  The following conditions are necessary and sufficient for an arbitrary
c\`adl\`ag local martingale to be a uniformly integrable martingale: (i) The
weak tail of the supremum of its modulus is zero; (ii) its jumps at the
first-exit times from compact intervals converge to zero in $L^1$, on the
events that those times are finite; and (iii) its almost sure limit is an
integrable random variable.
