Preface xiii

of σ-algebra. If a generalization of length has these three key prop-

erties, then it needs to be defined on a σ-algebra for these properties

to make sense.

In Chapter 2 the text introduces null sets and shows that any

generalization of length satisfying monotonicity and countable addi-

tivity must assign zero to them. We then define Lebesgue measurable

sets to be sets in the σ-algebra generated by open sets and null sets.

At this point we state a theorem which asserts that Lebesgue

measure exists and is unique, i.e., there is a function µ defined for

measurable subsets of a closed interval which satisfies monotonicity,

countable additivity, and translation invariance.

The proof of this theorem (Theorem 2.4.2) is included in an ap-

pendix where it is also shown that the more common definition of

measurable sets (using outer measure) is equivalent to being in the

σ-algebra generated by open sets and null sets.

Chapter 3 discusses bounded Lebesgue measurable functions and

their Lebesgue integral. The last section of this chapter, and some of

the exercises following it, focus somewhat pedantically on the concept

of “almost everywhere.” The hope is to develop suﬃcient facility with

the concept that it can be treated more glibly in subsequent chapters.

Chapter 4 considers unbounded functions and some of the stan-

dard convergence theorems. In Chapter 5 we introduce the Hilbert

space of

L2

functions on an interval and show several elementary

properties leading up to a definition of Fourier series.

Chapter 6 discusses classical real and complex Fourier series for

L2

functions on the interval and shows that the Fourier series of an

L2

function converges in

L2

to that function. The proof is based on

the Stone-Weierstrass theorem which is stated but not proved.

Chapter 7 introduces some concepts from measurable dynamics.

The Birkhoff ergodic theorem is stated without proof and results on

Fourier series from Chapter 6 are used to prove that an irrational

rotation of the circle is ergodic and the squaring map z → z2 on the

complex numbers of modulus 1 is ergodic.

Appendix A summarizes the needed prerequisites providing many

proofs and some exercises. There is some emphasis in this section