If I have the following expression:
$$E[M_{t\wedge n}|\mathcal{F}_s]$$
Where the set of random variables $M_{t\wedge n}$ is bounded in $L^2$, i.e.
$$\sup_{t} E[M^2_{t\wedge n}]<\infty$$
Hence they are uniformaly integrable. In fact $(M_t)$ is a local martingale. Now my question, why can I take the limit for $n\to \infty$ insight the expectation in the first expression to obtain
$$E[M_t|\mathcal{F}_s]$$ ?