Real Analysis Exchange

The Distribution Function and Measure Preserving Maps

Behrouz Emamizadeh

Full-text: Open access


Existence of measure preserving maps has been discussed in books on real analysis where the Axiom of Choice is instrumental. In this note we introduce a method to \textit{construct} such maps. For our construction we use the distribution function and elementary differential equations.

Article information

Real Anal. Exchange, Volume 36, Number 1 (2010), 161-168.

First available in Project Euclid: 14 March 2011

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54C30: Real-valued functions [See also 26-XX] 34B05: Linear boundary value problems
Secondary: 35J25: Boundary value problems for second-order elliptic equations

measure preserving maps distribution function differential equations Saint-Venant equation


Emamizadeh, Behrouz. The Distribution Function and Measure Preserving Maps. Real Anal. Exchange 36 (2010), no. 1, 161--168.

Export citation


  • Arnold, V. I. and Khesin, B. A., Topological methods in hydrodynamics. Applied Mathematical Sciences, 125, Springer-Verlag, New York, 1998.
  • Burton, G. R. and Douglas, R. J., Uniqueness of the polar factorisation and projection of a vector-valued mapping, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20(3) (2003), 405–418.
  • Brenier, Yann, Décomposition polaire et réarrangement monotone des champs de vecteurs, (French) [Polar decomposition and increasing rearrangement of vector fields], C. R. Acad. Sci. Paris Sér. I Math., 305(19) (1987), 805–808.
  • Brenier, Y., Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44(4) (1991), 375–417.
  • Kesavan, S., Symmetrization and applications, Series in Analysis 3, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.
  • McCann, R. J., Existence and uniqueness of monotone measure-preserving maps, Duke Math. J., 80(2) (1995), 309–323.
  • Royden, H. L., Real Analysis, The Macmillan Co., New York; Collier-Macmillan Ltd., London 1963.
  • Ryff, J. V., Measure preserving transformations and rearrangements, J. Math. Anal. Appl., 31 (1970), 449–458.