Computability on measurable functions

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 $\mathit{B}\mathrm{(}\mathit{a}\mathrm{,}\mathit{s}\mathrm{)}$ with center from the dense set and rational radius. From a given sequence ${\mathrm{(}{\mathit{\mu }}_{\mathit{k}}\mathrm{)}}_{\mathit{k}}$ of finite Borel measures we can compute a dense sequence ${\mathrm{(}{\mathit{r}}_{\mathit{j}}\mathrm{)}}_{\mathit{j}}$ of radii such that ${\mathit{\mu }}_{\mathit{k}}\mathrm{(}\mathit{S}\mathrm{)}=0$ for all k and all spheres $\mathit{S}=\mathit{S}\mathrm{(}\mathit{a}\mathrm{,}{\mathit{r}}_{\mathit{j}}\mathrm{)}$ with a from the dense set and $\mathit{j}\in \mathbb{N}$ . 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.