Let $\{\mathcal{F}_t\}_{0\le t\le T}$ be a filtration on $(\Omega,\mathcal{F},P)$, to which the process $\lambda_t$ is adapted. Let $\{W(t),\mathcal{F}_t,0\le t\le T\}$ be a Brownian motion and $\mathcal{G}_t\subset\mathcal{F}_t$ be a sub-filtration, suppose further that $W(t)$ is also adapted to $\mathcal{G}_t$.
Consider the following stochastic integral on $(\Omega,\mathcal{F},\mathcal{F}_t,P)$ $$Z_t=\int_0^t \lambda_s d W(s)\qquad 0\le t\le T$$ Is it true that (provided that every term is well-defined)$$\mathbf{E}[Z_t|\mathcal{G}_t]=\int_0^t \mathbf{E}[\lambda_s|\mathcal{G}_s] d W(s)\qquad 0\le t\le T$$If the answer is yes, how to prove it then?