Israel Journal of Mathematics. Volume 249, May 2022. Pages 501-551.
DOI: 10.1007/s11856-022-2312-8. PDF. arXiv. Bibtex.

 @ARTICLE{MSSW22,
   AUTHOR = {S. Müller and P. Schlicht and D. Schrittesser and T. Weinert},
   TITLE = {Lebesgue's density theorem and definable selectors for ideals},
   JOURNAL={Israel Journal of Mathematics},
   VOLUME = {249},
   YEAR = {2022},
   PAGES = {501--551},
   EPRINT = {1811.06489},
   DOI = {10.1007/s11856-022-2312-8} }

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.

As a contrasting result of independent interest, 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.