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This paper is concerned with a class of random jump systems represented by transition operators, which includes switched linear systems with strictly stationary switching signals in infinite modes space as its special case. A series of necessary and sufficient conditions are established for almost sure stability of this class of random jump systems under different scenarios. The stability criteria obtained are further extended to Markov jump linear systems with infinite states, and hence a unified approach to describing the almost sure stability of MJLSs is addressed under this context. All the results in the work are developed for both the continuous- and discrete-time systems.