Chapter 2. Functions of Bounded Variation
1. Monotone Functions
Let I be an interval in IR. A function f : I IR is said to be increasing (strictly
increasing) if f (x) f (y) (f (x) < f (y) whenever x, y I and x < y. A function
f : I IR is said to be decre
MATH 418
Assignment #5
1. Let X and Y be two nonempty sets, and let S and T be -algebras of subsets of X
and Y , respectively. Dene
S T := cfw_A B : A S, B T ,
and let be the -algebra generated by S T .
(a) Prove that for each E and each x X, Ex T .
Proof
MATH 418
Assignment #4
1. For a > 0, let fa be the function on [0, 1] dened by fa (0) := 0 and
,
x2
fa (x) := xa cos
0 < x 1.
(a) Prove that for a > 2, fa is absolutely continuous on [0, 1].
Proof . For 0 < x 1, we have
fa (x) = axa1 cos(/x2 ) + 2xa3 sin(
MATH 418
Assignment #2
1. Let f (x) := x ln(1 + 1/x), 0 < x < .
(a) Show that f is strictly increasing on (0, ).
Proof . For x > 0, we have
f (x) = ln 1 +
1
1
+x
x
1+
1
x
1
x2
= ln 1 +
1
1
x
x+1
and
f (x) =
1
1
1
+
=
< 0.
2
x(1 + x) (x + 1)
x(x + 1)2
Hen
MATH 418
Assignment #3
1. Let f be an increasing function from IR to IR, and let E be the set of those points at
which f is discontinuous.
(a) Prove that E is countable.
Proof . For x E, let ax := limyx f (y) and bx := limyx+ f (y). Then ax < bx .
For eac
MATH 418
Assignment #6
1. Let T be the linear transform on IR2 given by
(x, y) IR2 .
T (x, y) = (x + y, y),
Prove that for any Lebesgue measurable set E in IR2 , T (E) is Lebesgue measurable
and (T (E) = (E).
Proof . Suppose A is a rectangle of the form (
MATH 418
Assignment #7
1. Let r, s, t be positive real numbers such that r + s + t = 1. Suppose that f, g, h are
nonnegative measurable functions on a measure space (X, S, ). Prove that
f r g s ht d f
r
1
g
s
1
h t.
1
X
s
t
Proof . Let u := g s+t h s+t .
Chapter 1. Dierentiation
1. The Derivative
Let f be a real-valued function dened on an interval I in IR. The derivative of f
at a point a I is dened to be
f (x) f (a)
,
xa
xa
lim
provided this limit exists. The derivative f at a is denoted by f (a). We sa
Chapter 4. Product Measures
1. Product Measures
Let (X, S, ) and (Y, T, ) be two measure spaces. In order to dene the product of
and , we rst dene an outer measure on X Y in terms of and .
For each subset E of X Y , we dene
(Aj )(Bj ) : Aj S, Bj T j IN a
Chapter 5. Lp Spaces
1. Lp Spaces
Let (X, S, ) be a measure space. If f is a measurable function on X and 1 p < ,
we dene
1/p
f
p
p
|f | d
:=
.
X
If p = , we dene
f
:= infcfw_M 0 : |f (x)| M for almost every x X,
with the convention that inf = . Often f i
MATH 418
Assignment #1
1. Let X be a nonempty set, and let A, B, and C be subsets of X.
(a) Prove that ABC = A + B + C 2AB 2BC 2CA + 4ABC .
Proof . We have
EF = E + F 2EF .
It follows that
(AB)C = AB + C 2(AB)C
= AB + C 2[AC + BC 2ABC ]
= A + B + C 2AB 2B
Chapter 3. Absolutely Continuous Functions
1. Absolutely Continuous Functions
A function f : [a, b] IR is said to be absolutely continuous on [a, b] if, given
> 0, there exists some > 0 such that
n
|f (yi ) f (xi )| < ,
i=1
whenever cfw_[xi , yi ] : i =