6533b7cffe1ef96bd12598f4
RESEARCH PRODUCT
Space-filling vs. Luzin's condition (N)
Thomas Zürchersubject
28A75 (Primary) 54C10 26B35 28A12 28A20 (Secondary)General Mathematicsta111Hausdorff spaceMathematics::General TopologySpace (mathematics)Functional Analysis (math.FA)Mathematics - Functional AnalysisSurjective functionCombinatoricsSet (abstract data type)Metric spaceMathematics - Classical Analysis and ODEsHausdorff dimensionClassical Analysis and ODEs (math.CA)FOS: MathematicsMathematicsdescription
Let us assume that we are given two metric spaces, where the Hausdorff dimension of the first space is strictly smaller than the one of the second space. Suppose further that the first space has sigma-finite measure with respect to the Hausdorff measure of the corresponding dimension. We show for quite general metric spaces that for any measurable surjection from the first onto the second space, there is a set of measure zero that is mapped to a set of positive measure (both measures are the Hausdorff measures corresponding to the Hausdorff dimension of the first space). We also study more general situations where the measures on the two metric spaces are not necessarily the same and not necessarily Hausdorff measures.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2013-10-25 | Annales Academiae Scientiarum Fennicae Mathematica |