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The Elements of Integration and Lebesgue Measure, Robert G. Bartle

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About this book :-
The Elements of Integration and Lebesgue Measure written by Robert G. Bartle
It is hardly possible to overemphaize the importance of the theory of integration to mathemtical analysis, inded it is one of the twin pillars on which analysis is built. Granting that it is suprising that new devlopments continue to arise in this theory, which was originated by the great Newton and Leibuiz over three centruies ago, made rigorous by Riemann in the middle of the nineteenth century and extended by Lebesgue at the begining of the twentieth century.
(Robert G. Bartle)

Book Detail :-
Title: The Elements of Integration and Lebesgue Measure
Edition:
Author(s): Robert G. Bartle
Publisher: Wiley-Interscience
Series: Wiley Classics Library
Year: 1995
Pages: 193
Type: PDF
Language: English
ISBN: 0471042226, 9780471042228, 9781118164488
Country: US
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About Author :-
Author Robert Gardner Bartle (1927–2003) was an American mathematician. He was specializing in real analysis. He is known for writing the popular textbooks The Elements of Real Analysis (1964), The Elements of Integration (1966), and Introduction to Real Analysis (2011) published by John Wiley & Sons.
This book consists of two separate, but closely related, parts. The first part (Chapters 1-10) is subtitled The Elements of Integration; the second part (Chapters 11-17) is subtitled The Elements of Lebesgue measure. It is possible to read these two parts in either order, with only a bit of repetition.
The Elements of Lebesgue Measure is descended from class notes written to acquaint the reader with the theory of Lebesgue measure in the space RP. While it is easy to find good treatments of the case p = 1, the case p > 1 is not quite as simple and is much less frequently discussed. The main ideas of Lebesgue measure are presented in detail in Chapters 10-15, although some relatively easy remarks are left to the reader as exercises. The final two chapters venture into the topic of nonmeasurabl a sets and round out the subject.
There are many expositions of the Lebesgue integral from various points of view, but I believe that the abstract measure space approach used here strikes directly towards the most important results: the convergence theorems. Further, this approach is particularly wellsuited for students of probability and statistics, as well as students of analysis. Since the book is intended as an introduction, one do not follow all of the avenues that are encountered. However, it take pains not to attain brevity by leaving out important details, or assigning them to the reader.
Readers who complete this book are certainly not through, but if this book helps to speed them on their way, it has accomplished its purpose. In the References, some books that the author believe readers can profitably explore, as well as works cited in the body of the text.

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Book Contents :-
The Elements of Integration and Lebesgue Measure written by Robert G. Bartle cover the following topics.
The Elements of Integration
1. Introduction
2. Measurable I Cnctious
3. Measures
4. The Integral
5. Integrable Functions
6. The Lebesgue Spaces Lp
7. Modes of Convergence
8. Decomposition of Measures
9. Generation of Measures
10. Product Measures
11. Volumes of Cells and Intervals
12. The Outer Measure
13. Measurable Sets
14. Examples of Measurable Sets
15. Approximation of Measurable Sets
16. Additivity and Nonadditivity
17. NonmeasurabLe and Non-Borel Sets
References
Index


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