Recently, Lutz [14, 151 introduced a polynomial time bounded version of Lebesgue measure. He and others (see e.g. [ 11, 13-18,201) used this concept to investigate the quantitative structure of Exponential Time (E = DTIME(2"")). Previously, Ambos-Spies et al. [2,3] introduced polynomial time bounded genericity concepts and used them for the investigation of structural properties of NP (under appropriate assumptions) and E. Here we relate these concepts to each other. We show that, for any c 3 1, the class of n'-generic sets has p-measure 1. This allows us to simplify and extend certain p-measure l-results. To illustrate the power of generic sets we take the Small Span Theorem of Juedes and Lutz [ 111 as an example and prove a generalization for bounded query reductions. ' This work was supported in part by the Human Capital and Mobility program of the European Community under grant CHRXCT93.0415. The third author was supported by the Dutch VSB foundation during the time of this research.

Recently, Lutz [14, 151 introduced a polynomial time bounded version of Lebesgue measure.He and others (see e.g. [ 11,[13][14][15][16][17][18]201) used this concept to investigate the quantitative structure of Exponential Time (E = DTIME(2"")). Previously, Ambos-Spies et al. [2,3] introduced polynomial time bounded genericity concepts and used them for the investigation of structural properties of NP (under appropriate assumptions) and E. Here we relate these concepts to each other. We show that, for any c 3 1, the class of n'-generic sets has p-measure 1. This allows us to simplify and extend certain p-measure l-results. To illustrate the power of generic sets we take the Small Span Theorem of Juedes and Lutz [ 111 as an example and prove a generalization for bounded query reductions.