I want to show the uniformly integrability of a solution to the Skorokhod embedding Problem. i.e. given a centered probability measure $\mu$ with finite first moment, we want to construct a stopping time tau such that for a Brownian motion $B_t$ the stopped process $B_\tau \sim \mu$. Moreover, $B_{\tau \land t}$ should be uniformly integrable.
I have constructed the stopping time $\tau$ via Azema-Yor's solution but I struggle showing the uniform integrability of $B_{\tau \land t}$
While constructing we have a sequence of stopping times $\tau_n$ such that $B_{\tau_n} \sim \mu_n$ and $\mu_n$ converges to $\mu$.
So the problem consists of:
Given a Brownian motion $M_t$ and a stopping time $\tau$, such that $\mathbb {E} [\tau] = \mathbb {E}[M^2_{\tau}]$, moreover I know $\mathbb {E}[|M_\tau|] < \infty$ and that there exists a sequence of $\tau_n$ converging to $\tau$ and for each $n \quad \mathbb {E} [M^2_{\tau_n}] < \infty$.
I want to show that the family $\{M_{t \land \tau}| t \ge 0\}$ is uniformly integrable.
I have been trying show that $\mathbb {E}[|M_{t \land \tau}|1_{\{|M_{t \land \tau}|\ge K\}}] < \infty $.
For this I tried to show that $\mathbb {E}\left[|M_{t \land \tau}|1_{\{|M_{t \land \tau}|\ge K\}}\right] \le \mathbb {E}[|M_{ \tau}|1_{\{|M_{ \tau}|\ge K\}}] $ where one easily could find for any suitable $\epsilon$ a K such that the expectation is smaller.
Thank you for hints.