$\rho$ is a random (not stopping) time. For every uniformly integrable martingale $(M_t)$, we have $E[|M_{\rho}|]\leq E[|M_{\infty}|]$. We want to show that $E[M_{\rho}]=E[M_{\infty}]$. I do not understand the proof for this. First, it suffices to apply our hypothesis to any martingale $(M_t)$ taking values in $[0,1]$ (Why?). Then \begin{align} E[M_{\rho}] &\leq E[M_{\infty}]\\ E[1-M_{\rho}] &\leq E[1-M_{\infty}] \\ E[M_{\rho}] &= E[M_{\infty}]. \end{align} The last equality follows from the sums on both sides amounts to 1 (what does this mean?).
Secondly, what can we tell about a martingale belongs to $\mathcal{H}^1$ (Banach space with norm $\int_0^{\infty} \sup_{t\geq 0}|M_t|<\infty$), and another martingale is uniformly integrable, I mean how they are related?