  We establish two strong senses of universality of logarithmic loss as a
distortion criterion in lossy compression: For any fixed length lossy
compression problem under an arbitrary distortion criterion, we show that there
is an equivalent lossy compression problem under logarithmic loss. In the
successive refinement problem, if the first decoder operates under logarithmic
loss, we show that any discrete memoryless source is successively refinable
under an arbitrary distortion criterion for the second decoder.
