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A non-standard construction of Haar measure and weak König's lemma

Published online by Cambridge University Press:  12 March 2014

Kazuyuki Tanaka  and
Takeshi Yamazaki
Kazuyuki Tanaka
Affiliation:
Mathematical Institute, Tohoku University, Sendai 980-8578Japan
Takeshi Yamazaki
Affiliation:
Mathematical Institute, Tohoku University, Sendai 980-8578Japan
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Abstract

In this paper, we show within RCA0 that weak König's lemma is necessary and sufficient to prove that any (separable) compact group has a Haar measure. Within WKL0, a Haar measure is constructed by a non-standard method based on a fact that every countable non-standard model of WKL0 has a proper initial part isomorphic to itself [10].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 1 , March 2000 , pp. 173 - 186
Copyright
Copyright © Association for Symbolic Logic 2000

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References

REFERENCES

[1]Bishop, E., Foundations of constructive analysis, McGraw-Hill, 1967.Google Scholar
[2]Brown, D. K. and Simpson, S. G., Which set existence axioms are needed to prove the separable Hahn-Banach theorem?, Annals of Pure and Applied Logic, vol. 31 (1986), pp. 123–144.CrossRefGoogle Scholar
[3]Hausner, M., On a non-standard construction of Haar measure, Communications on Pure and Applied Mathematics, vol. 25 (1972), pp. 403–405.CrossRefGoogle Scholar
[4]Pontryagin, L. S., Topological groups, second ed., Gordon and Breach science, 1966, translated from Russian by A. Brown.Google Scholar
[5]Shioji, N. and Tanaka, K., Fixed point theory in weak second order arithmetic, Annals of Pure and Applied Logic, vol. 47 (1990), pp. 167–188.CrossRefGoogle Scholar
[6]Simpson, S. G., Which set existence axioms are needed to prove the Cauchy-Peano theorem for ordinary differential equations?, this Journal, vol. 49 (1984), pp. 783–802.Google Scholar
[7]Simpson, S. G., Subsystems of Second Order Arithmetic, Springer, 1998.Google Scholar
[8]Tanaka, K., Reverse mathematics and subsystems of second-order arithmetic, American Mathematical Society, Sugaku Expositions, vol. 5 (1992), pp. 213–234.Google Scholar
[9]Tanaka, K., Non-standard analysis in WKL0, Mathematical Logic Quarterly, vol. 43 (1997), pp. 396–400.CrossRefGoogle Scholar
[10]Tanaka, K., The self-embedding theorem of WKL0 and a non-standard method, Annals of Pure and Applied Logic, vol. 84 (1997), pp. 41–49.CrossRefGoogle Scholar
[11]Yu, X., Riesz representation theorem, Borel measures and subsystems of second-order arithmetic, Annals of Pure and Applied Logic, vol. 59 (1993), pp. 65–78.CrossRefGoogle Scholar
[12]Yu, X. and Simpson, S. G., Measure theory and weak KÖnig's lemma. Archive for Mathematical Logic, vol. 30 (1990), pp. 171–180.CrossRefGoogle Scholar