Lebesgue Convergence Theorems and Reverse Mathematics

Abstract

Concepts of L1 space, integrable functions and integrals are formalized in weak subsystems of second order arithmetic. They are discussed especially in relation with the combinatorial principle WWKL (weak-weak Konig's lemma and arithmetical comprehension. Lebesgue dominated convergence theorem is proved to be equivalent to arithmetical comprehension. A weak version of Lebesgue monotone convergence theorem is proved to be equivalent to weak-weak Konig's lemma. Mathematics Subject Classification: 03F35.