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

Lines Missing Every Random Point

We prove that there is, in every direction in Euclidean space, a line that misses every computably random point. We also prove that there exist, in every direction in Euclidean space, arbitrarily long line segments missing every double exponential time random point.Comment: Added a section: "Betting in Doubly Exponential Time.

Covering the Recursive Sets

Contains fulltext : 147448.pdf (preprint version ) (Open Access

Covering the recursive sets

Contains fulltext : 173462.pdf (preprint version ) (Open Access

Lowness for the class of schnorr random reals

10.1137/S0097539704446323SIAM Journal on Computing353647-657SMJC

Higher Kurtz randomness

10.1016/j.apal.2010.04.001Annals of Pure and Applied Logic161101280-1290APAL

Selection by recursively enumerable sets

10.1007/978-3-642-38236-9_14Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)7876 LNCS144-15