Let $(X_t : t \geq 0)$ be a stochastic process. Is it necessarily the case that
$$P (\limsup_{t \geq 0} X_t \leq a) \in \{ 0,1\}$$
as it is in discrete time? If some conditions are needed on the process, what are these?
Many thanks for your help.
EG What if $X_t$ is a Feller process.