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

be a complete separable metric space is defined by

is also а complete separable metric space. The corresponding open subsets generate a σ-algebra, the Borel sigma algebra.

A random compact set is а measurable function from а probability space.

is almost surely compact and

.

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

)

is obtained, which satisfies

Thus the covering function is given by

:

is called the support 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

References

• Matheron, G. (1975) Random Sets and Integral Geometry. J.Wiley & Sons, New York.

• Molchanov, I. (2005) The Theory of Random Sets. Springer, New York.

• Stoyan D., and H.Stoyan (1994) Fractals, Random Shapes and Point Fields. John Wiley & Sons, Chichester, New York.