30.2 Conditional Expectation, Part 1

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Motivation for conditional expectation, in a discrete setting. Relation to L^2 inner product.
  1. Todd Kemp
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@0:53: There's a misstatement here. L^1(P_A) is NOT equal to L^1(P), unless P(A)=1 (in which case P_A = P). You can take any random variable that is L^1 on A and then set it equal to infinity on A^c; then it will be in L^1(P_A), but will not be in L^1(P). That's not a problem: the correct statement is that L^1(P) is CONTAINED in L^1(P_A), which is correct and easy to check from the formula for the integral with respect to P_A that is the content of this Lemma. That means that any L^1(P) random variable X is one for which E[X|A] makes sense, for all events A. (For any given event, E[X|A] makes sense fr some random variables that aren't in L^1(P), as described above, but since we want to string these together into a function that varies from one partition bin A to another, the best place to start is L^1(P), which is contained in all of the L^1(P_A).)

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