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.
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