Computability of the Radon-Nikodym Derivative

Abstract

We show that a single application of the noncomputable operator EC, which transforms enumerations of sets (in N) to their characteristic functions, suffices to compute the Radon-Nikodym derivative dµ/dλ of a finite measure µ, which is absolutely continuous w.r.t. the σ-finite measure λ. We also give a condition on the two measures (in terms of computability of the norm of a certain linear operator involving the two measures) which is sufficient to compute the derivative.