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