The Lebesgue density theorem states that for almost every X in a
measurable class C, the relative measure of C along X converges to
one. Call such an X a density point of C. We ask how random must a
real X be to be a density point of every Pi^0_1 class that contains
it. Along the way, we prove that no K-trivial can be cupped to 0' by
an incomplete ML-random (one direction of the solution to the
ML-cupping problem) and together with the work of Bienvenu,
Greenberg, Kucera, Nies and Turetsky that Noam has presented, we
prove that there is an incomplete ML-random that computes every
K-trivial (solving the ML-covering problem).
Although this talk is a closely related to Noam's lectures, no
understanding of previous material should be required.
(Joint work variously with Bienvenu, Hölzl and Nies; Day; Andrews,
Cai, Diamondstone and Lempp.)