Effective Randomness of Unions and Intersections

摘要

We investigate the μ-randomness of unions and intersections of random sets under various notions of randomness corresponding to different probability measures. For example, the union of two relatively Martin-Loef random sets is not Martin-Lof random but is random with respect to the Bernoulli measure λ_(3/4) under which any number belongs to the set with probability 3/4. Conversely, any λ_(3/4) random set is the union of two Martin-Loef random sets. Unions and intersections of random closed sets are also studied.