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We establish some classical results on ergodicity of inhomogeneous Markov chains in a more general context. Madsen (Ann Math Stat 42(1):405–408, 1971) extended the characterization of weak ergodicity presented by Paz (Ann Math Stat 41(2):539–550, 1970) from the countable case to the continuous case, admitting, however, that the transition probability functions have associated density functions. Here,...