  In this article, we study several aspects of the intersections of
algorithmically random closed sets. First, we answer a question of Cenzer and
Weber, showing that the operation of intersecting relatively random closed sets
(with respect to certain underlying measures induced by Bernoulli measures on
the space of codes of closed sets), which preserves randomness, can be
inverted: a random closed set of the appropriate type can be obtained as the
intersection of two relatively random closed sets. We then extend the
Cenzer/Weber analysis to the intersection of multiple random closed sets,
identifying the Bernoulli measures with respect to which the intersection of
relatively random closed sets can be non-empty. We lastly apply our analysis to
provide a characterization of the effective Hausdorff dimension of sequences in
terms of the degree of intersectability of random closed sets that contain
them.
