# 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.