What’s the price of a nonmeasurable set?

Mirko Sardella; Guido Ziliotti

Mathematica Bohemica (2002)

  • Volume: 127, Issue: 1, page 41-48
  • ISSN: 0862-7959

Abstract

top
In this note, we prove that the countable compactness of { 0 , 1 } ℝ together with the Countable Axiom of Choice yields the existence of a nonmeasurable subset of ℝ . This is done by providing a family of nonmeasurable subsets of 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.

How to cite

top

Sardella, Mirko, and Ziliotti, Guido. "What’s the price of a nonmeasurable set?." Mathematica Bohemica 127.1 (2002): 41-48. <http://eudml.org/doc/249032>.

@article{Sardella2002,
abstract = {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.},
author = {Sardella, Mirko, Ziliotti, Guido},
journal = {Mathematica Bohemica},
keywords = {Lebesgue measure; nonmeasurable set; axiom of choice; Lebesgue measure; nonmeasurable set; axiom of choice},
language = {eng},
number = {1},
pages = {41-48},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {What’s the price of a nonmeasurable set?},
url = {http://eudml.org/doc/249032},
volume = {127},
year = {2002},
}

TY - JOUR
AU - Sardella, Mirko
AU - Ziliotti, Guido
TI - What’s the price of a nonmeasurable set?
JO - Mathematica Bohemica
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 127
IS - 1
SP - 41
EP - 48
AB - 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.
LA - eng
KW - Lebesgue measure; nonmeasurable set; axiom of choice; Lebesgue measure; nonmeasurable set; axiom of choice
UR - http://eudml.org/doc/249032
ER -

References

top
  1. 10.4064/fm-6-1-244-277, Fund. Math. 6 (1924), 244–277. (1924) DOI10.4064/fm-6-1-244-277
  2. How good is Lebesgue measure? Math, Intell. 11 (1984), 54–58. (1984) MR0994965
  3. 10.4064/cm-10-2-267-269, Coll. Math. 10 (1963), 267–269. (1963) MR0160858DOI10.4064/cm-10-2-267-269
  4. 10.4064/fm-138-1-13-19, Fund. Math. 138 (1991), 13–19. (1991) MR1122273DOI10.4064/fm-138-1-13-19
  5. Counterexamples in analysis, Holden-Day, San Francisco, 1964. (1964) MR0169961
  6. Measure theory, Second edition, Van Nostrand, Princeton, 1950. (1950) Zbl0040.16802MR0033869
  7. Non-measurable sets and the equation f ( x + y ) = f ( x ) + f ( y ) , Proc. Amer. Math. Soc. 2 (1951), 221–224. (1951) Zbl0043.11002MR0040387
  8. The Axiom of Choice, North-Holland, Amsterdam, 1973. (1973) Zbl0259.02052MR0396271
  9. General topology, Van Nostrand, New York, 1955. (1955) Zbl0066.16604MR0070144
  10. 10.1007/BF01835991, Aequationes Math. 14 (1976), 421–428. (1976) MR0409750DOI10.1007/BF01835991
  11. 10.4064/cm-48-1-127-134, Coll. Math. 48 (1984), 127–134. (1984) Zbl0546.39001MR0750764DOI10.4064/cm-48-1-127-134
  12. Zermelo’s Axiom of Choice, Springer-Verlag, New York, 1982. (1982) Zbl0497.01005MR0679315
  13. 10.4064/fm-54-1-67-71, Fund. Math. 54 (1964), 67–71. (1964) MR0161788DOI10.4064/fm-54-1-67-71
  14. 10.4064/fm-138-1-21-22, Fund. Math. 138 (1991), 21–22. (1991) Zbl0792.28006MR1122274DOI10.4064/fm-138-1-21-22
  15. The strength of Hahn-Banach’s Theorem, Victoria Symposium on Non-Standard Analysis vol. 369, Lecture Notes in Math. Springer, 1974, pp. 203–248. (1974) MR0476512
  16. 10.2307/2272118, J. Symbolic Logic 42 (1977), 179–190. (1977) MR0480028DOI10.2307/2272118
  17. 10.1007/BF02760523, Israel J. Math. 48 (1984), 48–56. (1984) MR0768265DOI10.1007/BF02760523
  18. 10.4064/fm-1-1-105-111, Fund. Math. 1 (1920), 105–111. (1920) DOI10.4064/fm-1-1-105-111
  19. 10.4064/fm-1-1-112-115, Fund. Math. 1 (1920), 112–115. (1920) DOI10.4064/fm-1-1-112-115
  20. 10.4064/fm-30-1-96-99, Fund. Math. 30 (1938), 96–99. (1938) DOI10.4064/fm-30-1-96-99
  21. 10.1080/00029890.1984.11971372, Math. Montly 91 (1984), 190–193. (1984) MR0734931DOI10.1080/00029890.1984.11971372
  22. 10.1007/BF02760522, Math. 48 (1984), 1–47. (1984) MR0768264DOI10.1007/BF02760522
  23. 10.2307/1970696, Ann. of Math. 92 (1970), 1–56. (1970) Zbl0207.00905MR0265151DOI10.2307/1970696
  24. 10.4064/fm-14-1-231-233, Fund. Math. 14 (1929), 231–233. (1929) DOI10.4064/fm-14-1-231-233
  25. Sul problema della misura dei gruppi di punti di una retta, Bologna, 1905. (1905) 
  26. The Banach-Tarski paradox, Cambridge University Press, Cambridge, 1986. (1986) MR0803509
  27. 10.4064/fm-1-1-82-92, Fund. Math. 1 (1920), 82–92. (1920) DOI10.4064/fm-1-1-82-92

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.