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 the measurable sets are closed under finite union and intersection and under countable union. 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) = 0f o ra l lk 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.