Random Compact Set
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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) be a complete separable metric space. Let mathcal{K} denote the set of all compact subsets of M. The Hausdorff metric h on mathcal{K} is defined by h(K_{1}, K_{2}) := max left{ sup_{a in K_{1}} inf_{b in K_{2}} d(a, b), sup_{b in K_{2}} inf_{a in K_{1}} d(a, b) right}.(mathcal{K}, h)...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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) be a complete separable metric space. Let mathcal{K} denote the set of all compact subsets of M. The Hausdorff metric h on mathcal{K} is defined by h(K_{1}, K_{2}) := max left{ sup_{a in K_{1}} inf_{b in K_{2}} d(a, b), sup_{b in K_{2}} inf_{a in K_{1}} d(a, b) right}.(mathcal{K}, h) is also complete separable metric space. The corresponding open subsets generate a -algebra on mathcal{K}, the Borel sigma algebra mathcal{B}(mathcal{K}) of mathcal{K}.A random compact set is measurable function K from probability space (Omega, mathcal{F}, mathbb{P}) into (mathcal{K}, mathcal{B} (mathcal{K}) ).Put another way, a random compact set is a measurable function K : Omega to 2^{Omega} such that K( ) is almost surely compact and omega mapsto inf_{b in K(omega)} d(x, b) is a measurable function for every x in M.
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