Lebesgue's density theorem and definable selectors for ideals

Philipp Schlicht , Sandra Müller, David Schrittesser, Thilo Weinert

Research output: Contribution to journalArticle (Academic Journal)peer-review

Abstract

We introduce a notion of density point and prove results analogous to Lebesgue’s density theorem for various well-known ideals on Cantor space and Baire space. In fact, we isolate a class of ideals for which our results hold.
In contrast to these results, we show that there is no reasonably definable selector that chooses representatives for the equivalence relation on the Borel sets of having countable symmetric difference. In other words, there is no notion of density which makes the ideal of countable sets satisfy an analogue to the density theorem.
The proofs of the positive results use only elementary combinatorics of trees, while the negative results rely on forcing arguments.
Original languageEnglish
Number of pages28
JournalIsrael Journal of Mathematics
Publication statusSubmitted - 2019

Fingerprint Dive into the research topics of 'Lebesgue's density theorem and definable selectors for ideals'. Together they form a unique fingerprint.

Cite this