The reverse mathematics of theorems of Jordan and Lebesgue

2017/04/04 by André Nies, Nies, André, Marcus A. Triplett +3 · 1 citation
Computer Science · Mathematics · #Benford’s Law and Fraud Detection #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.1704.00931

openalex publication_date 2017/04/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Jordan decomposition theorem states that every function f \colon [0,1] → 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 RCA₀, a stronger version of Jordan's result where all functions are continuous is equivalent to ACA₀, while the version stated is equivalent to WKL₀. The result that every function on [0,1] of bounded variation is almost everywhere differentiable is equivalent to WWKL₀. To state this equivalence in a meaningful way, we develop a theory of Martin-Löf randomness over RCA₀.

Cited by

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

Related

• Generalized α-variation and Lebesgue equivalence to differentiable functions
• About the general chain rule for functions of bounded variation
• Multivariate Bounded Variation Functions of Jordan-Wiener Type
• Sensitive Random Variables are Dense in Every L^p(ℝ, \mathscrB_ℝ, ℙ)
• Functions of bounded variation, signed measures, and a general\n Koksma-Hlawka inequality