The rate-distortion infimum is attained for lower semi-continuous distortions on locally compact Polish alphabets via one-point compactification for bounded distortions and concentration-compactness for unbounded coercive distortions.
Lossy compression of general random variables,
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Rate-distortion Theory with Lower Semi-continuous Distortion on Noncompact Alphabets
The rate-distortion infimum is attained for lower semi-continuous distortions on locally compact Polish alphabets via one-point compactification for bounded distortions and concentration-compactness for unbounded coercive distortions.