This work is a sequel to the authors’ publication in [SIAM J. Control Optimization 32, 480-500 (1994; Zbl 0799.90120)] and another one by the first author and D. Hernández-Hernández in [J. Math. Anal. Appl. 183, 335-351 (1994; Zbl 0820.90124)]. Or rather, this is a sort of synthesis. To the average-cost (AC) Markov control process, and for each \(\alpha \in (0,1)\), a discounted linear program pair \((P_\alpha) \& (P^*_\alpha)\) is introduced. The vanishing discounted approach is carried through for a related linear program pair \((MP_\alpha) \& (MP^*_\alpha)\). They established the solvability and the no-gap in duality: \[ \sup (MP^*_1) = \inf (MP_1) = (AC) \] without the usual hypothesis, and on a general Borel space.