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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 both a state based bisimulation 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 state base bisimulation for arbitrary NLMP. This logic, however, is infinitary as it contains a denumerable V. We then introduce a finitary sublogic that characterize both state and event bisimulation 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.