Lebesgue Density and ∏10 Classes

Analyzing the effective content of the Lebesgue density theorem played a crucial role in some recent developments in algorithmic randomness, namely, the solutions of the ML-covering and ML-cupping problems. Two new classes of reals emerged from this inquiry: the positive density points with respect to effectively closed (or Π1) sets of reals, and a proper subclass, the density-one points. Bienvenu, Hölzl, Miller, and Nies have shown that the Martin-Löf random positive density points are exactly the ones that do not compute the halting problem. Treating this theorem as our starting point, we present several new results that shed light on how density, randomness, and computational strength interact. 1. A generalization of 1-genericity The Lebesgue density theorem says that if A is any Lebesgue measurable set of reals, for almost every point x of A, the density of A at x is 1. Roughly speaking, the more we “zoom in” on x by looking at a smaller and smaller interval containing it, the closer to 1 is the fractional measure of A within that interval. Suppose that C is a countable collection of Lebesgue measurable subsets of the unit interval. We say x ∈ [0, 1] is a positive density point for C if for every P ∈ C that contains x, the density of P at x is positive. We say x is a density-one point for C if for every P ∈ C that contains x, the density of P at x is 1. It follows from the Lebesgue density theorem that almost every point in the unit interval is a density-one point for C. Of particular interest are the positive density and density-one points we obtain when C is the collection of effectively closed (or Π1) subsets of the unit interval. These have been at the heart of several recent developments in algorithmic randomness, such as the solutions of the ML-covering and MLcupping problems [1, 4, 5]. An interesting fact that emerged from this line 2010 Mathematics Subject Classification. 03D32, 03D78, 03F60, 26E40.