  It is well-known that well-posedness of a martingale problem in the class of
continuous (or r.c.l.l.) solutions enables one to construct the associated
transition probability functions. We extend this result to the case when the
martingale problem is well-posed in the class of solutions which are continuous
in probability. This extension is used to improve on a criterion for a
probability measure to be invariant for the semigroup associated with the
Markov process. We also give examples of martingale problems that are
well-posed in the class of solutions which are continuous in probability but
for which no r.c.l.l. solution exists.
