Let $M$ be progressively $\mathbb{F}$-adapted process, where index set is $[0,T]$. $(T<\infty)$
Claim If $E[M_{\tau}] = E[M_0]$ for all stopping times $\tau \in \mathcal{T}(\mathbb{F})$, then $M$ is a martingale.
If $M$ is uniformly integrable, then converse is also true.
Definition Process $M$ is called uniformly integrable if \begin{equation} \lim_{R\to\infty} \sup_{\tau\in \mathcal{T}(\mathbb{F})} E\left[|M_{\tau}|,|M_\tau|>R\right] = 0 \end{equation}
Question 1 What is the reason for taking supremum over stopping times? On the discrete setting, I have seen that u.i. is equivalent for $L^1$ convergence of $X_n \to X$ as $n\to\infty$. What is the characterization in this setting?
Question 2 Is the claim true? To show the converse, I tried to start with decreasing discrete stopping times $\tau_n \to \tau$, but without right continuity I am not sure how to proceed.