Conditional Expectation as an L² Projection

1. In which space does the projection interpretation of conditional expectation hold?

2. What property characterizes the projection?

3. Complete the statement: $E[X|\mathcal{G}] = \arg\min_{Y \in L^2(\mathcal{G})}$ ___

4. It is sufficient to check orthogonality only against indicator functions of G-sets.