Non-Additive Measure and Integral
Dieter Denneberg
Kluwer Academic Publishers
Non-Additive Measure and Integral is the first  systematic approach to the subject. Much of the additive theory  (convergence theorems, Lebesgue spaces, representation theorems) is  generalized, at least for submodular measures which are characterized  by having a subadditive integral. The theory is of interest for  applications to economic decision theory (decisions under risk and  uncertainty), to statistics (including belief functions, fuzzy  measures) to cooperative game theory, artificial intelligence,  insurance, etc.   Non-Additive Measure and Integral collects the results of  scattered and often isolated approaches to non-additive measures and  their integrals which originate in pure mathematics, potential theory,  statistics, game theory, economic decision theory and other fields of  application. It unifies, simplifies and generalizes known results and  supplements the theory with new results, thus providing a sound basis  for applications and further research in this growing field of  increasing interest. It also contains fundamental results of  sigma-additive and finitely additive measure and integration theory  and sheds new light on additive theory.  Non-Additive Measure  and Integral employs distribution functions and quantile  functions as basis tools, thus remaining close to the familiar  language of probability theory.   In addition to serving as an important reference, the book can be used  as a mathematics textbook for graduate courses or seminars, containing  many exercises to support or supplement the text.