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

interval.

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

concepts.

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

chapters.

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