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æ��æ��a.primer.of.lebesgue.integration

# æ��æ��a.primer.of.lebesgue.integration - PREFACE...

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PREFACE TO THE FIRST EDITION This text provides an introduction to the Lebesgue integral for advanced undergraduates or beginning graduate students in mathematics. It is also designed to furnish a concise review of the fundamentals for more advanced students who may have forgotten one or two details from their real analysis course and find that more scholarly treatises tell them more than they want to know. The Lebesgue integral has been around for almost a century, and the presentation of the subject has been slicked up con- siderably over the years. Most authors prefer to blast through the preliminaries and get quickly to the more interesting results. This very efficient approach puts a great burden on the reader; all the words are there, but none of the music. In this text we deliberately unslick the presentation and grub around in the fundamentals long enough for the reader to develop some in- tuition about the subject. For example, the Caratheodory def- inition of measurability is slick—even brilliant—but it is not intuitive. In contrast, we stress the importance of additivity for the measure function and so define a set £ e (0,1) to be mea- surable if it satisfies the absolutely minimal additivity condi- tion: m(E) + m{E) = 1, where E^ = (0,1) E and m is the outer measure in (0,1). We then show in easy steps that measurability of E is equivalent to the Caratheodory criterion, m(E n T) + m(E^ oT) = m(T) for all T. In this way we remove the magic from the Caratheodory condition, but retain its util- ity. After the measure function is defined in (0,1), it is extended to each interval (n, n + 1) in the obvious way and then to the whole line by countable additivity.

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viii A PRIMER OF LEBESGUE INTEGRATION We define the integral via the famihar upper and lower Darboux sums of the calculus. The only new wrinkle is that now a measurable set is partitioned into a finite number of measurable sets rather than partitioning an interval into a fi- nite number of subintervals. The use of upper and lower sums to define the integral is not conceptually different from the usual process of approximating a function by simple functions. How- ever, the customary approach to the integral tends to create the impression that the Lebesgue integral differs from the Riemann integral primarily in the fact that the range of the function is par- titioned rather than the domain. What is true is that a partition of the range induces an efficient partition of the domain. The real difference between the Riemann and Lebesgue integrals is that the Lebesgue integral uses a more sophisticated concept of length on the line. We take pains to show that both the Riemann-Darboux in- tegral and the Lebesgue integral are limits of Riemann sums, for that is the way scientists and engineers tend to think of the integral. This requires that we introduce the concept of a con- vergent net. Net convergence also allows us to make sense out of unordered sums and is in any case something every young mathematician should know.
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