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.