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. 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. Let ( M , d ) {displaystyle (M,d)} be a complete separable metric space. Let K {displaystyle {mathcal {K}}} denote the set of all compact subsets of M {displaystyle M} . The Hausdorff metric h {displaystyle h} on K {displaystyle {mathcal {K}}} is defined by ( K , h ) {displaystyle ({mathcal {K}},h)} is also а complete separable metric space. The corresponding open subsets generate a σ-algebra on K {displaystyle {mathcal {K}}} , the Borel sigma algebra B ( K ) {displaystyle {mathcal {B}}({mathcal {K}})} of K {displaystyle {mathcal {K}}} . A random compact set is а measurable function K {displaystyle K} from а probability space ( Ω , F , P ) {displaystyle (Omega ,{mathcal {F}},mathbb {P} )} into ( K , B ( K ) ) {displaystyle ({mathcal {K}},{mathcal {B}}({mathcal {K}}))} . Put another way, a random compact set is a measurable function K : Ω → 2 M {displaystyle Kcolon Omega o 2^{M}} such that K ( ω ) {displaystyle K(omega )} is almost surely compact and is a measurable function for every x ∈ M {displaystyle xin M} . 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 P ( X ⊂ K ) . {displaystyle mathbb {P} (Xsubset K).} ) For K = { x } {displaystyle K={x}} , the probability P ( x ∈ X ) {displaystyle mathbb {P} (xin X)} is obtained, which satisfies