A Sard type theorem for Borel mappings

ArticleOriginal scientific text

Title

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We find a condition for a Borel mapping

f : ℝ^{m} \to ℝ^{n}

which implies that the Hausdorff dimension of

f^{- 1} (y)

is less than or equal to m-n for almost all

y \in ℝ^{n}

.

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