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 .
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