xii Preface
other end is represented, for example, by the excellent graduate text
[Ru] by Rudin which introduces Lebesgue measure as a corollary of
the Riesz representation theorem. That is a sophisticated and elegant
approach, but, in my opinion, not one which is suited to a student’s
first encounter with Lebesgue integration.
In this text the less elegant, and more technical, classical con-
struction of Lebesgue measure due to Caratheodory is presented, but
is relegated to an appendix. The intent is to introduce the Lebesgue
integral as a tool. The hope is to present it in a quick and intuitive
way, and then go on to investigate the standard convergence theorems
and a brief introduction to the Hilbert space of L2 functions on the
This text should provide a good basis for a one semester course
at the advanced undergraduate level. It might also be appropriate
for the beginning part of a graduate level course if Appendices B
and C are covered. It could also serve well as a text for graduate
level study in a discipline other than mathematics which has serious
mathematical prerequisites.
The text presupposes a background which a student should pos-
sess after a standard undergraduate course in real analysis. It is terse
in the sense that the density of definition-theorem-proof content is
quite high. There is little hand holding and not a great number of
examples. Proofs are complete but sometimes tersely written. On
the other hand, some effort is made to motivate the definitions and
Chapter 1 provides a treatment of the “regulated integral” (as
found in Dieudonn´ e [D]) and of the Riemann integral. These are
treated briefly, but with the intent of drawing parallels between their
definition and the presentation of the Lebesgue integral in subsequent
As mentioned above the actual construction of Lebesgue measure
and proofs of its key properties are left for an appendix. Instead the
text introduces Lebesgue measure as a generalization of the concept of
length and motivates its key properties: monotonicity, countable ad-
ditivity, and translation invariance. This also motivates the concept
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