Let T be a Markov operator on an L¹-space. We study conditions under which T is mean ergodic and satisfies dim Fix(T) < ∞. Among other things we prove that the sequence $(n^{-1} ∑_{k=0}^{n-1} T^k)ₙ$ converges strongly to a rank-one projection if and only if there exists a function 0 ≠ h ∈ L¹₊ which satisfies $lim_{n→∞} ||(h - n^{-1} ∑_{k=0}^{n-1} T^k f)₊|| = 0$ for every density f. Analogous results for strongly continuous semigroups are given.