Representations of measurable sets in computable measure theory

TL;DR

This paper explores how to represent measurable sets in computable measure theory using the TTE framework, comparing various representations and identifying the most effective for studying computability of measurable functions.

Contribution

It introduces and compares multiple natural representations of measurable sets within the TTE framework, establishing their equivalence classes and identifying the most useful for computability analysis.

Findings

Several natural representations are admissible and form four equivalence classes.

One representation is identified as most useful for computability on measurable functions.

Comparison with previous representations shows improvements in studying computability.

Abstract

This article is a fundamental study in computable measure theory. We use the framework of TTE, the representation approach, where computability on an abstract set X is defined by representing its elements with concrete "names", possibly countably infinite, over some alphabet {\Sigma}. As a basic computability structure we consider a computable measure on a computable $\sigma$-algebra. We introduce and compare w.r.t. reducibility several natural representations of measurable sets. They are admissible and generally form four different equivalence classes. We then compare our representations with those introduced by Y. Wu and D. Ding in 2005 and 2006 and claim that one of our representations is the most useful one for studying computability on measurable functions.