  Let $(\xi_n)_{n=0}^\infty$ be a nonhomogeneous Markov chain taking values
from finite state-space of
  $\mathbf{X}=\{1,2,\ldots,b\}$. In this paper, we will study the generalized
entropy ergodic theorem with almost-everywhere and $\mathcal{L}_1$ convergence
for nonhomogeneous Markov chains, which generalizes the corresponding classical
results for the Markov chains.
