Efficient sphere-covering and converse measure concentration via generalized coding theorems

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.