Random Closed Sets

We investigate notions of randomness in the space C[2] of nonempty closed subsets of {0, 1}. A probability measure is given and a version of the Martin-Löf Test for randomness is defined. Π 2 random closed sets exist but there are no random Π 1 closed sets. It is shown that a random closed set is perfect, has measure 0, and has no computable elements. A closed subset of 2 may be defined as the set of infinite paths through a tree and so the problem of compressibility of trees is explored. This leads to some results on a Chaitin-style notion of randomness for closed sets.