The reverse mathematics of theorems of Jordan and Lebesgue

TL;DR

This paper investigates the logical strength of classical theorems by Jordan and Lebesgue within reverse mathematics, establishing their equivalences to various subsystems and developing a theory of randomness in this framework.

Contribution

It formalizes the reverse mathematical strength of Jordan's and Lebesgue's theorems, linking them to subsystems like ACA_0, WKL_0, and WWKL_0, and introduces a theory of Martin-Löf randomness over RCA_0.

Findings

Jordan's theorem with continuous functions is equivalent to ACA_0.

The standard Jordan decomposition is equivalent to WKL_0.

Almost everywhere differentiability of bounded variation functions is equivalent to WWKL_0.

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 $\mathsf{WKL}_0$. The result that every function on $[0,1]$ of bounded variation is almost everywhere differentiable is equivalent to $\mathsf{WWKL}_0$. To state this equivalence in a meaningful way, we develop a theory of Martin-L\"of randomness over $\mathsf{RCA}_0$.