Schnorr randomness and the Lebesgue differentiation theorem

We exhibit a close correspondence between L1-computable functions and Schnorr tests. Using this correspondence, we prove that a point x ∈ [0, 1]d is Schnorr random if and only if the Lebesgue Differentiation Theorem holds at x for all L1-computable functions f ∈ L1([0, 1]d).

[1]  V. V'yugin,et al.  Effective Convergence in Probability and an Ergodic Theorem forIndividual Random Sequences , 1998 .

[2]  Rodney G. Downey,et al.  Algorithmic Randomness and Complexity , 2010, Theory and Applications of Computability.

[3]  Wilfrid Hodges,et al.  Model Theory: The existential case , 1993 .

[4]  H. Towsner,et al.  LOCAL STABILITY OF ERGODIC AVERAGES , 2007, 0706.1512.

[5]  Mathieu Hoyrup,et al.  Computability of probability measures and Martin-Löf randomness over metric spaces , 2007, Inf. Comput..

[6]  Владимир Вячеславович Вьюгин,et al.  Эффективная сходимость по вероятности и эргодическая теорема для индивидуальных случайных последовательностей@@@Effective convergence in probability and an ergodic theorem for individual random sequences , 1997 .

[7]  Mathieu Hoyrup,et al.  Effective symbolic dynamics, random points, statistical behavior, complexity and entropy , 2007, Inf. Comput..

[8]  Denis R. Hirschfeldt,et al.  Algorithmic randomness and complexity. Theory and Applications of Computability , 2012 .

[9]  Richard L. Wheeden.  Measure and integral , 1977 .

[10]  A. Zygmund,et al.  Measure and integral : an introduction to real analysis , 1977 .

[11]  Johanna N. Y. Franklin,et al.  Martin-Löf random points satisfy Birkhoff’s ergodic theorem for effectively closed sets , 2012 .

[12]  C. R Gonzalez,et al.  Randomness and Ergodic Theory: An Algorithmic Point of View , 2008 .

[13]  Mathieu Hoyrup,et al.  An Application of Martin-Löf Randomness to Effective Probability Theory , 2009, CiE.

[14]  A. Nies.  Computability and randomness , 2009 .

[15]  Péter Gács,et al.  Randomness on Computable Probability Spaces - A Dynamical Point of View , 2009, STACS.

[16]  Stephen G. Simpson,et al.  Vitali's Theorem and WWKL , 2002, Arch. Math. Log..

[17]  Noopur Pathak.  A computational aspect of the Lebesgue differentiation theorem , 2009, J. Log. Anal..

[18]  André Nies,et al.  Algorithmic Aspects of Lipschitz Functions , 2014, Comput..

[19]  Mathieu Hoyrup,et al.  A constructive version of Birkhoff's ergodic theorem for Martin-Löf random points , 2010, Inf. Comput..

[20]  Mathieu Hoyrup,et al.  Computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems , 2010, CCA.