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  1. Riesz representation theorem, Borel measures and subsystems of second-order arithmetic.Xiaokang Yu - 1993 - Annals of Pure and Applied Logic 59 (1):65-78.
    Yu, X., Riesz representation theorem, Borel measures and subsystems of second-order arithmetic, Annals of Pure and Applied Logic 59 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 ATR0. Arithmetical transfinite recursion is enough to prove the measurability of Borel sets for (...)
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  • Measure theory and weak König's lemma.Xiaokang Yu & Stephen G. Simpson - 1990 - Archive for Mathematical Logic 30 (3):171-180.
    We develop measure theory in the context of subsystems of second order arithmetic with restricted induction. We introduce a combinatorial principleWWKL (weak-weak König's lemma) and prove that it is strictly weaker thanWKL (weak König's lemma). We show thatWWKL is equivalent to a formal version of the statement that Lebesgue measure is countably additive on open sets. We also show thatWWKL is equivalent to a formal version of the statement that any Borel measure on a compact metric space is countably additive (...)
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  • Lebesgue Convergence Theorems and Reverse Mathematics.Xiaokang Yu - 1994 - Mathematical Logic Quarterly 40 (1):1-13.
    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 König'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 König's lemma.
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  • A study of singular points and supports of measures in reverse mathematics.Xiaokang Yu - 1996 - Annals of Pure and Applied Logic 79 (2):211-219.
    Arithmetical comprehension is proved to be equivalent to the enumerability of singular points of any measure on the Cantor space. It is provable in ACA0 that any perfect closed subset of [0, 1] is the support of some continuous positive linear functional on C[0, 1].
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  • Classes of Recursively Enumerable Sets and Degrees of Unsolvability.Donald A. Martin - 1966 - Mathematical Logic Quarterly 12 (1):295-310.
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  • Vitali's Theorem and WWKL.Douglas K. Brown, Mariagnese Giusto & Stephen G. Simpson - 2002 - Archive for Mathematical Logic 41 (2):191-206.
    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.
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  • Reverse Mathematics 2001.Stephen G. Simpson - 2007 - Bulletin of Symbolic Logic 13 (1):106-109.
     
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  • N? Sets and models of wkl0.Stephen G. Simpson - 2005 - In Stephen Simpson (ed.), Reverse Mathematics 2001. pp. 21--352.
     
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