P}(B | A) = \mathbb{P}(B)</math> for all <math>A \in \mathcal{A}</math>. It is incorrect to conclude in general that the information in <math>\mathcal{A}</math> does not tell us anything about the probability of event <math>B</math> occurring. This can be shown with a counter-example:
Consider a probability space on the unit interval, <math>\Omega = [0, 1]</math>. Let <math>\mathcal{G}</math> be the sigma-field of all countable sets and sets whose complement is countable. So each set in <math>\mathcal{G}</math> has measure <math>0</math> or <math>1</math> and so is independent of each event in <math>\mathcal{F}</math>. However, notice that <math>\mathcal{G}</math> also contains all the singleton events in <math>\mathcal{F}</math> (those sets which contain only a single <math>\omega \in \Omega</math>). So knowing which of the events in <math>\mathcal{G}</math> occurred is equivalent to knowing exactly which <math>\omega \in \Omega</math> occurred! So in one sense, <math>\mathcal{G}</math> contains no information about <math>\mathcal{F}</math> (it is independent of it), and in another sense it contains all the information in <math>\mathcal{F}</math>.<ref>{{Cite book |last=Billingsley |first=Patrick |url=https://www.amazon.com/Probability-Measure-Patrick-Billingsley/dp/1118122372 |title=Probability and Measure |date=2012-02-28 |publisher=Wiley |isbn=978-1-118-12237-2 |edition= |___location=Hoboken, New Jersey |language=English}}</ref>
== See also ==
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