Skip to main content
SpringerLink
Log in

Extensions of the measurable choice theorem by means of forcing

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Using the method of forcing of set theory, we prove the following two theorems on the existence of measurable choice functions: LetT be the closed unit interval [0,1] and letm be the usual Lebesgue measure defined on the Borel subsets ofT. Theorem1. LetS⊂T×T be a Borel set such that for alltεT,S t def={x|(t,x)εS} is countable and non-empty. Then there exists a countable series of Lebesgue-measurable functionsf n: T→T such thatS t={fn(t)|nεω} for alltε[0,1],W x={y|(x,y)εW} is uncountable. Then there exists a functionh:[0,1]×[0,1]→W with the following properties: (a) for each xε[0,1], the functionh(x,·) is one-one and ontoW x and is Borel measurable; (b) for eachy, h(·, y) is Lebesgue measurable; (c) the functionh is Lebesgue measurable.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. J. Aumann,Measurable utility and the measurable choice theorem, Colloq. Int. Cent. Natl. Rech. Sci.171 (1967), 15–26.

    Google Scholar 

  2. P. Cohen,Set Theory and the Continuum Hypothesis, W. A. Benjamin, Inc., New York, 1966.

    MATH  Google Scholar 

  3. K. Kuratowski,Topology, Volume I, Academic Press, New York, London, 1966.

    Google Scholar 

  4. P. R. Halmos,Measure Theory, D. Van Nostrand Company, Inc., New York, 1950.

    Google Scholar 

  5. N. Lusin,Leçons sur les ensembles analytiques et leurs applications, Gauthier-Villars, Paris, 1930.

    MATH  Google Scholar 

  6. J. R. Shoenfield,Mathematical Logic, Addision-Wesley Publishing Company, Reading, Massachusetts, 1967.

    MATH  Google Scholar 

  7. J. R. Shoenfield,The problem of predicativity, In: Essays on the Foundations of Mathematics, Jerusalem, (1961), pp. 132–139.

  8. R. M. Solovay,A Model of Set theory in which every set of reals is Lebesuge measurable, Ann. of Math.92 (1970), 1–56.

    Article  MathSciNet  Google Scholar 

  9. J. Von Neumann,On Rings of operators, Reduction theory, Ann. of Math.50(1949), 401–485.

    Article  MathSciNet  Google Scholar 

  10. E. Wesley,Borel preference orders in markets with a continuum of traders, (to appear).

Download references

Author information

Authors and Affiliations

  1. The Hebrew University of Jerusalem, Jerusalem, Israel

    Eugene Wesley

Authors
  1. Eugene Wesley
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wesley, E. Extensions of the measurable choice theorem by means of forcing. Israel J. Math. 14, 104–114 (1973). https://doi.org/10.1007/BF02761539

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02761539

Keywords

Search

Navigation