probability – Next claim is true or not?

If $lim_{lambda rightarrow infty} mathbb{P}(sup_{s le t}|M_s|>lambda)=0$,then $mathbb{E}(|M_t|:sup_{sle t|M_s|} ge lambda)=0$.
This proposition is true or not? In my intuitively, it is true. Because conditional’s measure will banish. But I cannot prove it. Please help me.