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
$E[M_t|\mathcal{F}_s]$ ?