Invariant probabilities for Markov chains on a metric space

We consider Markov kernels on a locally compact separable metric space that satisfy the (weak) Feller property. We provide a very simple necessary and sufficient condition for existence of an invariant probability measure. We also prove that every Feller Markov kernel on a compact Hausdorff (not necessarily metric) has an invariant probability measure. An alternative sample-path criterion of existence is also provided, as well as a sufficient condition for uniqueness.