Measure theory and weak König's lemma |
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Authors: | Xiaokang Yu Stephen G. Simpson |
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Affiliation: | (1) Department of Mathematics, Pennsylvania State University, Altoona Campus, 16601 Altoona, PA, USA;(2) Department of Mathematics, Pennsylvania State University, 16802 University Park, PA, USA |
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Abstract: | 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 on open sets.The research of both authors was partially supported by NSF Grant DMS-8701481. |
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