Theory of Random Sets
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From the reviews:
"Together with the foundations of the modern probability theory Kolmogorov introduced the concept of a random set. . The book is written in a theorem-proof style, where the proofs are quite detailed and clearly presented. . to enable easier reading, the author provides visual illustrations where necessary. . the author provides a vast bibliography on the subject, which is completely searchable on author's website. The book should be read and even more studied by any researcher or a student considering research in this field." (Ita Cirovic Donev, MathDL, January, 2006)
"This book is part of the Springer-Verlag series 'Probability and its applications'. . The most remarkable aspect of the book is the reader-friendly structure and the style in which it has been written. There are masses of examples either worked out in the book or left for the reader. A number of facts are equipped with graphical illustrations. This book has a good index and index of notations, and a very detailed bibliography . . It will be an essential part of every mathematical library." (V. K. Oganyan, Mathematical Reviews, Issue 2006 b)
"The book under review develops . an approach in a self-contained and systematic manner. Full proofs are given, and many steps are illustrated by graphs and drawings. The interdisciplinary nature of the theory of random sets within mathematics is well shown. . The book will be an invaluable reference for probabilists, mathematicians, statisticians and electronic and electrical engineers in the fields of image analysis. . The book is highly recommended both for personal use and for libraries." (Janos Galambos, Zentralblatt MATH, Vol. 1109 (11), 2007)
"Random sets play an important role in many applications of mathematics . . This book is an important contribution to the mathematical theory and will surely serve as a valuable textbook for students as well as researchers. It presentsa self-contained survey of all the significant results . . A number of open problems are presented and each chapter concludes with a list of bibliographical notes." (EMS Newsletter, September, 2007)
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Ilya Molchanov is Professor of Probability Theory in the Department of Mathematical Statistics and Actuarial Science at the University of Berne, Switzerland.
Content
1 The selection expectation
The space F of closed sets (and also the space K of compact sets) is non-linear, so that conventional concepts of expectations in linear spaces are not directly applicable for random closed (or compact) sets. Sets have different features (that often are dif.cult to express numerically) and particular definitions of expectations highlight various features important in the chosen context.
To explain that an expectation of a random closed (or compact) set is not straightforward to de.ne, consider a random closed set X which equals [0, 1] with probability 1/2 and otherwise is {0, 1}. For another example, let X be a triangle with probability 1/2 and a disk otherwise. A "reasonable" expectation in either example is not easy to de.ne. Strictly speaking, the de.nition of the expectation depends on what the objective is, which features of random sets are important to average and which are possible to neglect.
This section deals with the selection expectation (also called the Aumann expectation), which is the best investigated concept of expectation for random sets. Since many results can be naturally formulated for random closed sets in Banach spaces, we assume that E is a separable Banach space unless stated otherwise. Special features inherent to expectations of random closed sets in Rd will be highlighted throughout. To avoid unnecessary complications, it is always assumed that all random closed sets are almost surely non-empty.
1.1 Integrable selections
The key idea in the de.nition of the selection expectation is to represent a random closed set as a family of its integrable selections. The concept of a selection of a random closed set was introduced in De.nition 1.2.2.While properties of selections discussed in Section 1.2.1 can be formulated without assuming a linear structure on E, now we discuss further features of random selections with the key issue being their integrability.
