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 Shannon capacity is the highest rate asymptotically achievable with arbitrarily small error probability. The outage capacity is the highest rate asymptotically achievable with a given probability of decoder-recognized outage. The expected capacity is the highest expected rate asymptotically achievable with a single encoder and multiple decoders, where the channel side information determines the decoder in use. Expected capacity equals Shannon capacity for channels governed by a stationary ergodic random process but is typically greater for general channels. These alternative definitions essentially relax the constraint that all transmitted information must be decoded at the receiver. We derive equations for these capacity definitions through information density. Examples are also provided to demonstrate their implications.