Maximal identity for a continuous nonnegative local martingale tending to zero ($\mathbb{P}({\sup }_{t\ge 0}{M}_t>a)=1/a$)

For $M_0=1$ and $M_t\to 0$, stop at the infimum of strict crossings above $a>1$. Continuity bounds the stopped process by $a$ and makes its value at a finite crossing time exactly $a$. Its expectation remains one, and bounded convergence at infinity yields $\mathbb{P}(\sup M>a)=1/a$. The maximum has density $m^{-2}$ on $m>1$, with no endpoint atom.