  Based on a weak convergence argument, we provide a necessary and sufficient
condition that guarantees that a nonnegative local martingale is indeed a
martingale. Typically, conditions of this sort are expressed in terms of
integrability conditions (such as the well-known Novikov condition). The weak
convergence approach that we propose allows to replace integrability conditions
by a suitable tightness condition. We then provide several applications of this
approach ranging from simplified proofs of classical results to
characterizations of processes conditioned on first passage time events and
changes of measures for jump processes.
