Let X X be a quasi-stonian space, and let T T be a σ \sigma -additive Markov operator on C ( X ) C(X) . Ando proved that if all T T -invariant probabilities are σ \sigma -additive, then T T is strongly ergodic (and the space of fixed points is finite-dimensional). We prove that if the set of σ \sigma -additive T T -invariant probabilities is weak-* dense in the set of all T T -invariant probabilities, then T T is strongly ergodic. This result is easy in case X X is hyperstonian. Our method of proof is to use an idea of Gordon to "hyperstonify" part of our quasi-stonian space.