8 0. Background

In the above, if the function F (x) is taken to be a constant, then the result is

known as the Bounded Convergence Theorem.

Just as the real numbers are complete, the spaces Lp([a, b]) are complete in the

following sense.

Theorem 0.26. Suppose that 1 ≤ p ∞, and that f1,f2,... is a sequence of

functions in Lp([a, b]) such that

lim

m→∞

n→∞

fm − fn

p

→ 0 .

Then there is an f ∈ Lp([a, b]) such that limn→∞ fn − f

p

= 0.

The above also holds for Lp(T) and for Lp(R).

Theorem 0.27. (Fubini’s Theorem) Let I = [a, b], J = [c, d], and let R be the

rectangle R = I × J. If f(x, y) is a measurable function on R such that

R

|f(x, y)| dA ∞,

then

R

f(x, y) dA =

b

a

d

c

f(x, y) dy dx =

d

c

b

a

f(x, y) dx dy .

The full definition of the measure of a set is a little complicated, but sets of

measure zero are easily described: If S is a set of real numbers, then S has measure

0 if for every ε 0 there exist intervals I1,I2,... such that

S ⊆

∞

n=1

In,

∞

n=1

|In| ε .

Theorem 0.28. If f is a bounded function on a finite interval [a, b], then f is

Riemann-integrable on this interval if and only if the set of points x ∈ [a, b] at

which f is discontinuous is a set of Lebesgue measure 0.

Theorem 0.29. If f is Lebesgue integrable and F (x) =

x

a

f(u) du, then F is

continuous, and F (x) = f(x) for almost all x.

In the above situation, we say that F is absolutely continuous. A continuous

function may have derivative 0 almost everywhere and yet be nonconstant. Such a

function is said to be singular.

If f is a Lebesgue integrable function, then we say that x is a Lebesgue point

of f if

lim

h→0+

1

h

x+h

x−h

|f(u) − f(x)| du = 0 .

Roughly, x is a Lebesgue point if f(u) is on average near f(x) when u is near x.

Theorem 0.30. If f is a Lebesgue integrable function, then almost all x are

Lebesgue points of f.