Random closed sets viewed as random recursions

It is known that the box dimension of any Martin-Löf random closed set of {0, 1}N is log2( 43 ). Barmpalias et al. [Journal of Logic and Computation, Vol. 17, No. 6 (2007)] gave one method of producing such random closed sets and then computed the box dimension, and posed several questions regarding other methods of construction. We outline a method using random recursive constructions for computing the Hausdorff dimension of almost every random closed set of {0, 1}N, and propose a general method for random closed sets in other spaces. We further find both the appropriate dimensional Hausdorff measure and the exact Hausdorff dimension for such random closed sets.