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In this note we study the stability of Markov jump linear systems with additive noise. We show in a rather direct manner that the system is mean square Lagrange asymptotic stable if and only if the long run average cost is bounded and the system is weak detectable, generalizing previous results employing observability notions. In control applications this means that, for detectable systems, closed loop controls incurring in bounded long run average cost are ensured to be stabilizing. A numerical example is included.