Effective Capacity and Randomness of Closed Sets

We investigate the connection between measure and capacity for the space of nonempty closed subsets of {0,1}*. For any computable measure, a computable capacity T may be defined by letting T(Q) be the measure of the family of closed sets 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 that characterize when the capacity of a random closed set equals zero or is >0. We construct for certain measures an effectively closed set with positive capacity and with Lebesgue measure zero

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

continuity of capping in c-bt

回顾眼红斑痤疮的临床表现、组织病理、病因及发病机制、诊断及治疗的进展。强调毛囊、皮脂腺蠕螨感染引起的迟发性超敏反应的作用,指出能作为诊断依据的常见和重要体征

Continuity of capping in EbT

A set A ⊆ ω is called computably enumerable (c.e., for short), if there is an algorithm to enumerate the elements of it. For sets A, B ⊆ ω, we say that A is bounded Turing reducible to B if there is a Turing functional, Φ say, with a computable bound of oracle query bits such that A is computed by Φ equipped with an oracle B, written A ≤bT B. Let EbT be the structure of the c.e. bT-degrees, the c.e. degrees under the bounded Turing reductions. In this paper we study the continuity properties in EbT. We show that for any c.e. bT-degree b � = 0, 0 ′ , there is a c.e. bT-degree a> b such that for any c.e. bT-degree x, b ∧ x = 0 if and only if a ∧ x = 0. This is the first continuity property of the c.e. bT-degrees.