Mathematical Notes

, Volume 58, Issue 3, pp 960–965 | Cite as

TheN−1-property of maps and Luzin's condition (N)

  • S. P. Ponomarev
Article

Abstract

A functionfG→ℝ n , whereG is an open set in ℝ n , has theN−1-property if for allE⊂ℝ n we have {¦E¦=0⇒¦f−1(E)¦=0} (¦·¦ is the Lebesgue measure). The article is concerned with the relations between theN−1-property of functions, the maximal rank of derivatives, and the differentiability almost everywhere of composite functions.

Keywords

Lebesgue Measure Composite Function Maximal Rank 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. K. Vodop'yanov and V.M. Goldshtein, “Quasiconformal mappings and spaces of functions with generalized first derivatives,”Sibirsk. Mat. Zh. [Siberian Math. J.],17, No. 3, 515–531 (1976).Google Scholar
  2. 2.
    S. Saks,Theory of the Integral, New York (1937).Google Scholar
  3. 3.
    T. Rado and P. V. Reichelderfer,Continuous Transformations in Analysis, Springer-Verlag, Berlin-Göttingen-Heidelberg (1955).Google Scholar
  4. 4.
    H. Federer,Geometric Measure Theory, Springer-Verlag, Berlin-Goettingen-Heidelberg (1969).Google Scholar
  5. 5.
    Elias M. Stein,Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton (1970).Google Scholar
  6. 6.
    D. R. MacMillan, “Taming Cantor sets inE n,”Bull. Amer. Math. Soc.,70, No. 5, 706–708 (1964).MathSciNetGoogle Scholar
  7. 7.
    V. K. A. M. Gugenheim, “Piecewise-linear isotopy and embedding of elements and spheres, I,”Proc. London Math. Soc.,3, 29–53 (1953).MATHMathSciNetGoogle Scholar
  8. 8.
    S. P. Ponomarev, “An example of an ACTL p-homeomorphism which is not Banach absolutely continuous,”Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.],201, No. 5, 1053–1054 (1971).MATHMathSciNetGoogle Scholar
  9. 9.
    C. P. Rourke and B. J. Sanderson,Introduction to Piecewise-Linear Topology, Springer-Verlag, Berlin-Göttingen-Heidelberg (1982).Google Scholar
  10. 10.
    J. C. Oxtoby,Measure and Category: A Survey of the Analogies between Topological and Measure Spaces, Springer-Verlag, New York-Berlin (1980).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • S. P. Ponomarev
    • 1
  1. 1.Moscow Institute of Steel and AlloysUSSR

Personalised recommendations