Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 39, 2014, 831-857

SPACE-FILLING VS. LUZIN'S CONDITION (N)

Thomas Zürcher

University of Jyväskylä, Department of Mathematics and Statistics
P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland; thomas.t.zurcher 'at' jyu.fi

Abstract. Let us assume that we are given two metric spaces (X,d_X) and (Y,d_Y) where the Hausdorff dimension s of X is strictly smaller than that of Y. Suppose that X is σ-finite with respect to H^s. Then we show that for quite general metric spaces, if f : X → Y is a measurable surjection, there is a set N ⊂ X with H^s(N) = 0 and H^s(f(N)) > 0. If f is continuous, then we investigate whether N can be chosen to be perfect. We also study more general situations where the measures on X and Y are not necessarily the same and not necessarily Hausdorff measures.

2010 Mathematics Subject Classification: Primary 28A75, 54C10, 26B35, 28A12, 28A20.

Key words: Space-fillings, Luzin's condition (N), mappings between spaces of different dimensions.

Reference to this article: T. Zürcher: Space-filling vs. Luzin's condition (N). Ann. Acad. Sci. Fenn. Math. 39 (2014), 831-857.

Full document as PDF file

doi:10.5186/aasfm.2014.3950