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We prove the existence of codebooks for d-semifaithful lossy compression thatare simultaneously universal with respect to both the class of finite-alphabetmemoryless sources and the class of all bounded additive distortion measures.By applying independent random selection of the codewords according to amixture of all memoryless sources, we achieve redundancy rates that are withinO(log n/n) close to the empirical rate-distortion function of every givensource vector with respect to every bounded distortion measure. As outlined inthe last section, the principal ideas can also be extended significantly beyondthe class of memoryless sources, namely, to the setting of individual sequencesencoded by finite-state machines.

$D$-semifaithful codes that are universal over both memoryless sources and distortion measures