A zero-one law for RP and derandomization of AM if NP is not small

We show that if RP does not have p-measure zero then ZPP = EXP. As corollaries we obtain a zero-one law for RP in EXP, and that both probabilistic classes ZPP and RP have the same measure in EXP. We also prove that if NP does not have p-measure zero then NP = AM.