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random compact set ：ウィキペディア英語版

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

(M, d)

\mathcal

denote the set of all compact subsets of

M

. The Hausdorff metric

h

\mathcal

is defined by
:

h(K_, K_) := \max \left\} d(a, b), \sup_} d(a, b) \right\}.

(\mathcal, h)

is also а complete separable metric space. The corresponding open subsets generate a σ-algebra on

\mathcal

\mathcal(\mathcal)

\mathcal

.
A random compact set is а measurable function

K

(\Omega, \mathcal, \mathbb)

into

(\mathcal, \mathcal (\mathcal) )

.
Put another way, a random compact set is a measurable function

K \colon \Omega \to 2^

such that

K(\omega)

is almost surely compact and
:

\omega \mapsto \inf_ d(x, b)

is a measurable function for every

x \in M

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