Convergence results related to the equalizing property in a Markov decision process

In this paper we consider a Markov decision process with countable state and time spaces. Rewards have the so called charge structure and the optimality criterion is the total expected reward. It is proved, that when an optimal decision rule is applied, the value of the state at time t converges for $ t \rightarrow \infty $ both to zero in L^1 norm and almost surely