  Suppose A is a finite set equipped with a probability measure P and let M be
a ``mass'' function on A. We give a probabilistic characterization of the most
efficient way in which A^n can be almost-covered using spheres of a fixed
radius. An almost-covering is a subset C_n of A^n, such that the union of the
spheres centered at the points of C_n has probability close to one with respect
to the product measure P^n. An efficient covering is one with small mass
M^n(C_n); n is typically large. With different choices for M and the geometry
on A our results give various corollaries as special cases, including Shannon's
data compression theorem, a version of Stein's lemma (in hypothesis testing),
and a new converse to some measure concentration inequalities on discrete
spaces. Under mild conditions, we generalize our results to abstract spaces and
non-product measures.
