A general formula for compound channel capacity

A general formula for the capacity of arbitrary compound channels, which are not necessarily ergodic, stationary or information-stable, is obtained using the information density approach. A direct (constructive) proof is given. To prove achievability, we generalize Feinstein Lemma to the compound channel setting, and to prove converse, we generalize Verdu-Han Lemma to the same compound setting. This extends the general formula for channel capacity in [8] to arbitrary compound channels (not necessarily finite-state or countable).