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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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    1. Questions on MSE should be as self-contained as possible. I recommend that you copy here Folland's relevant passage. 2. The Hilbert setting is nice but definitely not the only one nor does it provide *the best characterization*, whatever that means. One flaw is that it ccan only encompass the $L^1$ case by approximation. 3. Since your background seems to be more in analysis than in probability, you could try an accessible reference on conditioning such as the small and wonderful book *Probability with martingales* by David Williams.2011-12-19

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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 $Z=E[X|\mathcal{B}]$.