Resource-Bounded Randomness and Compressibility with Respect to Nonuniform Measures

Most research on resource-bounded measure and randomness has focused on the uniform probability density, or Lebesgue measure, on {0, 1}∞; the study of resource-bounded measure theory with respect to a nonuniform underlying measure was recently initiated by Breutzmann and Lutz [1]. In this paper we prove a series of fundamental results on the role of nonuniform measures in resource-bounded measure theory. These results provide new tools for analyzing and constructing martingales and, in particular, offer new insight into the compressibility characterization of randomness given recently by Buhrman and Longpré [2]. We give several new characterizations of resource-bounded randomness with respect to an underlying measure μ: the first identifies those martingales whose rate of success is asymptotically optimal on the given sequence; the second identifies martingales which induce a maximal compression of the sequence; the third is a (nontrivial) extension of the compressibility characterization to the nonuniform case. In addition we prove several technical results of independent interest, including an extension to resource-bounded measure of the classical theorem of Kakutani on the equivalence of product measures; this answers an open question in [1].