THE REVERSE MATHEMATICS OF THEOREMS OF JORDAN AND LEBESGUE

• DOI: 10.1017/jsl.2021.16
• Publication Date: 2021-02-01T07:01:53Z
• Authors: ANDRÉ NIES; MARCUS A. TRIPLETT; KEITA YOKOYAMA
• Keywords: Almost everywhere; Bounded variation; Lebesgue–Stieltjes integration; Reverse mathematics; Lebesgue measure

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 two non-decreasing functions. Combining this fact with a result Lebesgue, is differentiable almost everywhere in sense Lebesgue measure. We analyze strength these theorems setting reverse mathematics. Over $\mathsf {RCA}_{0}$ , stronger version Jordan’s where all functions are continuous equivalent to {ACA}_0$ while stated ${\textsf {WKL}}_{0}$ . on $[0,1]$ {WWKL}}_{0}$ To state equivalence meaningful way, we develop theory Martin–Löf randomness over {RCA}_0$