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Let $X$ be a random variable on the measurable space $ (\Omega, \mathcal{A})$ and $\mathcal{B} $ be a sub-$\sigma$-field of $\mathcal{A}$.

Question 1: how to prove that $ \mathbb{E}(X |\mathcal{B})\in L^2(\Omega,\mathcal{B})$ is solution of the variational problem $$\min\{\mathbb{E}(X-Y)^2 : Y\quad \mathcal{B}-\text{measurable}\}$$ is $ \mathbb {E}(X | \mathcal {B}) $ ?

Question 2: Is that solution unique in $L^2(\Omega,\mathcal{B})$ ?

Question 3: What is the best characterization for $\mathbb {E}(X | \mathcal {B})\,$?

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You want to apply to the general case of $L^1$ random variables a characterization of conditional expectation as a projection, which is valid for $L^2$ random variables only. For example, if $X$ is in $L^1$, there is no guarantee that $X-E(X\mid B)$ is even in $L^2$. – Did Dec 18 '11 at 19:07
Didier Piau understood! Thank you for watching. I made the necessary corrections. – MathOverview Dec 18 '11 at 19:19
Now, the answer to Q2 is YES, as explained in every good textbook on the subject, the answer to Q1 depends on your definition of conditional expectation, and the meaning of Q3 is unclear. – Did Dec 18 '11 at 20:02
Hello, Didier Piau. In q1 question the definition is the same exercise 17 on page 92 of the book of Folland (Real Analysis: Modern Techniques and Their Applications). – MathOverview Dec 19 '11 at 0:05
The best characterization refers to that which provides the best geometric interpretation of the conditional expectation. That is, ownership similar to a Hilbert space. – MathOverview Dec 19 '11 at 0:09

I think you need $X \in L^2$.

Anyway you can write, for Q1, $$E[(X-Y)^2]=E[\{(X-Z)+(Z-Y)\}^2]$$ where $Z=E[X|\mathcal{B}]$.

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