LIMIT COMPUTABILITY AND CONSTRUCTIVE MEASURE

In this paper we study constructive measure and dimension in the class of limit computable sets. We prove that the lower cone of any Turing-incomplete set in has -dimension 0, and in contrast, that although the upper cone of a noncomputable set in always has -measure 0, upper cones in have nonzero -dimension. In particular the -dimension of the Turing degree of (the Halting Problem) is 1. Finally, it is proven that the low sets do not have -measure 0, which means that they do not form a small subset of . This result has consequences for the existence of bi-immune sets.