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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This book presents a simple and novel theory of integration, both real and vectorial, particularly suitable for the study of PDEs. This theory allows for integration with values in a Neumann space E, i.e. in which all Cauchy sequences converge, encompassing Neumann and Fréchet spaces, as well as "weak" spaces and distribution spaces.
We integrate "integrable measures", which are equivalent to "classes of integrable functions which are a.e. equals" when E is a Fréchet space. More precisely, we associate the measure f with a class f, where f(u) is the integral of fu for any test function u. The classic space Lp(Ω;E) is the set of f, and ours is the set of f; these two spaces are isomorphic.
Integration studies, in detail, for any Neumann space E, the properties of the integral and of Lp(Ω;E): regularization, image by a linear or multilinear application, change of variable, separation of multiple variables, compacts and duals. When E is a Fréchet space, we study the equivalence of the two definitions and the properties related to dominated convergence.
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Seitenzahl: 683
Veröffentlichungsjahr: 2026
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Table of Contents
Cover
Table of Contents
Dedication Page
Title Page
Copyright Page
Introduction
List of Notations and Figures
PART 1: Integration
Chapter 1: Integration of Continuous Functions
1.1. Neumann spaces
1.2. Continuous mappings
1.3. Cauchy integral of a uniformly continuous function
1.4. Some properties of the integral
1.5. Dependence of the integral on the domain of integration
1.6. Continuity of the integral
1.7. Successive integration
Chapter 2: Measurable Sets
2.1. Why introduce measurable sets?
2.2. Some properties of the measure of an open set
2.3. Definition of measurable sets and their measure
2.4. First properties of the measure
2.5. Additivity of the measure
2.6. Countable union and countable intersection of measurable sets
2.7. Continuity of the measure
2.8. Translation invariance and the product measure
2.9. Negligible sets
Chapter 3: Measures
3.1. Space of measures ℳ(Ω;
E
)
3.2. Equicontinuity of bounded subsets of ℳ(Ω;
E
)
3.3. Sequential completeness of ℳ(Ω;
E
)
3.4. Continuity of the ⟨ , ⟩ mapping
3.5. Identification of continuous functions with measures
3.6. Regularization of measures
3.7. Regularization of functions
Chapter 4: Integrable Measures
4.1. Definition of integrable measures
4.2. Space of integrable measures
L
1
(Ω;
E
)
4.3. Some properties of
L
1
(Ω;
E
)
4.4. Regularization in
L
1
(Ω;
E
)
4.5. Sequential completeness of
L
1
(Ω;
E
)
Chapter 5: Integration of Integrable Measures
5.1. Integral of an integrable measure
5.2. Linearity and continuity of the integral
5.3. Positive measures, real-valued integrals
5.4. Examples of value spaces
5.5. The case where
E
is not a Neumann space
Chapter 6: Properties of the Integral
6.1. Additivity with respect to the domain of integration
6.2. Continuity with respect to the domain of integration
6.3. Contribution of negligible sets
6.4. Image of a measure under a linear mapping
6.5. Image under a linear mapping
6.6. Restriction and support
6.7. Differentiation under the integral sign
Chapter 7: Change of Variables
7.1. Image of a measurable set
7.2. Determinant of
d
vectors
7.3. Measure of a parallelepiped
7.4. Change of variable in the Cauchy integral
7.5. Change of variable in a measure
7.6. Change of variable in an integrable measure
7.7. Product of a measure with a continuous function
7.8. Change of variable in an integral
7.9. Affine change of variables
Chapter 8: Multivariable Integration
8.1. Permutation of variables in a measure of measures
8.2. Integration of an integrable measure of measures
8.3. Separation of variables in an integral of a continuous function
8.4. Separation of variables of a measure
8.5. Separation of variables
8.6. Fubini’s theorem
PART 2: Lebesgue Spaces
Chapter 9: Inequalities
9.1. Elementary inequalities
9.2. Inequalities for continuous functions
9.3. Young’s convolution inequality
9.4. Properties of regularizations of continuous functions
Chapter 10:
L
p
(Ω;
E
) Spaces
10.1. Definition of
L
p
(Ω;
E
)
10.2. Separability of
L
p
(Ω;
E
)
10.3. Some properties of
L
p
(Ω;
E
)
10.4. Properties of
L
∞
(Ω;
E
)
10.5. Approximation via regularizations and density
