Preface
"Mathematics presented as a closed, linearly ordered, system of truths
without reference to origin and purpose has its charm and satisfies
a philosophical need. But the attitude of introverted science is un-
suitable for students who seek intellectual independence rather than
indoctrination; disregard for applications and intuition leads to isola-
tion and atrophy of mathematics. It seems extremely important that
students and instructors should be protected from smug purism."
Richard Courant and Fritz John
(Introduction to Calculus and Analysis)
This text presents a motivated introduction to the subject which goes under
various headings such as Real Analysis, Lebesgue Measure and Integration,
Measure Theory, Modern Analysis, Advanced Analysis, and so on.
The subject originated with the doctoral dissertation of the French
mathematician Henri Lebesgue and was published in 1902 under the ti-
tle Integrable, Longueur, Aire. The books of C. Caratheodory [8] and [9],
S. Saks [35], LP. Natanson [27] and P.R. Halmos [14] presented these ideas
in a unified way to make them accessible to mathematicians. Because of its
fundamental importance and its applications in diverse branches of mathe-
matics, the subject has become a part of the graduate level curriculum.
Historically, the theory of Lebesgue integration evolved in an effort to
remove some of the drawbacks of the Riemann integral (see Chapter 1).
However, most of the time in a course on Lebesgue measure and integra-
tion, the connection between the two notions of integrals comes up only
XI
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