Measure and Integral
1
Algebras
Denition 1.1. An algebra on the set is a collection F of subsets of
with the properties:
(i) F
(ii) C F C c = C F
(iii) A, B F A B F .
Note that if F is an algebra on then = c F and if A, B F then
A B = (Ac B c )c F . Note

Modern Analysis 1
Homework 07
1. Let X R and let f : X R map Cauchy sequences to Cauchy sequences.
Prove that if X is bounded then f is uniformly continuous; also, show by
example that the boundedness hypothesis may not be dropped.
Generalize?
1

Modern Analysis 1
Homework 08
1. Let f (a, b) R be continuous at all points and dierentiable except
perhaps at p (a, b). Assume that the limit
lim f (t)
tp
exists (as a real number). Must f be dierentiable at p? Give proof or
counterexample as appropriate

Measure and Integral
1
Algebras
Denition 1.1. An algebra on the set is a collection F of subsets of
with the properties:
(i) F
(ii) C F C c = C F
(iii) A, B F A B F .
Note that if F is an algebra on then = c F and if A, B F then
A B = (Ac B c )c F . Note

Measure and Integration
Problems
1
Problem 1.1 Let C be a collection of subsets of the set . For each A (C )
show that there is a countable subcollection CA of C such that A (CA ).
Problem 1.2 Let (, F ) and ( , F ) be measurable spaces and f :
a functio

Measure and Integration
Solutions
1
Problem 1.1 Let C be a collection of subsets of the set . For each A (C )
show that there is a countable subcollection CA of C such that A (CA ).
Solution Let D comprise all countable subcollections in C . Consider the

Modern Analysis 1
Midterm
Answer at most ve questions.
1. Let S be a complete ordered set. For each let A S be nonempty
and let A = A be bounded above; let = sup A and for each
let = sup A . Prove or disprove:
= supcfw_ : .
2. (i) State the Cauchy-Schwa

Analysis
Midterm 1
Solutions
1. Let (, F , ) be a measure space on which f, f1 , f2 , . . . are nonnegative
integrable functions. Show that if fn f almost everywhere and fn d
f d then |fn f |d 0.
Solution Notice that 0 (fn f ) f since f, fn are nonnegati

Analysis
Midterm 2
Solutions
Write solutions in a neat and logical fashion, giving complete reasons for
all steps.
1. Let Z be a closed subspace of the Banach space X . Prove that if X is
reexive then so is Z .
Solution Recall that if T : Z X denotes incl

Modern Analysis 1
Homework 06
1. Let (an ) be a sequence of strictly positive real numbers.
n=0
(i) Prove that if n 0 an converges then so does n 0 an an+1 .
(ii) Prove the converse when the sequence (an ) is monotonic.
n=0
(iii) Show by example that the

Modern Analysis 1
Homework 05
1. Let M be a compact metric space. Let f : M M be isometric in the
sense
a, b M d(f (a), f (b) = d(a, b).
Prove that f is surjective.
Suggestion: Let z M . By considering the sequence (zn ) given by
n=0
zn = f f (z ) (with n

Analysis
Midterm 1
Write solutions in a neat and logical fashion, giving complete reasons for
all steps.
1. Let H be a complex Hilbert space.
(a) Prove that the set F (H) comprising all nite-rank operators on H is a
minimal ideal in B (H).
(b) Prove that

Analysis
Midterm 2
Write solutions in a neat and logical fashion, giving complete reasons for
all steps.
1. Let (, F ) be a measurable space on which , and are (nite) measures.
Show that if
and
then
; further, carefully verify the
relation
d d
d
=
d

Analysis
PhD Examination
Sample Solutions
1. Let X and Y be normed linear spaces. Prove that the normed space
B (X, Y ) (comprising all bounded linear maps from X to Y ) is complete if
and only if Y is complete.
Solution () This is the familiar direction.

Analysis
Final Sketch Solutions
Write solutions in a neat and logical fashion, giving complete reasons for
all steps.
1. State what it means for a Banach space X to be (i) uniformly convex
(ii) strictly convex. Prove carefully that if X uniformly convex t

Modern Analysis 1
Homework 01
1. Let F be an ordered eld. Let A and B be subsets of F for which sup A
and sup B exist (in F ); dene subsets A + B and AB of F by
A + B = cfw_a + b : a A, b B ,
AB = cfw_ab : a A, b B .
(i) Does sup(A + B ) exist in F ? If s

Modern Analysis 1
Homework 02
1. Let F be a complete ordered eld and Q its prime subeld. Fix a > 1 in
F and for x X dene
A(x) = cfw_aq : Q
q < x.
(i) Show that A(x) is nonempty and bounded above, whence it is possible to
dene
ax = sup A(x).
(ii) Show that

Modern Analysis 1
Homework 03
1. Let (M, d) be a metric space.
(i) Show that a new metric D is dened on M by the rule
x, y M D(x, y ) =
d(x, y )
.
1 + d(x, y )
(ii) Show that a subset of M is D-open if and only if it is d-open.
Solution
It is convenient t

Modern Analysis 1
Homework 04
1. For x M and A M dene
d(x, A) = inf cfw_d(x, a) : a A.
Show that if > 0 then
B (A) = cfw_x M : d(x, A) <
is an open subset of M .
2. Let the compact subset K M and the closed subset F M be disjoint.
Prove that there exist

Q
R
Here follows one route from the rationals Q to the reals R.
The basic idea is to fashion each real number from the rational sequences
that converge to it; without having the actual real number in hand, we
recognize convergence from the Cauchy conditio