Effective Capacity and Randomness of Closed Sets

We investigate the connection between measure and capacity for the space C of nonempty closed subsets of 2N. For any computable measure μ∗, 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 the Choquet’s 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.