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We investigate the connection between measure and capacity for the space ofnonempty closed subsets of {0,1}*. For any computable measure, a computablecapacity T may be defined by letting T(Q) be the measure of the family ofclosed sets which have nonempty intersection with Q. We prove an effectiveversion of Choquet's capacity theorem by showing that every computable capacitymay be obtained from a computable measure in this way. We establish conditionsthat characterize when the capacity of a random closed set equals zero or is>0. We construct for certain measures an effectively closed set with positivecapacity and with Lebesgue measure zero.

Effective Capacity and Randomness of Closed Sets