In the book Stochastic calculus of Baldi, there is the following lemma :
For the last equality, they made a quite long and complicate proof. But, don't we simply have that $$\int_a^b X_sdW_s\quad \text{and}\quad \int_a^bX_s^2ds$$ are independent of $\mathcal F_a$ ? And thus, using Itô isometry $$\mathbb E\left[\left(\int_a^bX_sdW_s\right)^2\mid \mathcal F_a\right]=\mathbb E\left[\int_a^bX_s^2ds\right]=\mathbb E\left[\int_a^b X_s^2ds\mid \mathcal F_a\right]\ \ ?$$
$M^2(0,T)$ is the set of process s.t. $X_t$ is progressively measurable and s.t. $$\mathbb E\int_0^TX_s^2ds<\infty .$$