Measure on small complexity classes, with applications for BPP

We present a notion of resource-bounded measure for P and other subexponential-time classes. This generalization is based on Lutz´s notion of measure, but overcomes the limitations that cause Lutz´s definitions to apply only to classes at least as large as E. We present many of the basic properties of this measure, and use it to explore the class of sets that are hard for BPP. Bennett and Gill showed that almost all sets are hard for BPP; Lutz improved this from Lebesgue measure to measure on ESPACE. We use our measure to improve this still further, showing that for all ε>0, almost every set in E_ε is hard for BPP, where E_ε=∪_δ<εDTIME(2(n ^δ)), which is the best that can be achieved without showing that BPP is properly contained in E. A number of related results are also obtained in this way