Ville inequality ($\mathrm{Pr}({\sup }_t{M}_t\ge a)\le \mathbb{E}{M}_0/a$)

For a nonnegative martingale, the displayed maximal probability bound follows by stopping at the first crossing of $a$. The stopped expectation is at most its initial expectation, and on the crossing event its value is at least $a$. Bound the stopping time first and pass to the limit using nonnegativity. This form is useful for the Poisson maximal concentration bound.