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 RiemannDarboux 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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 Spring '08
 Mark
 Math, Lebesgue integration, Lebesgue, measurable sets

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