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Finding subsets of positive measure
| Content Provider | Semantic Scholar |
|---|---|
| Author | Kjos-Hanssen, Bjørn Reimann, Jan |
| Copyright Year | 2014 |
| Abstract | An important theorem of geometric measure theory (first proved by Besicovitch and Davies for Euclidean space) says that every analytic set of non-zero $s$-dimensional Hausdorff measure $\mathcal H^s$ contains a closed subset of non-zero (and indeed finite) $\mathcal H^s$-measure. We investigate the question how hard it is to find such a set, in terms of the index set complexity, and in terms of the complexity of the parameter needed to define such a closed set. Among other results, we show that given a (lightface) $\Sigma^1_1$ set of reals in Cantor space, there is always a $\Pi^0_1(\mathcal{O})$ subset on non-zero $\mathcal H^s$-measure definable from Kleene's $\mathcal O$. On the other hand, there are $\Pi^0_2$ sets of reals where no hyperarithmetic real can define a closed subset of non-zero measure. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://arxiv.org/pdf/1408.1999v1.pdf |
| Alternate Webpage(s) | http://www.math.hawaii.edu/~bjoern/Publications/RegularityHausdorff_TOCS.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
