Riesz representation theorem, Borel measures and subsystems of second-order arithmetic

Abstract

Yu, X., Riesz representation theorem, Borel measures and subsystems of second-order arithmetic, Annals of Pure and Applied Logic 59 (1993) 65-78. Formalized concept of finite Borel measures is developed in the language of second-order arithmetic. Formalization of the Riesz representation theorem is proved to be equivalent to arithmetical comprehension. Codes of Borel sets of complete separable metric spaces are defined and proved to be meaningful in the subsystem ATR 0. Arithmetical transfinite recursion is enough to prove the measurability of Borel sets for any finite Borel measure on a compact complete separable metric space.