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Lebesgue Integration and Measure - Alan J. Weir

Lebesgue Integration and Measure

Paperback Published: 9th July 1973
ISBN: 9780521097512
Number Of Pages: 296

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Lebesgue integration is a technique of great power and elegance which can be applied in situations where other methods of integration fail. It is now one of the standard tools of modern mathematics, and forms part of many undergraduate courses in pure mathematics.

Dr Weir's book is aimed at the student who is meeting the Lebesgue integral for the first time. Defining the integral in terms of step functions provides an immediate link to elementary integration theory as taught in calculus courses. The more abstract concept of Lebesgue measure, which generalises the primitive notions of length, area and volume, is deduced later.

The explanations are simple and detailed with particular stress on motivation. Over 250 exercises accompany the text and are grouped at the ends of the sections to which they relate: notes on the solutions are given.

Industry Reviews

'The book is easy to read, partly because of the treatment adopted, and partly because of the quality of the exposition. Dr Weir's style is clear, friendly and informal; he shows how the results fit in with the reader's intuition; he highlights the important things and warns of the difficult things (these warnings when a hard bit is coming up are most confidence-preserving). He does not aim at maximum generality at the cost of understanding. The examples are chosen with care, many of them being, in effect, lemmas that will be needed later in the proofs of theorems.' Mathematical Gazette

Prefacep. ix
The Completeness of the Realsp. 1
The Axiom of Completenessp. 2
Infima and supremap. 8
Postscript on the axioms for Rp. 11
Null Setsp. 15
Countable setsp. 15
Null setsp. 18
Cantor's ternary setp. 20
The Lebesgue Integral on Rp. 22
Step functionsp. 23
Construction of the Lebesgue integralp. 30
Relation to the 'definite integral'p. 44
Relation to the 'indefinite integral'p. 54
Some further resultsp. 63
The Lebesgue Integral on Rp. 70
Step functions on Rp. 70
The Lebesgue integral on Rp. 77
Fubini's Theoremp. 83
The Convergence Theoremsp. 93
The Monotone Convergence Theoremp. 94
The Dominated Convergence Theoremp. 106
Measurable Functions and Lebesgue Measurep. 119
Measurable functionsp. 119
Lebesgue measure on Rp. 124
The geometry of Lebesgue measurep. 134
Transformation of integralsp. 146
The Spaces L[superscript p]p. 162
The completeness of R as a metric spacep. 162
The spaces L[superscript p]p. 164
The geometry of L[superscript 2]p. 172
Bounded linear functionals on L[suprescript 2]p. 181
Orthonormal sets in L[superscript 2]p. 188
Classical Fourier seriesp. 202
Reflections on Hilbert spacep. 219
The elements of topologyp. 223
Solutionsp. 241
Referencesp. 277
Indexp. 279
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521097512
ISBN-10: 0521097517
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 296
Published: 9th July 1973
Country of Publication: GB
Dimensions (cm): 21.79 x 16.46  x 1.78
Weight (kg): 0.44