Measure and Integration: Publications 1997-2011

ISBN-13: 9783034803816 / Angielski / Twarda / 2012 / 512 str.

Heinz K. Nig

Opis

This collection of Heinz Konig's publications connects to his book of 1997 "Measure and Integration" and presents significant developments in the subject from then up to the present day. The result is a consistent new version of measure theory, including selected applications. The basic step is the introduction of the inner - (bullet) and outer - (bullet) premeasures and their extension to unique maximal measures. New "envelopes" for the initial set function (to replace the traditional Caratheodory outer measures) have been created, which lead to much simpler and more explicit treatment. In view of these new concepts, the main results are unmatched in scope and plainness, as well as in explicitness. Important examples are the formation of products, a unified Daniell-Stone-Riesz representation theorem, and projective limits.

Recenzje

From the reviews:

"The author took up several of the crucial topics of measure theory and developed them according to his 'new foundations of measure theory'. The author is offering twenty of his papers in this tome ... . This collection of papers along with MI are a must on the bookshelves of any measure theorist. ... On the whole the present volume will serve as a useful resource ... in MI." (K. P. S. Bhaskara Rao, Mathematical Reviews, April, 2013)

"The main body of this impressive volume is a collection of twenty-five papers whose sole author is Heinz König, an esteemed analyst. ... the volume can be recommended to all those interested in the foundations of measure theory and stochastic processes." (Zbigniew Lipecki, zbMATH, Vol. 1267, 2013)

Spis treści

Image measures and the so-called image measure catastrophe.- The product theory for inner premeasures.- Measure and Integration: Mutual generation of outer and inner premeasures.- Measure and Integration: Integral representations of isotone functionals.- Measure and Integration: Comparison of old and new procedures.- What are signed contents and measures?- Upper envelopes of inner premeasures.- On the inner Daniell-Stone and Riesz representation theorems.- Sublinear functionals and conical measures.- Measure and Integration: An attempt at unified systematization.- New facts around the Choquet integral.- The (sub/super)additivity assertion of Choquet.- Projective limits via inner premeasures and the trueWiener measure.- Stochastic processes in terms of inner premeasures.- New versions of the Radon-Nikodým theorem.- The Lebesgue decomposition theorem for arbitrary contents.- The new maximal measures for stochastic processes.- Stochastic processes on the basis of new measure theory.- New versions of the Daniell-Stone-Riesz representation theorem.- Measure and Integral: New foundations after one hundred years.- Fubini-Tonelli theorems on the basis of inner and outer premeasures.- Measure and Integration: Characterization of the new maximal contents and measures.- Notes on the projective limit theorem of Kolmogorov.- Measure and Integration: The basic extension theorems.- Measure Theory: Transplantation theorems for inner premeasures.​  

O autorze

Heinz König is a distinguished analyst, who has given lasting contributions to functional analysis, distribution theory, convex analysis, mathematical economics and many other fields of mathematics. Typical of his work is the analysis or creation of basic new concepts from most original viewpoints. Heinz König gave a large number of original, short and elegant proofs of fundamental results in mathematics. Most remarkable is the new theory of measure and integration he developed in the last two decades.

Born in Stettin (Szczecin/Poland), Heinz König has been a professor at the University of Saarland (Germany) since 1965 and a visiting professor at many prestigious universities around the world.

Blurb

This volume presents a collection of twenty-five of Heinz König’s recent and most influential works. Connecting to his book of 1997 “Measure and Integration”, the author has developed a consistent new version of measure theory over the past years. For the first time, his publications are collected here in one single volume.

Key features include:

- A first-time, original and entirely uniform treatment of abstract and topological measure theory

- The introduction of the inner • and outer • premeasures and their extension to unique maximal measures

- A simplification of the procedure formerly described in Chapter II of the author’s previous book

- The creation of new “envelopes” for the initial set function (to replace the traditional Carathéodory outer measures), which lead to much simpler and more explicit treatment

- The formation of products, a unified Daniell-Stone-Riesz representation theorem, and projective limits, which allows to obtain the Kolmogorov type projective limit theorem for even huge domains far beyond the countably determined ones

- The incorporation of non-sequential and of inner regular versions, which leads to much more comprehensive results

- Significant applications to stochastic processes.

“Measure and Integration: Publications 1997–2011” will appeal to both researchers and advanced graduate students in the fields of measure and integration and probabilistic measure theory.