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

First- and Second-Order Coding Theorems for Mixed Memoryless Channels with General Mixture