Fri Oct 19, 2007
60 Evans Hall, 4:10–6 PM
Jan Reimann (Mathematics, UC Berkeley)
Never continuously random reals - an intriguing Π11 set
The duality between measures and the sets they “charge” is a central theme in modern analysis. An effective analogue of this question is: Given a real x, does there exist a (probability) measure relative to which x is effectively random (so that x is not an atom of the measure)? And if such a measure exists, can we ensure that it has certain properties (nonatomic, of a certain minimum capacity, etc)? While every noncomputable real is random with respect to some measure, there exists a nested sequence of countable Π11 sets of reals that are not n-random with respect to any continuous measure. This sequence exhibits a number of interesting properties. For instance, the proof that all of the sets are countable requires the existence of infinitely many iterates of the power set of ω, similar to Borel determinacy. Furthermore, it seems quite hard to find a natural notion of rank for such reals.
I will first survey the basic results on continuous randomness, before discussing more recent results on 1-randomness. Techniques are drawn from various areas of logic and analysis, such as Turing degrees, Π01 classes, determinacy, fine structure theory, or Hausdorff measures and capacities.
This is an ongoing joint project with Theodore Slaman.