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We investigate the relation between the e-property and the eventualcontinuity, or called the asymptotic equicontinuity, which is a generalizationof the e-property. We prove that, for any discrete-time or strongly continuouscontinuous-time eventually continuous Markov-Feller semigroup with an ergodicmeasure, if the interior of the support of the ergodic measure is nonempty,then the e-property is satisfied on the interior of the support. In particular,it implies that, restricted on the support of each ergodic measure, thee-property and the eventual continuity are equivalent for the discrete-time andthe strongly continuous continuous-time Markov-Feller semigroups.

Relation between the eventual continuity and the e-property for Markov-Feller semigroups