Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Theory of fuzzy integrals and its applications

Aggregation of information is of primary importance in the construction of knowledge based systems in various domains, ranging from medicine, economics, and engineering to decision-making processes, artificial intelligence, robotics, and machine learning. This book gives a broad introduction into the topic of aggregation functions, and provides a concise account of the properties and the main classes of such functions, including classical means, medians, ordered weighted averaging functions, Choquet and Sugeno integrals, triangular norms, conorms and copulas, uninorms, nullnorms, and symmetric sums. It also presents some state-of-the-art techniques, many graphical illustrations and new interpolatory aggregation functions. A particular attention is paid to identification and construction of aggregation functions from application specific requirements and empirical data. This book provides scientists, IT specialists and system architects with a self-contained easy-to-use guide, as well as examples of computer code and a software package. It will facilitate construction of decision support, expert, recommender, control and many other intelligent systems.

Aggregation Functions: A Guide for Practitioners

Probability and Measure

Convex Analysis: (pms-28)

Preface. 1. Introduction. 2. Null-additive set functions. 3. Autocontinuity of null-additive set functions. 4. Atoms of null-additive set functions. 5. Choquet and Sugeno integral. 6. Lebesgue decomposition and regularity of null-additive set functions. 7. The range of null-additive set function. 8. k-Triangular set functions. 9. Uniform space valued set functions. 10. Integrals based on decomposable measures. References. Index.

Null-Additive Set Functions

There is given a short overview of the monograph rdquoAggregation Functionsrdquo (M. Grabisch, J. L. Marichal, R. Mesiar, E. Pap), Cambridge University Press (in press).

”Aggregation Functions”, Cambridge University Press

Introduction 1 Triangular norms 2 Probabilistic metric spaces 3 Probabilistic GBPIiGBP-contraction principles for single-valued mappings 4 Probabilistic GBPIiGBP-contraction principles for multi-valued mappings 5 Hicks' contraction principle 6 Fixed point theorems in topological vector spaces and applications to random normed spaces Bibliography Index

/pdf/fixed-point-theory-in-probabilistic-metric-spaces-5a37iax9k5.pdf

Fixed Point Theory in Probabilistic Metric Spaces

The Choquet and the Sugeno integral provide a useful tool in many problems in engineering and social choice where the aggregation of data is required. However, their applicability is restricted because of the special operations used in the construction of these integrals. Therefore, we provide a concept of integrals generalizing both the Choquet and the Sugeno case. For functions with values in the nonnegative real numbers, universal integrals are introduced and investigated, which can be defined on arbitrary measurable spaces and for arbitrary monotone measures. For a fixed pseudo-multiplication on the nonnegative real numbers, the smallest and the greatest universal integrals are given. Finally, another construction method for obtaining universal integrals is introduced, and the restriction to the unit interval, i.e., to fuzzy integrals, is considered.

A Universal Integral as Common Frame for Choquet and Sugeno Integral

http://www.flll.jku.at/sites/default/files/44_2004_KMesPap_TNormsPosPaperI_FSSPrintedVersion.pdf

Endre Pap

Papers

Null-Additive Set Functions

”Aggregation Functions”, Cambridge University Press

Fixed Point Theory in Probabilistic Metric Spaces

A Universal Integral as Common Frame for Choquet and Sugeno Integral

Triangular norms. Position paper I: basic analytical and algebraic properties