Further remarks on absorbing Markov decision processes

In this note, based on the recent remarkable results of Dufour and Prieto-Rumeau, we deduce that for an absorbing MDP with a given initial state, under a standard compactness-continuity condition, the space of occupation measures has the same convergent sequences, when it is endowed with the weak topology and with the weak-strong topology. We provided two examples demonstrating that imposed condition cannot be replaced with its popular alternative, and the above assertion does not hold for the space of marginals of occupation measures on the state space. Moreover, the examples also clarify some results in the previous literature.