  We study a wide class of metrics in a Lebesgue space with a standard measure,
the class of so-called admissible metrics. We consider the cone of admissible
metrics, introduce a special norm in it, prove compactness criteria, define the
"-entropy of a measure space with an admissible metric, etc. These notions and
related results are applied to the theory of transformations with invariant
measure; namely, we study the asymptotic properties of orbits in the cone of
admissible metrics with respect to a given transformation or a group of
transformations. The main result of this paper is a new discreteness criterion
for the spectrum of an ergodic transformation: we prove that the spectrum is
discrete if and only if the "-entropy of the averages of some (and hence any)
admissible metric over fragments of its trajectory is uniformly bounded.
