  Necessary and sufficient conditions for approximation of a general channel by
a general source are proved. For the special case in which the channel input is
deterministic, which corresponds to source simulation, we prove a stronger
necessary condition. As the approximation criteria, vanishing variational
distance between the original and the approximated quantity is used for both of
the problems. Both necessary and sufficient conditions for the two problems are
based on some individual properties of the sources and the channel and are
relatively easy to evaluate. In particular, unlike prior results for this
problem, our results do not require solving an optimization problem to test
simulatability. The results are illustrated with several non-ergodic examples.
