Kurzweil-Henstock Integral in Riesz spaces - Bentham Science

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PREFACE
Kurzweil and Henstock’s idea to construct a new type of integral turned out to
be both surprisingly successful and extremely useful, not only from the didactic but
also from the scientific point of view. It has very promising applications, for
example in differential equations and surface integrals. Riesz spaces, on the other
hand, offer a very important tool in modern mathematics and have many practical
applications, for example in economics. Recall that the eminent mathematician and
Nobel Prize L. V. Kantorovich was the founder of the theory of Riesz spaces. This
monograph is concerned with both the theory of the Kurzweil-Henstock integral
and the basic facts on Riesz spaces.
Another important application of this theory was discovered recently. In 2002 D.
Kahneman received the Nobel Prize in Economics. While he was looking for a
theoretical basis on his economical theory, he found an appropriate mathematical
model: the so-called ipo integral, one of the topics we present in this
monograph.
It is well-known that integration theory with values in ordered spaces cannot be
reduced to the analogous theory for locally convex spaces. This fact justifies the
main goal of this book: to investigate and develop a measure and integration theory
of the Kurzweil-Henstock type for functions with values in ordered spaces.
The first chapter offers to the reader a self-contained treatment of the realvalued
theory of the Kurzweil-Henstock integral. Namely, the extremely simple
definition enables us to use a very concise and effective theory.
The following chapter on Riesz spaces should also be accessible to a large class
of readers. We not only mention slightly more general structures such as lattice
ordered groups, but also some basic facts about MV-algebras: these are important
for multivalued logic. The general theory of the Kurzweil-Henstock integral in
Riesz spaces is presented in the third chapter. As discovered by J. D. M. Wright
and D. Fremlin, there exists a sufficient and necessary condition for the possibility
to extend Riesz space-valued Daniell integrals from the set of simple functions to
the set of integrable functions, or a Riesz space-valued measure from an algebra to
the generated -algebra. This condition, which is called weak -distributivity,
holds in any probability MV-algebra.
Chapters 4 - 6 contain new and, in our opinion, important results on
convergence theorems and multiple integrals. These chapters also contain a
systematic exposition of the Kurzweil-Henstock integral theory for functions
defined on abstract topological spaces. Recall that most papers on the Kurzweil-
Henstock integral use as a domain only Euclidean spaces.
Some more special topics are treated in chapters 7 - 10 and 13, namely improper
integrals, SL-integrals, the Pettis and Choquet approach, and integration in metric
semigroups. The Choquet integral (or its ipo symmetric variant) is of particular
importance in non-additive measures.