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 simple way of assigning to each Borel set a representative modulo the equivalence relation of having countable symmetric difference. This implies that the ideal of countable sets does not satisfy an analogue to the density theorem for any notion of density.
A preprint of this paper will be uploaded here soon.