The reverse mathematics of theorems of Jordan and lebesgue
André Nies, Marcus A. Triplett & Keita Yokoyama
Journal of Symbolic Logic 86 (4):1657-1675 (2021)
@article{Nies2021-NIETRM, volume = {86}, journal = {Journal of Symbolic Logic}, year = {2021}, doi = {10.1017/jsl.2021.16}, number = {4}, author = {Andr\'e Nies and Marcus A. Triplett and Keita Yokoyama}, pages = {1657--1675}, title = {The Reverse Mathematics of Theorems of Jordan and Lebesgue} }
Nies, André ; Triplett, Marcus A. & Yokoyama, Keita (2021). The reverse mathematics of theorems of Jordan and lebesgue. Journal of Symbolic Logic 86 (4):1657-1675.
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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10.1017/jsl.2021.16
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