Arithmetical Measure

We develop arithmetical measure theory along the lines of Lutz [10]. This yields the same notion of “measure 0 set” as considered before by Martin-Löf, Schnorr, and others. We prove that the class of sets constructible by r.e.-constructors, a direct analogue of the classes Lutz devised his resource bounded measures for in [10], is not equal to RE, the class of r.e. sets, and we locate this class exactly in terms of the common recursion-theoretic reducibilities below K. We note that the class of sets that bounded truth-table reduce to K has r.e.-measure 0, and show that this cannot be improved to “truth-table.” For ∆2-measure the borderline between measure zero and measure nonzero lies between weak truth-table reducibility and Turing reducibility to K. It follows that there exists a Martin-Löf random set that is tt-reducible to K, and that no such set is btt-reducible to K. In fact, by a result of Kautz, a much more general result holds. Mathematical Subject Classification: 03D55, 28A99, 68Q15