In the problem of variable-length $\delta$-channel resolvability, the channeloutput is approximated by encoding a variable-length uniform random numberunder the constraint that the variational distance between the target andapproximated distributions should be within a given constant $\delta$asymptotically. In this paper, we assume that the given channel input is amixed source whose components may be general sources. To analyze the minimumachievable length rate of the uniform random number, called the$\delta$-resolvability, we introduce a variant problem of the variable-length$\delta$-channel resolvability. A general formula for the$\delta$-resolvability in this variant problem is established for a generalchannel. When the channel is an identity mapping, it is shown that the$\delta$-resolvability in the original and variant problems coincide. Thisrelation leads to a direct derivation of a single-letter formula for the$\delta$-resolvability when the given source is a mixed memoryless source. Weextend the result to the second-order case. As a byproduct, we obtain thefirst-order and second-order formulas for fixed-to-variable length sourcecoding allowing error probability up to $\delta$.
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