A generalization of Lutz's measure to probabilistic classes

We extend Lutz's measure to probabilistic classes, and obtain notions of measure on probabilistic complexity classes C such as BPP, BPE and BPEXP. Unlike former attempts, all our measure notions satisfy all three Lutz's measure axioms, that is every singleton fLg has measure zero in C, the whole space C has measure one in C, and "easy innnite unions" of measure zero sets have measure zero. Finally we prove a conditional time hierarchy theorem for probabilistic classes, and show that under the same assumption, both the class of p T-autoreducible sets and the class of p T-complete sets for EXP have measure zero in BPE.