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Brenden Tran 2022-06-14 Answered
Let ( Σ , F , P ) be a probability space, let X be an F-random variable, let G be a sub-sigma algebra of F, let ω Ω, let A be the intersection of all the sets in G that contain ω, and suppose that A F. Then is it true that E ( X | G ) ( ω ) = E ( X | A )?
I just wrote this statement down based on my intuition of what conditional expectation means, and I want to verify if it’s actually true. Note that sigma algebras need not be closed under uncountable intersections, so A need not be an element of G. But I assumed A is at least an element of F so that E ( X | A ) is meaningful.
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Answers (1)

Anika Stevenson
Answered 2022-06-15 Author has 19 answers
No, when you condition on a sigma field, there is little connection to the basic definition of conditionings. The only result I know is when you condition on a discrete random variable. In other words, let Z be an integrable random variable and let X be a discrete random variable. Suppose all the values X take with positive probability are ( x k ) . Then
E ( Z X ) = k E ( Z X = x k ) 1 { X = x k }
and here E ( Z X = x k ) is the usual conditional expectation over an event A with positive probabiltiy E ( Z A ) = 1 P ( A ) A Z d P . (Note that when Z = 1 { Y B } then you have
E ( Z A ) = P ( Y B , X = x k ) P ( X = x k ) = P ( Y B X = x k ) . )
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