Algorithmic randomness was originally defined for Cantor space with the fair-coin measure. Recent work has examined algorithmic randomness in new contexts, in particular closed subsets of 2ω ([2] and [8]). In this paper we use the probability theory of closed set-valued random variables (RACS) to extend the definition of Martin-Löf randomness to spaces of closed subsets of locally compact, Haus...

We shall show that the union-closed families conjecture holds for a random union-closed family with high probability. This strengthens a recent result of Bruhn and Schaudt.

We investigate the notion of K-triviality for closed sets and continuous functions in 2N. For every K-trivial degreee d, there exists a closed set of degree d and a continuous function of degree d. Every K-trivial closed set contains a K-trivial real. There exists a K-trivial Π1 class with no computable elements. A closed set is K-trivial if and only if it is the set of zeroes of some K-trivial...

We investigate notions of randomness in the space C(2N) of continuous functions on 2N. A probability measure is given and a version of the Martin-Löf Test for randomness is defined. Random ∆2 continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set...

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...

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...

Geometrical and topological properties of the set of all belief functions generated by a random closed interval are studied. It is shown that this set is a metrizable (noncompact) simplex and its extreme points are completely characterized.