Introduction

The purpose of Reverse Mathematics is to study the role of set existence axioms, with an eye to determining which axioms are needed in order to prove specific mathematical theorems. In many cases, it is shown that a specific mathematical theorem is equivalent to the set existence axiom which is needed to prove it. Such equivalences are often proved in the weak base theory $\mathsf{RCA}_0$. $\mathsf{RCA}_0$ may be viewed as a kind of formalized constructive or recursive mathematics, with full classical logic but severely restricted comprehension and induction. The program of Reverse Mathematics has been developed in many publications; see for instance [5,10,11,12,19].

In this paper we carry out a Reverse Mathematics study of some aspects of classical Lebesgue measure theory. Historically, the subject of measure theory developed hand in hand with the nonconstructive, set-theoretic approach to mathematics. Errett Bishop has remarked that the foundations of measure theory present a special challenge to the constructive mathematician. Although our program of Reverse Mathematics is quite different from Bishop-style constructivism, we feel that Bishop's remark implicitly raises an interesting question: Which nonconstructive set existence axioms are needed for measure theory? This paper, together with earlier papers of Yu and others [21,22,23,24,25,26], constitute an answer to that question.

The results of this paper build upon and clarify some early results of Yu and Simpson. The reader of this paper will find that familiarity with Yu-Simpson [26] is desirable but not essential. We begin in section 2 by exploring the extent to which measure theory can be developed in $\mathsf{RCA}_0$. We show that pairwise disjoint countable additivity for open sets of reals is provable in $\mathsf{RCA}_0$. This is in contrast to a result of Yu-Simpson [26]: countable additivity for open sets of reals is equivalent over $\mathsf{RCA}_0$ to a nonconstructive set existence axiom known as Weak Weak König's Lemma (WWKL). We show in sections 3 and 4 that several other basic propositions of measure theory are also equivalent to WWKL over $\mathsf{RCA}_0$. Finally in section 5 we show that the Vitali Covering Theorem is likewise equivalent to WWKL over $\mathsf{RCA}_0$.