10.6. Completeness of
L
p
(Ω;
E
)
10.7. Remarks on methods of construction
Chapter 11: Dependence on
p
and Ω, Local Spaces
11.1. Dependence on
p
11.2. (Ω;
E
) spaces
11.3. Localization–extension
11.4. Dependence on Ω
11.5. Infinite gluing on Ω and continuity in
p
Chapter 12: Image Under a Linear Mapping
12.1. Image under a linear mapping and dependence on
E
12.2. Image under a multilinear mapping
12.3. Images in Banach and Hilbert spaces
12.4. Images in local spaces
Chapter 13: Various Operations
13.1. Image under a semi-norm of
E
13.2. Powers
13.3. Extensions
13.4. Step measures
13.5. Density and separability
13.6. Limit of a bounded sequence in
L
∞
(Ω;
E
)
Chapter 14: Change of Variable, Weightings
14.1. Change of variable
14.2. Regrouping and separation of variables
14.3. Permutation of variables
14.4. Weightings of measures
14.5. Weightings
Chapter 15: Compact Sets
15.1. Preliminaries
15.2. Compact subsets of
L
p
(Ω;
E
)
15.3. Special cases of compactness
15.4. Compact subsets of (Ω;
E
)
15.5. Compactness in intermediate spaces
Chapter 16: Duals
16.1. Uniform convexity of
L
p
(Ω;
E
)
16.2. Canonical injection from
L
p′
(Ω;
E
′) into the dual of
L
p
(Ω;
E
)
16.3. Riesz representation theorems
16.4. Riesz–Fréchet theorem
16.5. Weak topology of
L
p
(Ω;
E
)
16.6. ∗Weak topology of
L
∞
(Ω;
E
)
PART 3: Integrable Functions
Chapter 17: Measurable Functions
17.1. Measurable functions
17.2. Integral of a positive measurable function
17.3. Dominated convergence of positive functions
17.4. Spaces of classes of integrable functions
17.5. Completion and approximation in spaces of classes of functions
17.6. Some properties of spaces of classes of functions
17.7. Lebesgue points
17.8. Measures associated to classes of functions
17.9. Identity of the spaces of measures
Chapter 18: Applications
18.1. Equi-integrability
18.2. Dominated convergence
18.3. Image under a continuous mapping
18.4. Continuity with respect to increasing
p
(again)
18.5. Riesz representation theorem (again)
Appendix: Reminders
A.1. Notation and numbering
A.2. Banach, Hilbert, Fréchet, and Neumann spaces
A.3. Continuous mappings
A.4. Duals, weak topology, and reflexivity
A.5. Differentiable mappings
Bibliography
Index
Other titles from iSTE in Mathematics and Statistics
End User License Agreement
List of Illustrations
Chapter 1
Figure 1.1.
Tiling of ω by
∆
s,n
Chapter 7
Figure 7.1. Equivalent decompositions, to “measure” a parallelepiped...
Figure 7.2. Change of variable. The boundary of T (∆s,n) is in bold, wh...
Chapter 14
Figure 14.1. Domain of definition ΩD of the weighting f ⋄ µ...
Figure 14.2. Domain ΩK going up to a section of the boundary. The bound...
Chapter 15
Figure 15.1.
Lipschitz boundary and cone property
Guide
Cover Page
Table of Contents
Dedication
Title Page
Copyright Page
Introduction
List of Notations and Figures
Begin Reading
Appendix: Reminders
Bibliography
Index
Other titles from iSTE in Mathematics and Statistics
Wiley End User License Agreement
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To Enrique Fernández Vara,
Who, for over twenty years, has tirelessly reread the countless versions of this work, and has kindly suggested so many much-needed improvements. How arid those early drafts were!
He introduced me to the fruitful alliance of work in the Ecuaciones Diferenciales y Analísis Numérico group that he directed with kindness and good humor, of the Feria, temple of the Sevillana and manzanilla, and of emotion, even for a nonbeliever, during Semana Santa, at the poignant sound of the trumpets announcing the Virgen del Baratillo.
Analysis for PDEs Set
coordinated byJacques Blum
Volume 4
Integration
Jacques Simon
First published 2026 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK
www.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA
www.wiley.com
© ISTE Ltd 2026The rights of Jacques Simon to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.
British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78630-013-3
The manufacturer’s authorized representative according to the EU General Product Safety Regulation is Wiley-VCH GmbH, Boschstr. 12, 69469 Weinheim, Germany, e-mail: [email protected].
PART 1Integration
An ungrateful country constantly tells its learned menthat they are useless citizens,and while it enjoys the fruits of their long labours,it asks them what they have done with them.
Charles de Secondat, baron de la Brède et de MONTESQUIEUMes Pensées
