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.
Contents
Definition
Let
A random compact set is а measurable function
Put another way, a random compact set is a measurable function
is a measurable function for every
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
Thus the covering function
Of course,
The covering function takes values between
and