Format:
Online-Ressource (XI, 508 p. 1 illus. in color, digital)
ISBN:
9783034803823
,
128080291X
,
9781280802911
Series Statement:
SpringerLink
Content:
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. .
Content:
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.
Note:
Description based upon print version of record
,
Measure and Integration; Preface; Contents; Introduction; Image Measures and the So-Called Image Measure Catastrophe; 1. The Natural Domain of a Measure; 2. ImageMeasures and Their Restrictions; 3. The ImageMeasure Theorem; Acknowledgement; References; THE PRODUCT THEORY FOR INNER PREMEASURES; 1.Finite Products; 2. Preparations for Infinite Products; 3. Infinite Products; 4. The General Situation; References; MEASURE AND INTEGRATION: MUTUAL GENERATION OF OUTER AND INNER PREMEASURES; I. TAME PAIRS OF PREMEASURES; 1. RECOLLECTON OF THE MAIN EXTENSION THEOREMS; 2. FROM OUTER TO INNER PREMEASURES
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3. FROM INNER TO OUTER PREMEASURES4. THE MAIN THEOREM; 5. THE FUNDAMENTAL THEOREM OF SAPOUNAKIS-SION; 6. DEFINITION AND MAIN THEOREM; 7. PROOF OF THE MAIN THEOREM; 8. INTERMEDIATE EXTENSIONS OF INNER PREMEASURES; 9. THE ESSENTIAL FORMATION FOR FUNCTIONALS; ACKNOWLEDGEMENT; REFERENCES; MEASURE AND INTEGRATION: INTEGRAL REPRESENTATIONS OF ISOTONE FUNCTIONALS; 1. NEW VERSION OF A RESULT OF CHOQUET; 2. THE REPRESENTATION THEOREM OF GRECO; 3. PREPARATIONS ON FUNCTION CLASSES AND FUNCTIONALS; 4. FUNCTIONALS WITH CONTINUOUS SOURCES; 5. THE MAIN REPRESENTATION THEOREMS; 6. THE INTEGRABILITY THEOREMS
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ACKNOWLEDGEMENTREFERENCES; Measure and integration: comparison of old and new procedures; 1. The level of set functions.; 1.1. Outer main theorem.; 1.2. Special case.; 1.3. Inner main theorem.; 1.4. Special case.; 1.5. Example.; 1.6. Remark.; 1.7. Examples.; 1.8. Theorem.; 2. The horizontal integral.; 2.1. Theorem.; 3. The level of functionals.; 3.1. Proposition.; 3.2. Outer main theorem.; 3.3. Addendum.; 3.4. Special case.; 3.5. Proposition.; 3.6. Inner main theorem.; 3.7. Addendum.; 3.8. Special case.; 4. The so-called essential formation.; 4.1. Properties.; 4.2. Addendum.; 4.3. Theorem.
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4.4. Example.4.5. Properties.; 4.6. Proposition.; 5. Comparison with the traditional theories on the level of functionals.; 5.1. Properties.; 5.2. Proposition.; 5.3. Proposition.; References; What Are signed Contents and Measures?; 1. The new difference formation; 2. Singular pairs of contents and measures; 3. Examples and further properties of singular pairs; 4. Signed contents and measures; 5. Signed contents and measures as set functions; 6. From singular to lattice singular via extension; Acknowledgements; References; UPPER ENVELOPES OF INNER PREMEASURES; 1. The Hahn-Banach type theorems.
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2. Some preliminaries.3. The basic results.; 4. Extension of the basic consequence.; 5. Some further comments and counterexamples.; BIBLIOGRAPHY; ON THE INNER DANIELL-STONE AND RIESZ REPRESENTATION THEOREMS; 1. INNER PREINTEGRALS; 2. THE INNER DANIELL-STONE THEOREM; 3. THE RIESZ REPRESENTATION THEOREM; 4. COMPARISON WITH ANOTHER APPROACH; ACKNOWLEDGEMENT; REFERENCES; Sublinear functionals and conical measures; 1. Some basic properties of sublinear functionals.; 2. The main results.; References; Measure and Integration: an Attempt at Unified Systematization; 1. Introduction
,
2. The Fundamentals for Set Functions
Additional Edition:
ISBN 9783034803816
Additional Edition:
Buchausg. u.d.T. König, Heinz, 1929 - 2024 Measure and integration Basel : Birkhäuser, 2012 ISBN 9783034803823
Additional Edition:
ISBN 9783034803816
Additional Edition:
ISBN 3034803818
Additional Edition:
Buchausg.: König, Heinz, 1929 - 2024 Measure and integration Basel : Birkhäuser, 2012 ISBN 9783034803823
Additional Edition:
ISBN 9783034803816
Additional Edition:
ISBN 3034803818
Language:
English
Subjects:
Mathematics
Keywords:
Maßtheorie
;
Integrationstheorie
;
Maßtheorie
;
Integrationstheorie
DOI:
10.1007/978-3-0348-0382-3
URL:
Volltext
(lizenzpflichtig)
URL:
Volltext
(lizenzpflichtig)
Author information:
König, Heinz 1929-2024
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