Algorithmic Randomness and Capacity of Closed Sets

TL;DR

This paper explores the relationship between measure, capacity, and algorithmic randomness in the space of closed sets, providing effective versions of classical theorems and new insights into computable measures and randomness.

Contribution

It establishes an effective version of Choquet's capacity theorem and characterizes when the capacity of an m-random closed set is zero, advancing the understanding of algorithmic randomness and measure theory.

Findings

Effective characterization of capacities via computable measures

Conditions for zero capacity of m-random closed sets

Construction of effectively closed sets with positive capacity and zero Lebesgue measure

Abstract

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.