Effective Capacity and Randomness of Closed Sets

Douglas Cenzer
(University of Florida)
Paul Brodhead
(Virginia State University)

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.

In Xizhong Zheng and Ning Zhong: Proceedings Seventh International Conference on Computability and Complexity in Analysis (CCA 2010), Zhenjiang, China, 21-25th June 2010, Electronic Proceedings in Theoretical Computer Science 24, pp. 67–76.
Published: 3rd June 2010.

ArXived at: http://dx.doi.org/10.4204/EPTCS.24.11 bibtex PDF

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