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 $\mathbb {E}(X | \mathcal {B})\,$?