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Heinz König’s recent and most influential works in one single volume
For the first time ever: Entirely uniform treatment of abstract and topological measure theory
New “envelopes” for the initial set function (to replace the traditional Carathéodory outer measures), which leads to much simpler and more explicit treatment
The incorporation of non-sequential and of inner regular versions leads to much more comprehensive results
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.
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.