  Let $X$ be a progressively measurable, almost surely right-continuous
stochastic process such that $X_\tau \in L^1$ and $E[X_\tau] = E[X_0]$ for each
finite stopping time $\tau$. In 2006, Cherny showed that $X$ is then a
uniformly integrable martingale provided that $X$ is additionally nonnegative.
Cherny then posed the question whether this implication also holds even if $X$
is not necessarily nonnegative. We provide an example that illustrates that
this implication is wrong, in general. If, however, an additional integrability
assumption is made on the limit inferior of $|X|$ then the implication holds.
Finally, we argue that this integrability assumption holds if the stopping
times are allowed to be randomized in a suitable sense.
