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
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