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 /spl epsiv/>0, almost every set in E/sub /spl epsiv// is hard for BPP, where E/sub /spl epsiv//=/spl cup//sub /spl delta/ >
['Eric Allender', 'Martin J. Strauss']
Measure on small complexity classes, with applications for BPP