Computability of the Radon-Nikodym Derivative
We show that a single application of the non-computable 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.
Unable to display preview. Download preview PDF.
- 1.Brattka, V., de Brecht, M., Pauly, A.: Closed choice and a uniform low basis theorem. arXiv:1002.2800v1 [math.LO] (February 14. 2010)Google Scholar
- 4.Bogachev, V.I.: Measure Theory. Springer, Heidelberg (2006)Google Scholar
- 11.Hoyrup, M., Rojas, C., Weihrauch, K.: The Radon-Nikodym operator is not computable. In: CCA 2011 (2011)Google Scholar
- 16.Weihrauch, K.: The degrees of discontinuity of some translators between representations of the real numbers. Technical Report TR-92-050, International Computer Science Institute, Berkeley (July 1992)Google Scholar
- 17.Weihrauch, K.: The degrees of discontinuity of some translators between representations of the real numbers. Informatik Berichte 129, FernUniversität Hagen, Hagen (July 1992)Google Scholar