Effective Capacity and Randomness of Closed Sets

TL;DR

This paper explores the relationship between measure and capacity in the space of closed sets, proving an effective version of Choquet's theorem and analyzing properties of random closed sets.

Contribution

It establishes an effective version of Choquet's capacity theorem linking computable measures and capacities, and characterizes when random closed sets have positive or zero capacity.

Findings

Effective capacity equals zero or positive for certain random closed sets.

Constructed effectively closed sets with positive capacity but zero Lebesgue measure.

Abstract

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.