I have the following problem regarding an inequality of conditional expectation.
Given that $X$ a R.V. such that $E|X|< \infty$ and $B_1 \subset B_2 \subset \mathcal{F}$ are sigma-algebras. Show that $$ E(X - E(X|B_1))^2 \geq E(X - E(X|B_2))^2. $$ This is quite obvious to prove if we have $E|X|^2 < \infty$ since we can use the definition of conditional expectation for square integrable R.V. My question is does this also hold for the integrable assumption of $X$ provided in this question? I think it can run into the trouble of doing things like $\infty - \infty$ which is problematic and I feel like the assumption for this problem is lacking. Thank you!