Let's say that I have a martingale $M_t$ with respect to a filtration $(\mathcal{F}_t)_{t \geq 0}$. Can I properly encounter the expectation of the martingale? What I mean is, is the following true?
$$\mathbb{E}[M_t] =^? \lim_{t \to \infty}\mathbb{E}[M_t|\mathcal{F}_0] = M_0$$
If not how can I encounter $\mathbb{E}[M_t]$?