##
**Fuzzy measure theory.**
*(English)*
Zbl 0812.28010

New York: Plenum Press. ix, 354 p. $ 69.50/hbk (1992).

A fuzzy measure is a set-function vanishing at \(\emptyset\), monotone and continuous. The notion is a very natural generalization of a positive, \(\sigma\)-additive measure. Of course, although the term fuzzy measure has been fixed in the literature, its use in this relation is problematic, actually it is not a fuzzy notion. On the other hand, among many examples of monotone and continuous functions there are many which are inspired and motivated by the problems of the fuzzy sets theory and its applications. In this direction more adequate to fuzzy sets is the matter presented in the Appendix E (New directions in fuzzy measure theory), where some functions on families of fuzzy sets are considered.

After two introductory chapters, fuzzy measures are studied in Chapters 3-5. The main attention is devoted to some important special cases (\(\lambda\)-fuzzy measures, quasimeasures, belief and plausibility measures, possibility and necessity measures) together with extension theories for some of them. Also null-additive functions are exposed (i.e., such functions that \(m(F)=0\), \(E\cap F= \emptyset\) implies \(m(E\cup F)= m(E)\)). On the other hand the results from the theory of so- called non-additive measures (where the usual operations \(+\) in \(R\) is replaced by a more general operation) are mentioned only marginally. (Of course, the terminology here is also problematic, because according to it every additive measure is an example of a non-additive measure.)

In the additive case one has \(m(E\cap A)=0\) if and only if \(m(A\setminus E)= m(A)\). Of course, for not necessarily additive functions it needs not to be true. Therefore, in the theory of measurable functions (Chapter 6) there is reasonable to consider, e.g., almost everywhere convergence on \(A\) (the set of those \(x\) from \(A\), for which the sequence does not converge, has measure zero) and so-called pseudo almost everywhere convergence on \(A\) (the set of those \(x\) from \(A\), for which the given sequence converges, has the same measure as \(A\)). Therefore, the theory of measurable functions is reexamined.

The culminating point of the book is the theory of fuzzy integrals (Chapters 7 and 8). Here the theory of Sugeno integral is developed (roughly speaking, it is based on infimum and supremum instead of usually used product and sum) and then the Pan integral is exposed (the usual sum and product of non-negative real numbers are substituted by some axiomatically characterized operations).

Although the purpose of the book is to present the mathematical foundations of fuzzy measure theory, the authors discuss applicability of the theory (Chapter 9 as well as Appendix F). Let us mention some areas of utility of fuzzy measure theory: systems theory, computer sciences, information sciences, cognitive sciences, artificial intelligence, quantitative management, mathematical social sciences, and some areas of engineering. Appendix \(F\) consists of three recent papers published by M. Strat, H. Prade and C. Testemale, and H. Tahari and J. M. Keller, respectively. Recall that also Appendix E consists of three recent papers published by Z. Wang, Q. Zhang, and J. Yen. In every new theory, some of its ideas can be found in some previous papers. In this relation we should like only to mention some papers by J. Šipoš [Math. Slovaca 29, 141–155 (1979; Zbl 0423.28003); ibid. 29, 257–270 (1979; Zbl 0442.28004); ibid. 29, 333–345 (1979; Zbl 0452.28007)].

The book is self-contained (some necessary knowledge is presented in Appendices A and B) and can serve not only for specialists, but also for graduate courses.

After two introductory chapters, fuzzy measures are studied in Chapters 3-5. The main attention is devoted to some important special cases (\(\lambda\)-fuzzy measures, quasimeasures, belief and plausibility measures, possibility and necessity measures) together with extension theories for some of them. Also null-additive functions are exposed (i.e., such functions that \(m(F)=0\), \(E\cap F= \emptyset\) implies \(m(E\cup F)= m(E)\)). On the other hand the results from the theory of so- called non-additive measures (where the usual operations \(+\) in \(R\) is replaced by a more general operation) are mentioned only marginally. (Of course, the terminology here is also problematic, because according to it every additive measure is an example of a non-additive measure.)

In the additive case one has \(m(E\cap A)=0\) if and only if \(m(A\setminus E)= m(A)\). Of course, for not necessarily additive functions it needs not to be true. Therefore, in the theory of measurable functions (Chapter 6) there is reasonable to consider, e.g., almost everywhere convergence on \(A\) (the set of those \(x\) from \(A\), for which the sequence does not converge, has measure zero) and so-called pseudo almost everywhere convergence on \(A\) (the set of those \(x\) from \(A\), for which the given sequence converges, has the same measure as \(A\)). Therefore, the theory of measurable functions is reexamined.

The culminating point of the book is the theory of fuzzy integrals (Chapters 7 and 8). Here the theory of Sugeno integral is developed (roughly speaking, it is based on infimum and supremum instead of usually used product and sum) and then the Pan integral is exposed (the usual sum and product of non-negative real numbers are substituted by some axiomatically characterized operations).

Although the purpose of the book is to present the mathematical foundations of fuzzy measure theory, the authors discuss applicability of the theory (Chapter 9 as well as Appendix F). Let us mention some areas of utility of fuzzy measure theory: systems theory, computer sciences, information sciences, cognitive sciences, artificial intelligence, quantitative management, mathematical social sciences, and some areas of engineering. Appendix \(F\) consists of three recent papers published by M. Strat, H. Prade and C. Testemale, and H. Tahari and J. M. Keller, respectively. Recall that also Appendix E consists of three recent papers published by Z. Wang, Q. Zhang, and J. Yen. In every new theory, some of its ideas can be found in some previous papers. In this relation we should like only to mention some papers by J. Šipoš [Math. Slovaca 29, 141–155 (1979; Zbl 0423.28003); ibid. 29, 257–270 (1979; Zbl 0442.28004); ibid. 29, 333–345 (1979; Zbl 0452.28007)].

The book is self-contained (some necessary knowledge is presented in Appendices A and B) and can serve not only for specialists, but also for graduate courses.

Reviewer: Beloslav Riečan (Bratislava)

### MSC:

28E10 | Fuzzy measure theory |

28-02 | Research exposition (monographs, survey articles) pertaining to measure and integration |

03E72 | Theory of fuzzy sets, etc. |

28-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration |