Advertisement

Israel Journal of Mathematics

, Volume 151, Issue 1, pp 61–110 | Cite as

Measured creatures

  • Andrzej Rosłanowski
  • Saharon Shelah
Article

Abstract

We prove that two basic questions on outer measure are undecidable. First we show that consistently every sup-measurable functionf: ℝ2 → ℝ is measurable. The interest in sup-measurable functions comes from differential equations and the question for which functionsf: ℝ2 → ℝ the Cauchy problemy′=f(x,y), y(x0)=y0 has a unique almost-everywhere solution in the classACt(ℝ) of locally absolutely continuous functions on ℝ. Next we prove that consistently every functionf: ℝ → ℝ is continuous on some set of positive outer Lebesgue measure. This says that in a strong sense the family of continuous functions (from the reals to the reals) is dense in the space of arbitrary such functions.

For the proofs we discover and investigate a new family of nicely definable forcing notions (so indirectly we deal with nice ideals of subsets of the reals—the two classical ones being the ideal of null sets and the ideal of meagre ones).

Concerning the method, i.e., the development of a family of forcing notions, the point is that whereas there are many such objects close to the Cohen forcing (corresponding to the ideal of meagre sets), little has been known on the existence of relatives of the random real forcing (corresponding to the ideal of null sets), and we look exactly at such forcing notions.

Keywords

Borel Function Outer Measure Force Notion Positive Lebesgue Measure Finite Tree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    U. Abraham, M. Rubin and S. Shelah,On the consistency of some partition theorems for continuous colorings, and the structure of ℵ 1 -dense real order types, Annals of Pure and Applied Logic29 (1985), 123–206.MathSciNetCrossRefGoogle Scholar
  2. [2]
    M. Balcerzak,Some remarks on sup-measurability, Real Analysis Exchange17 (1991/92), 597–607.MathSciNetzbMATHGoogle Scholar
  3. [3]
    M. Balcerzak and K. Ciesielski,On the sup-measurable functions problem, Real Analysis Exchange23 (1997/98), 787–797.MathSciNetzbMATHGoogle Scholar
  4. [4]
    T. Bartoszyński and H. Judah,Set Theory: On the Structure of the Real Line, A K Peters, Wellesley, Mass., 1995.zbMATHGoogle Scholar
  5. [5]
    J. E. Baumgartner,Iterated forcing, inSurveys in Set Theory (A. Mathias, ed.), Volume 87 of London Mathematical Society Lecture Notes, Cambridge University Press, 1978, pp. 1–59.Google Scholar
  6. [6]
    H. Blumberg,New properties of all real functions, Transactions of the American Mathematical Society24 (1922), 113–128.MathSciNetCrossRefGoogle Scholar
  7. [7]
    K. Ciesielski,Set theoretic real analysis, Journal of Applied Analysis3 (1997), 143–190.MathSciNetCrossRefGoogle Scholar
  8. [8]
    K. Ciesielski and S. Shelah,Category analogue of sup-measurability problem, Journal of Applied Analysis6 (2000), 159–172. math.LO/9905147.MathSciNetCrossRefGoogle Scholar
  9. [9]
    D. G. Fremlin,Problem list, Circulated notes. See http://www.essex.ac.uk/maths/staff/fremlin/problems.htm.Google Scholar
  10. [10]
    D. H. Fremlin, E-mail message to A. Roslanowski, October 16, 2000. Message-Id: <E131E5a-0007QQ-00@serultra0.essex.ac.uk>.Google Scholar
  11. [11]
    D. H. Fremlin,Measure Theory, Torres Fremlin, Colchester, England, 2004. See <|http://www.essex.ac.uk/maths/staff/fremlin/mstsales.html|URL|>zbMATHGoogle Scholar
  12. [12]
    M. Goldstern,Tools for your forcing construction, inSet Theory of the Reals, Israel Mathematical Conference Proceedings6, (1993), 305–360.Google Scholar
  13. [13]
    E. Grande and Z. Grande,Quelques remarques sur la superposition F(x, f (x)), Fundamenta Mathematicae121 (1984), 199–211.MathSciNetCrossRefGoogle Scholar
  14. [14]
    Z. Grande and J. S. Lipiński,Un exemple d'une fonction sup-mesurable qui n'est pas mesurable. Colloquium Mathematicum39 (1978), 77–79.MathSciNetCrossRefGoogle Scholar
  15. [15]
    E. Grzegorek,On some results of Darst and Sierpiński concerning universal null and universally measurable sets, Bulletin de l'Académie Polonaise des Séries Sciences Mathématiques29 (1981), 1–5.MathSciNetzbMATHGoogle Scholar
  16. [16]
    A. S. Kechris,Classical Descriptive Set Theory, Volume 156 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1994.Google Scholar
  17. [17]
    J. Kellner,Preserving non-null with Suslin + forcing, Archive for Mathematical Logic, to appear. math.LO/0211385.Google Scholar
  18. [18]
    J. Kellner,Definable forcings PhD thesis, Universität Wien, Austria, 2004.Google Scholar
  19. [19]
    J. Kellner and S. Shelah,Preserving preservation, Journal of Symbolic Logic, to appear. math.LO/0405081.Google Scholar
  20. [20]
    A. B. Kharazishvili,Sup-measurable and weakly sup-measurable mappings in the theory of ordinary differential equations, Journal of Applied Analysis3 (1997), 211–223.MathSciNetzbMATHGoogle Scholar
  21. [21]
    A. Roslanowski and S. Shelah,Norms on possibilities I: forcing with trees and creature. Memoirs of the American Mathematical Society141 (671) (1999), xii+167pp. math.LO/9807172.Google Scholar
  22. [22]
    A. Rosłanowski and S. Shelah,Norms on possibilities II: More ccc ideals on 2 w, Journal of Applied Analysis3 (1997), 103–127. math.LO/9703222.MathSciNetCrossRefGoogle Scholar
  23. [23]
    A. Rosłanowski and S. Shelah,Sweet & Sour and other flavours of ccc forcing notions, Archive for Mathematical Logic43 (2004), 583–663. math. LO/9909115.MathSciNetCrossRefGoogle Scholar
  24. [24]
    S. Shelah,Possibly every real function is continuous on a non-meagre set, Publications de L'Institute Mathématique, Beograd, Nouvelle Série,57(71) (1995), 47–60. math.LO/9511220.MathSciNetzbMATHGoogle Scholar
  25. [25]
    S. Shelah,Properness without elementaricity, Journal of Applied Analysis10 (2004), 169–289. math.LO/9712283.MathSciNetCrossRefGoogle Scholar
  26. [26]
    S. Shelah,Proper and Improper Forcing, Perspectives in Mathematical Logic, Springer, Berlin, 1998.CrossRefGoogle Scholar
  27. [27]
    H. von Weizsäcker,Remark on extremal measure extensions, inMeasure Theory, Oberwolfach 1979, Lecture Notes in Mathematics794, Springer, Berlin, 1980, pp. 79–80.CrossRefGoogle Scholar

Copyright information

© The Hebrew University Magnes Press 2006

Authors and Affiliations

  • Andrzej Rosłanowski
    • 1
    • 2
  • Saharon Shelah
    • 3
    • 4
  1. 1.Department of MathematicsUniversity of Nebraska at OmahaOmahaUSA
  2. 2.Mathematical Institute of Wroclaw UniversityWroclawPoland
  3. 3.Institute of MathematicsThe Hebrew University of Jerusalem Givat RamJerusalemIsrael
  4. 4.Department of MathematicsRutgers UniversityNew BrunswickUSA

Personalised recommendations