Effectively Closed Set of Measures and Randomness
نویسنده
چکیده
We show that if a real x ∈ 2 is Hausdorff H-random, where h is an order function, then it is also random for a continuous probability measure μ such that the μ-measure of the basic open cylinders shrinks according to h. We use this result to derive a characterization of effective Hausdorff dimension similar to Frostman’s Theorem.
منابع مشابه
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 f...
متن کاملEffectively closed sets of measures and randomness
We show that if a real x ∈ 2 is strongly Hausdorff Hh-random, where h is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure μ such that the μ-measure of the basic open cylinders shrinks according to h. The proof uses a new method to construct measures, based on effective (partial) continuous transformations and a basis theorem for Π...
متن کاملDegrees of Weakly Computable Reals
A real α is left-recursively enumerable (left-r.e. for short) if we can effectively generate α from below. That is, the left Dedkind cut of α, L(α) = {q ∈ Q : q ≤ α}, forms a r.e. set. Equivalently, a real α is left-r.e. if it is the limit of a converging recursive increasing sequence. If we can also compute the radius of convergence effectively, then α is recursive. Left-r.e. reals are the mea...
متن کاملAlgorithmic Randomness and Capacity of Closed Sets
We investigate the connection between measure, capacity and algorithmic randomness for the space of closed sets. For any computable measure m, a computable capacity T may be de ned by letting T (Q) be the measure of the family of closed sets K which have nonempty intersection with Q. We prove an e ective version of Choquet's capacity theorem by showing that every computable capacity may be obta...
متن کاملE ective Randomness of Unions and Intersections
We investigate the -randomness of unions and intersections of random sets under various notions of randomness corresponding to di erent probability measures. For example, the union of two relatively MartinL of random sets is not Martin-L of random but is random with respect to the Bernoulli measure 3 4 under which any number belongs to the set with probability 3 4 . Conversely, any 3 4 random...
متن کامل