0.3. Lebesgue measure theory 7

Theorem 0.20. (Green’s Theorem) Suppose that P (x, y), Q(x, y),

∂

∂y

P (x, y) and

∂

∂x

Q(x, y) are continuous on and inside a simple closed curve C with interior R.

Then

C

P dx + Q dy =

R

∂Q

∂x

−

∂P

∂y

dA .

A discussion of summation by parts, and its application to Abel’s Test and

Dirichlet’s Test, is found in §4.2.

0.3. Lebesgue measure theory

We begin by noting Littlewood’s three principles:

• Every [measurable] set of real numbers is nearly a finite union of intervals.

• Every [measurable] function is nearly continuous.

• Every convergent sequence of [measurable] functions is nearly uniformly con-

vergent.

Our first task is to formulate precise theorems that embody these principles.

Theorem 0.21. Let E be a measurable set of real numbers. Then for every ε 0

there is an open set O such that E ⊆ O and meas (O \ E) ε. Also, there is a

closed set F such that F ⊆ E and meas (E \ F ) ε.

Theorem 0.22. Continuous functions are dense in

L1([a,

b]) in the sense that if

f ∈

L1([a,

b]) and ε 0 are given, then there is a continuous function g(x) such

that

b

a

|f(x) − g(x)| dx ε.

For 1 ≤ p ∞, the same theorem holds for

Lp([a,

b]), for

Lp(T),

and for

Lp(R).

Theorem 0.23. (Egorov’s Theorem) Let f and a sequence {fn} of functions all

be defined on I = [a, b], and suppose that limn→∞ fn(x) = f(x) for almost all

x ∈ [a, b]. Then for any ε 0 there is a set A ⊆ I with meas(A) ε such that

fn(x) tends uniformly to f(x) as n → ∞, for all x ∈ I \ A.

Theorem 0.24. (Monotone Convergence Theorem) Suppose that 0 ≤ f1(x) ≤

f2(x) ≤ · · · for all x ∈ [a, b], and that there is a function f such that limn→∞ fn(x) =

f(x) for all x ∈ [a, b]. We assume that the functions fn are measurable. Then

lim

n→∞

b

a

fn(x) dx =

b

a

f(x) .

Theorem 0.25. (The Principle of Dominated Convergence) Let f1,f2,... be mea-

surable functions on [a, b] with limn→∞ fn(x) = f(x) for all x ∈ [a, b]. Suppose that

there is a function F (x) such that F (x) ≥ 0 for all x ∈ [a, b], with

b

a

F (x) dx ∞,

and such that |fn(x)| ≤ F (x) and |f(x)| ≤ F (x) for all n and all x ∈ [a, b]. Then

b

a

f(x) dx =

b

a

lim

n→∞

fn(x) dx = lim

n→∞

b

a

fn(x) dx .