A computational aspect of the Lebesgue differentiation theorem

Given an L1 -computable function, f , we identify a canonical represen- tative of the equivalence class of f , where f and g are equivalent if and only if R jf gj is zero. Using this representative, we prove a modified version of the Lebesgue Differentiation Theorem. Our theorem is stated in terms of Martin-L¨ random points in Euclidean space. 2000 Mathematics Subject Classification 03D80 (primary); 26A24 (secondary)

• Publication year

  2009

• Venue

  Journal of Logic and Analysis

• Publication date

  2009-08-20

• Fields of study

  Mathematics, Computer Science

• Identifiers
  DOI 10.4115/JLA.2009.1.9
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Almost everywhere domination and superhighness

  2007influential reference
• Vitali's Theorem and WWKL

  2002influential reference
• Computability in analysis and physics

  1989cited by this paper
• The Definition of Random Sequences

  1966cited by this paper
• Measure and Integral

  1957influential reference

• On the close interaction between algorithmic randomness and constructive/computable measure theory

  2018cites this paper
• Algorithmic randomness and analysis

  2017cites this paper
• Lowness, Randomness, and Computable Analysis

  2016influential citation
• Polynomial Space Randomness in Analysis with Application to the Lebesgue Differentiation Theorem

  2015cites this paper
• Polynomial Space Randomness in Analysis

  2015cites this paper
• On Non-metric Covering Lemmas and Extended Lebesgue-differentiation Theorem

  2015cites this paper
• Algorithmic Aspects of Lipschitz Functions

  2014cites this paper
• Logic Blog 2011

  2014influential citation
• Differentiability of polynomial time computable functions

  2014cites this paper
• ALGORITHMIC RANDOMNESS, MARTINGALES AND DIFFERENTIABILITY

  2013influential citation
• Schnorr randomness and the Lebesgue differentiation theorem

  2013influential citation
• Topics in algorithmic randomness and computable analysis

  2013influential citation
• ALGORITHMIC RANDOMNESS, MARTINGALES AND DIFFERENTIABILITY I (PRELIMINARY DRAFT)

  2012cites this paper
• Computability and analysis: the legacy of Alan Turing

  2012cites this paper
• Open Questions in Reverse Mathematics

  2011cites this paper
• Algorithmic randomness, reverse mathematics, and the dominated convergence theorem

  2011cites this paper
• Randomness and Differentiability

  2011influential citation
• Pathak ’ s “ computational ” Lebesgue differentiation theorem in context

  2009influential citation