Vitali's Theorem and WWKL

Abstract. Continuing the investigations of X. Yu and others, we study the role of set existence axioms in classical Lebesgue measure theory. We show that pairwise disjoint countable additivity for open sets of reals is provable in RCA0. We show that several well-known measure-theoretic propositions including the Vitali Covering Theorem are equivalent to WWKL over RCA0.

[1]  Per Martin-Löf,et al.  The Definition of Random Sequences , 1966, Inf. Control..

[2]  Antonín Kučera,et al.  Remarks on Constructive Mathematical Analysis , 1979 .

[3]  S. G. Simpson,et al.  Which set existence axioms are needed to prove the separable Hahn-Banach theorem? , 1986, Ann. Pure Appl. Log..

[4]  Antonín Kučera,et al.  On Relative Randomness , 1993, Ann. Pure Appl. Log..

[5]  Xiaokang Yu,et al.  Measure theory and weak König's lemma , 1990, Arch. Math. Log..

[6]  Stephen G. Simpson,et al.  Partial realizations of Hilbert's program , 1988, Journal of Symbolic Logic.

[7]  Xiaokang Yu,et al.  Riesz Representation Theorem, Borel Measures and Subsystems of Second-Order Arithmetic , 1993, Ann. Pure Appl. Log..

[8]  Stephen G. Simpson,et al.  Subsystems of second order arithmetic , 1999, Perspectives in mathematical logic.

[9]  Stephen G. Simpson,et al.  The Baire category theorem in weak subsystems of second-order arithmetic , 1993, Journal of Symbolic Logic.

[10]  Stephen G. Simpson,et al.  Separable Banach space theory needs strong set existence axioms , 1996 .

[11]  Stephen G. Simpson,et al.  Countable algebra and set existence axioms , 1983, Ann. Pure Appl. Log..

[12]  Stephen G. Simpson,et al.  Addendum to "countable algebra and set existence axioms" , 1985, Ann. Pure Appl. Log..

[13]  Xiaokang Yu,et al.  Lebesgue Convergence Theorems and Reverse Mathematics , 1994, Math. Log. Q..

[14]  Wilfried Sieg,et al.  Fragments of arithmetic , 1985, Ann. Pure Appl. Log..

[15]  Gerald E. Sacks,et al.  Recursion Theory Week , 1985 .

[16]  Antonín Kučera,et al.  Randomness and generalizations of fixed point free functions , 1990 .

[17]  A. Kucera.  Measure, Π10-classes and complete extensions of PA , 1985 .

[18]  Xiaokang Yu.  A Study of Singular Points and Supports of Measures in Reverse Mathematics , 1996, Ann. Pure Appl. Log..