  We consider three capacity definitions for general channels with channel side
information at the receiver, where the channel is modeled as a sequence of
finite dimensional conditional distributions not necessarily stationary,
ergodic, or information stable. The {\em Shannon capacity} is the highest rate
asymptotically achievable with arbitrarily small error probability. The {\em
capacity versus outage} is the highest rate asymptotically achievable with a
given probability of decoder-recognized outage. The {\em expected capacity} is
the highest average rate asymptotically achievable with a single encoder and
multiple decoders, where the channel side information determines the decoder in
use. As a special case of channel codes for expected rate, the code for
capacity versus outage has two decoders: one operates in the non-outage states
and decodes all transmitted information, and the other operates in the outage
states and decodes nothing. Expected capacity equals Shannon capacity for
channels governed by a stationary ergodic random process but is typically
greater for general channels. These alternative capacity definitions
essentially relax the constraint that all transmitted information must be
decoded at the receiver. We derive capacity theorems for these capacity
definitions through information density. Numerical examples are provided to
demonstrate their connections and differences. We also discuss the implication
of these alternative capacity definitions for end-to-end distortion,
source-channel coding and separation.
