  This paper investigates the first- and second-order maximum achievable rates
of codes with/without cost constraints for mixed {channels} whose channel law
is characterized by a general mixture of (at most) uncountably many stationary
and memoryless discrete channels. These channels are referred to as {mixed
memoryless channels with general mixture} and include the class of mixed
memoryless channels of finitely or countably memoryless channels as a special
case. For mixed memoryless channels with general mixture, the first-order
coding theorem which gives a formula for the $\varepsilon$-capacity is
established, and then a direct part of the second-order coding theorem is
provided. A subclass of mixed memoryless channels whose component channels can
be ordered according to their capacity is introduced, and the first- and
second-order coding theorems are established. It is shown that the established
formulas reduce to several known formulas for restricted scenarios.
