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A nonstandard proof of a lemma from constructive measure theory
Author(s) -
Ross David A.
Publication year - 2006
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200610008
Subject(s) - mathematics , constructive proof , lemma (botany) , constructive , mathematical proof , hausdorff space , discrete mathematics , measure (data warehouse) , combinatorics , space (punctuation) , metric space , sequence (biology) , ecology , genetics , geometry , poaceae , process (computing) , database , computer science , biology , operating system , linguistics , philosophy
Suppose that f n is a sequence of nonnegative functions with compact support on a locally compact metric space, that T is a nonnegative linear functional, and that $ \sum ^\infty _{n=1}$ T f n < T f 0 . A result of Bishop, foundational to a constructive theory of functional analysis, asserts the existence of a point x such that $ \sum ^\infty _{n=1}$ f n ( x ) < f 0 ( x ). This paper extends this result to arbitrary Hausdorff spaces, and gives short proofs using nonstandard analysis. While such arguments used are not themselves constructive, they can illuminate where the difficulty lies in finding the point x . An algorithm for constructing x is then given, with a nonstandard proof that the algorithm converges to a correct value. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)