In mathematics, a random compact set is essentially a compact set-valued random variable. Random compact sets are useful in the study of attractors for random dynamical systems.

Definition

Let (M, d) be a complete separable metric space. Let \mathcal{K} denote the set of all compact subsets of M. The Hausdorff metric h on \mathcal{K} is defined by is also а complete separable metric space. The corresponding open subsets generate a σ-algebra on \mathcal{K}, the Borel sigma algebra of \mathcal{K}. A random compact set is а measurable function K from а probability space into. Put another way, a random compact set is a measurable function such that K(\omega) is almost surely compact and is a measurable function for every x \in M.

Discussion

Random compact sets in this sense are also random closed sets as in Matheron (1975). Consequently, under the additional assumption that the carrier space is locally compact, their distribution is given by the probabilities (The distribution of а random compact convex set is also given by the system of all inclusion probabilities ) For K = { x }, the probability is obtained, which satisfies Thus the covering function p_{X} is given by Of course, p_{X} can also be interpreted as the mean of the indicator function : The covering function takes values between 0 and 1. The set b_{X} of all x \in M with is called the support of X. The set k_X, of all x \in M with p_X(x)=1 is called the kernel, the set of fixed points, or essential minimum e(X). If, is а sequence of i.i.d. random compact sets, then almost surely and converges almost surely to e(X).