Let \({\mathcal X}\) be the possibly infinite set of channel inputs and \({\mathcal Y}\) the output alphabet where \({\mathcal Y}\) is the Borel \(\sigma\)-field of a separable metric space \(({\mathcal Y},d)\). A channel \({\mathcal C}\) is a family of probability measures on \({\mathcal Y}\), i.e., \({\mathcal C}=(P^x)_{x\in{\mathcal X}}\) where \(P^x\in{\mathcal P}\), \({\mathcal P}\) denotes the collection of probability measures on \({\mathcal Y}\). Now consider a sequence of memoryless channels \(({\mathcal C}_n)^\infty_{n=1}\) with common alphabets which converges in the sense of convergence of distribution to a limiting channel \({\mathcal C}\). The relationship between weak convergence of channel probability measures, channel capacity, and error probability of block codes is examined for memoryless channels with general input and output alphabets. It is shown that the channel capacity is a lower semi-continuous function and that every block code with maximal probability of error \(\delta\) for a nominal channel for any \(\varepsilon>0\) can be modified such that the modification has probability of error less than \(\delta+\varepsilon\) for all channels in a sufficiently small neighborhood of the nominal channel.