Algorithmic Randomness and Capacity of Closed Sets

We investigate the connection between measure, capacity and algorithmic randomness for the space of closed sets. For any computable measure m, a computable capacity T may be defined by letting T(Q) be the measure of the family of closed sets K which have nonempty intersection with Q. We prove an effective version of Choquet's capacity theorem by showing that every computable capacity may be obtained from a computable measure in this way. We establish conditions on the measure m that characterize when the capacity of an m-random closed set equals zero. This includes new results in classical probability theory as well as results for algorithmic randomness. For certain computable measures, we construct effectively closed sets with positive capacity and with Lebesgue measure zero. We show that for computable measures, a real q is upper semi-computable if and only if there is an effectively closed set with capacity q

Evolving Computability

We consider the degrees of non-computability (Weihrauch degrees) of finding winning strategies (or more generally, Nash equilibria) in infinite sequential games with certain winning sets (or more generally, outcome sets). In particular, we show that as the complexity of the winning sets increases in the difference hierarchy, the complexity of constructing winning strategies increases in the effective Borel hierarchy.Comment: An extended abstract of this work has appeared in the Proceedings of CiE 201

HOUSTON JOURNAL OF MATHEMATICS, Volume 6, No. 1,1980. FAITHFUL EXTENSIONS OF ANALYTIC SETS TO BOREL SETS

ABSTRACT. A faithful extension property is a property P such that for any analytic subset A of the product X XY of two Polish spaces, X and Y, such that each Y-section of A possesses property P, there is a Borel set B including A so that each Y-section of B possesses property P. Lyapunov and others showed that various properties are faithful extension. In this paper a uniform method is given for showing that these and many other properties are faithful extension. Introduction. In this paper we consider the following general problem: Given two Polish spaces X and Y and an analytic subset A of the product X X Y such that each section Ay (defined to be { x ' (x,y) • A)) satisfies a certain property P-- Is there a Borel set B D_A such that each section By satisfies P? Such a B will be called a faithful extension of A. P is called a faithful extension (or FE) property of subsets o