Abstract: In this paper, we consider the Markov decision processes under an average cost criterion with non- uniformly bounded cost,  denumerable state and arbitrary action spaces. Some sufficient conditions are given under which every average optimal policy is strong average optimal. We improve the main results obtained by Cavazos-Cadena R.and Fernandez-Gaucheran E. (Math. Meth. Oper. Res., 1996, 43: 281-300).