  We provide a coupling proof of Doob's theorem which says that the transition
probabilities of a regular Markov process which has an invariant probability
measure $\mu$ converge to $\mu$ in the total variation distance. In addition we
show that non-singularity (rather than equivalence) of the transition
probabilities suffices to ensure convergence of the transition probabilities
for $\mu$-almost all initial conditions.
