  We extend the theory of labeled Markov processes with internal
nondeterminism, a fundamental concept for the further development of a process
theory with abstraction on nondeterministic continuous probabilistic systems.
We define nondeterministic labeled Markov processes (NLMP) and provide three
definition of bisimulations: a bisimulation following a traditional
characterization, a state based bisimulation tailored to our "measurable"
non-determinism, and an event based bisimulation. We show the relation between
them, including that the largest state bisimulation is also an event
bisimulation. We also introduce a variation of the Hennessy-Milner logic that
characterizes event bisimulation and that is sound w.r.t. the other
bisimulations for arbitrary NLMP. This logic, however, is infinitary as it
contains a denumerable $\lor$. We then introduce a finitary sublogic that
characterize all bisimulations for image finite NLMP whose underlying measure
space is also analytic. Hence, in this setting, all notions of bisimulation we
deal with turn out to be equal. Finally, we show that all notions of
bisimulations are different in the general case. The counterexamples that
separate them turn to be non-probabilistic NLMP.
