Computability on measurable functions

Article type: Research Article

Authors: Weihrauch, Klaus

Affiliations: Mathematics and Computer Science, University of Hagen, 58084 Hagen, Germany. klaus.weihrauch@fernuni-hagen.de. http://www.fernuni-hagen.de/weihrauch/

Abstract: For a computable measure space with computable σ-finite measure we study computability on the space of measurable functions. First we prove that for a natural multi-representation finite union and intersection and countable union are computable. We introduce a natural multi-representation of the measurable functions to a computable topological space and prove that composition with continuous functions is computable. Canonically, for a computable metric space the associated computable topological space has a basis of balls B(a,s) with center from the dense set and rational radius. From a given sequence (μk)k of finite Borel measures we can compute a dense sequence (rj)j of radii such that μk(S)=0 for all k and all spheres S=S(a,rj) with a from the dense set and j∈N. For measurable functions to computable compact computable metric spaces the natural multi-representation is equivalent to a multi-representation via fast uniformly converging sequences of simple functions. For non-negative measurable numerical functions the sequences can be chosen to be non-decreasing. Some results on integration are added.

Keywords: Computable analysis, measurable functions

DOI: 10.3233/COM-160058

Journal: Computability, vol. 6, no. 1, pp. 79-104, 2017