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.
be a complete separable metric space.
denote the set of all compact subsets of
The Hausdorff metric
is defined by
is also а complete separable metric space.
The corresponding open subsets generate a σ-algebra on
, the Borel sigma algebra
A random compact set is а measurable function
from а probability space
Put another way, a random compact set is a measurable function
( ω )
is almost surely compact and is a measurable function for every
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
, the probability
is obtained, which satisfies Thus the covering function
can also be interpreted as the mean of the indicator function
: The covering function takes values between
The set
is called the support of
is called the kernel, the set of fixed points, or essential minimum
, is а sequence of i.i.d.
random compact sets, then almost surely and
converges almost surely to