Let {X n , n 0) be a Markov chains with the state space S = {1,2, , m}, and the probability distribution P(x 0 ) n k = 1 P k (x k x k - 1 ), whereP k (j i) is the transition probabilityP (X k = j X k - 1 = i). Letg k (i,j) be the functions defined on S S, and let F n (ω) = (1n) n k = 1 g k (X k - 1 , X k ). In this paper the limit properties of F n (ω) and the relative entropy densityf n (ω) = -(1n) [log P(X 0 ) + n k = 1 logP k (X k X k - 1 )] are studied, and some theorems on a.e. convergence for {X n , n 0} are obtained, and the Shannon-McMillan theorem is extended to the case of nonhomogeneous Markov chains.