On the Finiteness of the Capacity of Continuous Channels

Evaluating the channel capacity is one of many key problems in information theory. In this work, we derive rather-mild sufficient conditions under which the capacity of continuous channels is finite and achievable. These conditions are derived for generic, memoryless, and possibly nonlinear additive noise channels. The results are based on a novel sufficient condition that guarantees the convergence of differential entropies under point-wise convergence of probability density functions. Perhaps surprisingly, the finiteness of channel capacity holds for the majority of setups, including those where inputs and outputs have possibly infinite second-moments.