  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.
