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THE REVERSE MATHEMATICS OF THEOREMS OF JORDAN AND LEBESGUE

Published online by Cambridge University Press:  01 February 2021

ANDRÉ NIES
Affiliation:
SCHOOL OF COMPUTER SCIENCE UNIVERSITY OF AUCKLANDAUCKLAND, NEW ZEALANDE-mail: andre@cs.auckland.ac.nz
MARCUS A. TRIPLETT
Affiliation:
CENTER FOR THEORETICAL NEUROSCIENCE COLUMBIA UNIVERSITYNEW YORK, NY, USAE-mail: marcus.triplett@uq.edu.au
KEITA YOKOYAMA
Affiliation:
MATHEMATICAL INSTITUTE TOHOKU UNIVERSITYSENDAI, JAPANE-mail: keita.yokoyama.c2@tohoku.ac.jp

Abstract

The Jordan decomposition theorem states that every function $f \colon \, [0,1] \to \mathbb {R}$ of bounded variation can be written as the difference of two non-decreasing functions. Combining this fact with a result of Lebesgue, every function of bounded variation is differentiable almost everywhere in the sense of Lebesgue measure. We analyze the strength of these theorems in the setting of reverse mathematics. Over $\mathsf {RCA}_{0}$ , a stronger version of Jordan’s result where all functions are continuous is equivalent to $\mathsf {ACA}_0$ , while the version stated is equivalent to ${\textsf {WKL}}_{0}$ . The result that every function on $[0,1]$ of bounded variation is almost everywhere differentiable is equivalent to ${\textsf {WWKL}}_{0}$ . To state this equivalence in a meaningful way, we develop a theory of Martin–Löf randomness over $\mathsf {RCA}_0$ .

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Copyright
© Association for Symbolic Logic 2021

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