Abstract: Notions of stochastic kernels and norm defined on a σ-finite measure space (S, ℱ, μ) are introduced. To obtain a necessary and sufficient condition for ergodic behavior of inhomogeneous Markov chains with continuous states, the stochastic transition matrices in countable inhomogeneous Markov chains are replaced by stochastic transition kernels densities p_n(x,y)_n∈ℕ. And the ergodic behavior of inhomogeneous Markov chains is generalized.