## On the Quantitative Structure of ... (2000)

### BibTeX

@MISC{Terwijn00onthe,

author = {Sebastiaan A. Terwijn},

title = {On the Quantitative Structure of ...},

year = {2000}

}

### OpenURL

### Abstract

We analyze the quantitative structure of 0 2 . Among other things, we prove that a set is Turing complete if and only if its lower cone is nonnegligible, and that the sets of r.e.-degree form a small subset of 0 2 . Mathematical Subject Classification: 03D15, 03D30, 28E15 Keywords: Computable measure theory, Turing degrees, completeness. 1 Introduction We study an eective measure theory suited for the study of 0 2 , the second level of the arithmetical hierarchy (alternatively, the sets computable relative to the halting problem K). This work may be seen as part of the constructivist tradition in mathematics as documented in [6]. The framework for eectivizing measure theory that we employ uses martingales. Martingales were rst applied to the study of random sequences by J. Ville [22]. Recursive martingales were studied in Schnorr [19], and became popular in complexity theory in more recent years through the work of Lutz [14, 15]. Lutz Research supported by a Ma...