A Sard type theorem for Borel mappings

Piotr Hajłasz

Colloquium Mathematicae (1994)

  • Volume: 67, Issue: 2, page 217-221
  • ISSN: 0010-1354

Abstract

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We find a condition for a Borel mapping f : ℝ m → ℝ n which implies that the Hausdorff dimension of f - 1 ( y ) is less than or equal to m-n for almost all y ∈ ℝ n .

How to cite

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Hajłasz, Piotr. "A Sard type theorem for Borel mappings." Colloquium Mathematicae 67.2 (1994): 217-221. <http://eudml.org/doc/210274>.

@article{Hajłasz1994,
abstract = {We find a condition for a Borel mapping $f:ℝ^m → ℝ^n$ which implies that the Hausdorff dimension of $f^\{-1\}(y)$ is less than or equal to m-n for almost all $y ∈ ℝ^n$.},
author = {Hajłasz, Piotr},
journal = {Colloquium Mathematicae},
keywords = {Hausdorff dimension; Sard type theorem; Borel mappings; Hausdorff measure},
language = {eng},
number = {2},
pages = {217-221},
title = {A Sard type theorem for Borel mappings},
url = {http://eudml.org/doc/210274},
volume = {67},
year = {1994},
}

TY - JOUR
AU - Hajłasz, Piotr
TI - A Sard type theorem for Borel mappings
JO - Colloquium Mathematicae
PY - 1994
VL - 67
IS - 2
SP - 217
EP - 221
AB - We find a condition for a Borel mapping $f:ℝ^m → ℝ^n$ which implies that the Hausdorff dimension of $f^{-1}(y)$ is less than or equal to m-n for almost all $y ∈ ℝ^n$.
LA - eng
KW - Hausdorff dimension; Sard type theorem; Borel mappings; Hausdorff measure
UR - http://eudml.org/doc/210274
ER -

References

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  1. [1] B. Bojarski and P. Hajłasz, in preparation. 
  2. [2] Y. Burago and V. Zalgaller, Geometric Inequalities, Grundlehren Math. Wiss. 285, Springer, 1988. 
  3. [3] S. Eilenberg, On ϕ -measures, Ann. Soc. Polon. Math. 17 (1938), 252-253. 
  4. [4] S. Eilenberg and O. Harold, Continua of finite linear measure I, Amer. J. Math. 65 (1943), 137-146. Zbl0063.01227
  5. [5] H. Federer, Geometric Measure Theory, Springer, 1969. Zbl0176.00801
  6. [6] P. Hajłasz, A note on weak approximation of minors, Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear. Zbl0910.49025
  7. [7] P. Hajłasz, Sobolev mappings, co-area formula and related topics, preprint. Zbl0988.28002
  8. [8] T. Jech, Set Theory, Acad. Press, 1978. 
  9. [9] K. Kuratowski, Topology, Vol. 1, Acad. Press, 1966. 
  10. [10] N. Lusin, Sur les ensembles analytiques, Fund. Math. 10 (1927), 1-95. 
  11. [11] N. Lusin et W. Sierpiński, Sur quelques propriétés des ensembles (A), Bull. Acad. Cracovie 4-5A (1918), 35-48. 
  12. [12] P. Mattila, Hausdorff dimension, orthogonal projections and intersections with planes, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), 227-244. Zbl0348.28019
  13. [13] J. Milnor, Topology from the Differentiable Viewpoint, The Univ. Press of Virginia, 1965. Zbl0136.20402
  14. [14] A. Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc. 48 (1942), 883-890. Zbl0063.06720
  15. [15] S. Sternberg, Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, N.J., 1964. 

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