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Integration - A Functional Approach Book

Integration - A Functional Approach
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  • Integration - A Functional Approach
  • Written by author Klaus Bichteler
  • Published by Springer-Verlag New York, LLC, July 1998
  • This book covers Lebesgue integration and its generalizations from Daniell's point of view, modified by the use of seminorms. Integrating functions rather than measuring sets is posited as the main purpose of measure theory. From this point of view Leb
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Authors

Preface
Ch. IReview1
I.1Introduction1
I.2Notation3
I.3The Theorem of Stone-Weierstrass8
I.4The Riemann Integral17
I.5An Integrability Criterion21
I.6The Permanence Properties23
I.7Seminorms28
Ch. IIExtension of the Integral31
II.1[Sigma]-additivity33
II.2Elementary Integrals35
II.3The Daniell Mean42
II.4Negligible Functions and Sets48
II.5Integrable Functions52
II.6Extending the Integral58
II.7Integrable Sets61
II.8Example of a Non-Integrable Function66
Ch. IIIMeasurability69
III.1Littlewood's Principles70
III.2The Permanence Properties73
III.3The Integrability Criterion75
III.4Measurable Sets81
III.5Baire and Borel Functions85
III.6Further Properties of Daniell's Mean92
III.7The Procedures of Lebesgue and Carathedory95
Ch. IVThe Classical Banach Spaces101
IV.1The p-norms102
IV.2The [actual symbol not reproducible]-spaces105
IV.3The L[superscript p] spaces108
IV.4Linear Functionals112
IV.5The Dual of L[superscript p]118
IV.6The Hilbert space L[superscript 2]122
Ch. VOperations on Measures125
V.1Products of Elementary Integrals125
V.2The Theorems of Fubini and Tonelli130
V.3An Application: Convolution133
V.4An Application: Marcinkiewicz Interpolation134
V.5Signed Measures137
V.6The Space of Measures144
V.7Measures with Densitites151
V.8The Radon-Nikodym Theorem154
V.9An Application: Conditional Expectation155
V.10Differentiation157
App. A: Answers to Selected Problems165
References180
Index of Notations181
Index184


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