Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
Lebesgue measure; nonmeasurable set; axiom of choice
Lebesgue measure; nonmeasurable set; axiom of choice
Summary:
In this note, we prove that the countable compactness of $\lbrace 0,1\rbrace ^{{\mathbb{R}}}$ together with the Countable Axiom of Choice yields the existence of a nonmeasurable subset of ${\mathbb{R}}$. This is done by providing a family of nonmeasurable subsets of ${\mathbb{R}}$ whose intersection with every non-negligible Lebesgue measurable set is still not Lebesgue measurable. We develop this note in three sections: the first presents the main result, the second recalls known results concerning non-Lebesgue measurability and its relations with the Axiom of Choice, the third is devoted to the proofs.
References:
[1] S. Banach, A. Tarski: Sur la dΘcomposition des ensembles de points en parties respectivement congruentes. Fund. Math. 6 (1924), 244–277. DOI 10.4064/fm-6-1-244-277
[2] K. Ciesielski: How good is Lebesgue measure? Math. Intell. 11 (1984), 54–58. MR 0994965
[3] P. Erdös: On some properties of Hamel bases. Coll. Math. 10 (1963), 267–269. DOI 10.4064/cm-10-2-267-269 | MR 0160858
[4] M. Foreman, F. Wehrung: The Hahn-Banach Theorem implies the existence of a non-Lebesgue measurable set. Fund. Math. 138 (1991), 13–19. DOI 10.4064/fm-138-1-13-19 | MR 1122273
[5] B. R. Gelbaum, J. M. H. Olmstead: Counterexamples in analysis. Holden-Day, San Francisco, 1964. MR 0169961
[6] P. H. Halmos: Measure theory. Second edition, Van Nostrand, Princeton, 1950. MR 0033869 | Zbl 0040.16802
[7] I. Halperin: Non-measurable sets and the equation $f(x+y) = f(x)+f(y)$. Proc. Amer. Math. Soc. 2 (1951), 221–224. MR 0040387 | Zbl 0043.11002
[8] T. Jech: The Axiom of Choice. North-Holland, Amsterdam, 1973. MR 0396271 | Zbl 0259.02052
[9] J. L. Kelley: General topology. Van Nostrand, New York, 1955. MR 0070144 | Zbl 0066.16604
[10] M. Kuczma, J. Smítal: On measures connected with the Cauchy equation. Aequationes Math. 14 (1976), 421–428. DOI 10.1007/BF01835991 | MR 0409750
[11] M. Kuczma: On some properties of Erdös sets. Coll. Math. 48 (1984), 127–134. DOI 10.4064/cm-48-1-127-134 | MR 0750764 | Zbl 0546.39001
[12] G. Moore: Zermelo’s Axiom of Choice. Springer-Verlag, New York, 1982. MR 0679315 | Zbl 0497.01005
[13] J. Mycielski, S. Swierczkowski: On the Lebesgue measurability and the axiom of determinateness. Fund. Math. 54 (1964), 67–71. DOI 10.4064/fm-54-1-67-71 | MR 0161788
[14] J. Pawlikowski: The Hahn-Banach Theorem implies the Banach-Tarski paradox. Fund. Math. 138 (1991), 21–22. DOI 10.4064/fm-138-1-21-22 | MR 1122274 | Zbl 0792.28006
[15] D. Pincus: The strength of Hahn-Banach’s Theorem. Victoria Symposium on Non-Standard Analysis vol. 369, Lecture Notes in Math. Springer, 1974, pp. 203–248. MR 0476512
[16] D. Pincus, R. Solovay: Definability of measures and ultrafilters. J. Symbolic Logic 42 (1977), 179–190. DOI 10.2307/2272118 | MR 0480028
[17] J. Raissonier: A mathematical proof of S. Shelah’s theorem on the measure problem and related results. Israel J. Math. 48 (1984), 48–56. DOI 10.1007/BF02760523 | MR 0768265
[18] W. Sierpiński: Sur la question de la mesurabilité de la base de M. Hamel. Fund. Math. 1 (1920), 105–111. DOI 10.4064/fm-1-1-105-111
[19] W. Sierpiński: Sur un problème concernant les ensembles mesurables superficiellement. Fund. Math. 1 (1920), 112–115. DOI 10.4064/fm-1-1-112-115
[20] W. Sierpiński: Fonctions additives non complétement additives et fonctions mesurables. Fund. Math. 30 (1938), 96–99. DOI 10.4064/fm-30-1-96-99
[21] A. Simonson: On two halves being two wholes. Math. Montly 91 (1984), 190–193. DOI 10.1080/00029890.1984.11971372 | MR 0734931
[22] S. Shelah: Can you take Solovay’s inaccessible away? Israel J. Math. 48 (1984), 1–47. DOI 10.1007/BF02760522 | MR 0768264
[23] R. Solovay: A model of set theory in which every set of reals is Lebesgue-measurable. Ann. of Math. 92 (1970), 1–56. DOI 10.2307/1970696 | MR 0265151 | Zbl 0207.00905
[24] S. Ulam: Concerning functions of sets. Fund. Math. 14 (1929), 231–233. DOI 10.4064/fm-14-1-231-233
[25] G. Vitali: Sul problema della misura dei gruppi di punti di una retta. Bologna, 1905.
[26] S. Wagon: The Banach-Tarski paradox. Cambridge University Press, Cambridge, 1986. MR 0803509
[27] W. Wilkosz: Sugli insiemi non-misurabili (L). Fund. Math. 1 (1920), 82–92. DOI 10.4064/fm-1-1-82-92
Search
Partner of
