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This paper addresses the robust stability and stabilization problems for a class of infinite Markov jump linear systems. It is assumed that, besides the jump parameters, the controller may only have access to an output of the system. Our Linear Matrix Inequality approach explores the connection between stability radii theory and some recent results on the Hinfin control of this class of systems. The novel characterization of robust stabilizing controllers gives rise to a rather flexible framework for design which, in a parallel to our previous work in an output feedback Hinfin control context, presents two algorithms for the practical construction of controllers.