The Lebesgue density theorem says that if A (a subset of 2^omega)
is measurable, then A is almost equal to D(A), the points of 2^omega
where A has density 1.
Therefore D selects a set from each measure class.
It turns out that D(A) Pi^0_3 --- in fact it can be complete Pi^0_3.
I will present some partial results on the complexity of D(A)
for various sets A.
This is joint work with Riccardo Camerlo.