PREFACE :
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.
Galois understood how this symmetry group can be used to characterize
The Lebesgue integral has been around for almost a century,
and the presentation of the subject has been slicked up
considerably 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
intuition about the subject.
We define the integral via the familiar 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
finite 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.
However, 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
partitioned 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.
Title : A primer on Lebesgue integration
author(s) : H. S. Bear
size : 0.8 Mb
file type : DJVU
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