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Generalized measure theory. (English) Zbl 1184.28002
IFSR International Series on Systems Science and Engineering 25. New York, NY: Springer (ISBN 978-0-387-76851-9/hbk). xv, 381 p. (2009).
In classical measure theory the crucial requirement for a measure is its \(\sigma\)-additivity. In particular, this can be said for the probability measures in A. N. Kolmogorov’s axiomatic setting of probability theory [Grundbegriffe der Wahrscheinlichkeitsrechnung (1933; Zbl 0007.21601)], which has beem widely accepted since that time. Starting with about the second half of the last century, this basic additivity requirement became a subject of controversy when some mathematicians felt that additivity was too restrictive in certain applications and therefore should be replaced by weaker requirements. A first challenge along these lines was produced by G. Choquet [Ann. Inst. Fourier 5, 131–295 (1953–1954; Zbl 0064.35101)] in his theory of capacities, which grew out of potential theory but was applicable far beyond potential theory, e.g. in the formalization of imprecise probabilites by G. J. Klir [cf. Int. J. Gen. Syst. 35, No. 1, 29–44 (2006; Zbl 1119.68196)]. M. Sugeno [Fuzzy Sets Appl. cogn. Decis. Processes, U.S.-Japan Semin., Berkeley 1974, 151–170 (1975; Zbl 0316.60005)] introduced his fuzzy measures, where he replaced the additivity requirement of classical measures by the weaker one of increasing monotonicity and continuity. Later on, the latter was found too restrictive and was replaced by the weaker semicontinuity. In various theories of imprecise probabilities for example, lower and upper probabilites are considered which are only semicontinuous. As R. Viertl [Contributions to stochastics, Hon. 75th Birthday W. Eberl sen., 236–247 (1987; Zbl 0634.62026)] noticed, additivity is inadequate to characterize most measurements under real, physical conditions, i.e., when measurement errors occur. The same applies for measurements connected with subjective judgements or non-repeatable experiments. It can be said that the book is written with a view on these applications, but it should be mentioned that prior to Choquet, non-additive measures appear in the so-called ‘submeasure problem’ from a paper of D. Maharam [Ann. Math. (2) 48, 154–167 (1947; Zbl 0029.20401)]. It seems that the authors were not interested in this circle of ideas, which, by the way, is excellently covered in the monograph of D. H. Fremlin [Measure theory. Vol. 3. Measure algebras. Colchester: Torres Fremlin (2004; Zbl 1165.28002)].
For covering the above applications, the authors were led to introduce the notion of a ‘general measure’ as a positive extended real-valued function defined on a class of subsets over a fixed basic set and characterized by the only requirement that it vanishes for the empty set. In the usual way, signed general measures are defined also. Of course, general measures form an extremely broad class and additional requirements have to be imposed for getting classes of general measures of any significance. So the term ‘generalized measure theory’ in the sense of this monograph is meant to study certain classes of general measures delimited by the two extremes of classical measure theory and of the class of general measures, with focus on non-additive measures which are not fully general. The first interest is in the broad class of monotone general measures and their large subclasses of sub- and superadditive measures. The latter classes are refined by Choquet capacities of various order, including infinite order. Choquet capacities form the core of general measure theory, by using them as benchmarks against which other classes of measures can be compared. Next, classes of general measures are studied, which originate from generalisation of classical measures, beginning with \(\lambda\)-measures. The latter class is enlarged to the class of the so-called quasi-measures. Each member in the latter class is still connected to additive measures via a particular type of reversible transformation. Choquet capacities of order infinity, the strongest Choquet capacities, and their dual measures, referred to as alternating capacities of order infinity, form a basis for a well-known theory of uncertainty, which is known in the literature as the Dempster-Shafer or evidence theory. The theory of graded possibilities is treated as an example based on non-additive measures, while belief and plausibilty measures are examples of lower and upper probabilities.
There is a chapter on extensions and another one on structural characteristics of general measures. For measurable functions on monotone measure spaces, corresponding to the generality of the measures, the Lebesgue integral, the Sugeno integral, and as a common generalization, the Pan-integral are discussed. Also, for non-negative functions, the Choquet integral is developed on the basis of monotone measure spaces and the upper and lower integrals on the basis of general measure spaces. The latter integrals are used for a definition of uncertainty, carried by a monotone measure. Finally, the relations between the different integrals are closely checked. Because of the importance of the Fubini theorem in classical measure theory, the reader would have appreciated a remark concerning the possibility or failure of a product theory for generalized measures and their corresponding integrals. From the above mentioned monograph of Fremlin it is known that a product theory for submeasures fails dramatically. The last chapter is devoted to applications of the generalized measure theory, amongst which are applications to generalized information theory, imprecise probability theory, possibility theory, information fusion, multiregression, decision making, economics, image processing and computer vision.
In 1992 [Z. Wang and G.J. Klir, Fuzzy measure theory, New York: Plenum Press (1992; Zbl 0812.28010)] the same authors published a book on ‘Fuzzy measure theory’, in which the term ‘fuzzy measure’ was used for set functions obtained by replacing the additivity requirement of classical measures by monotonicity and continuity. Due to recent developments in the literature, the authors decided to abandon this terminology, and also, since there was no fuzziness in these measures. Instead they define in the present monograph a fuzzification for arbitrary general measures by restricting them to fuzzy sets. The monograph contains many original results of the authors and provides an excellent introduction to the emerging field of generalized measure theory. The presentation of the large material is given with didactical skill, numerous examples serve for motivation and illustration of results. Each chapter is ended by a great number of exercises and by numerous bibliographical and historical notes. The monograph has an appendix with a glossary of key concepts and another one with a glossary of symbols which make the material easily accessible, and it is ended by an extensive bibliography. As it is known from experience with classical measure theory, this book should be easy to read by everyone with a solid background in mathematical analysis, and on this basis in particular for students in mathematics and engineering at the graduate level. Due to the completeness of available material on the topic, this book can serve as a starting point for research in this area.

MSC:
28-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration
28E99 Miscellaneous topics in measure theory
28E10 Fuzzy measure theory
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