Random compact set

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 formula_1 be a complete separable metric space. Let formula_2 denote the set of all compact subsets of formula_3. The Hausdorff metric formula_4 on formula_2 is defined by
formula_7 is also а complete separable metric space. The corresponding open subsets generate a σ-algebra on formula_2, the Borel sigma algebra formula_9 of formula_2.
A random compact set is а measurable function formula_11 from а probability space formula_12 into formula_13.
Put another way, a random compact set is a measurable function formula_14 such that formula_15 is almost surely compact and
is a measurable function for every formula_17.
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
For formula_21, the probability formula_22 is obtained, which satisfies
Thus the covering function formula_24 is given by
Of course, formula_24 can also be interpreted as the mean of the indicator function formula_28:
The covering function takes values between formula_30 and formula_31. The set formula_32 of all formula_17 with formula_34 is called the support of formula_35. The set formula_36, of all formula_37 with formula_38 is called the kernel, the set of fixed points, or essential minimum formula_39. If formula_40, is а sequence of i.i.d. random compact sets, then almost surely
and formula_42 converges almost surely to formula_43