Modern Analysis 2
Midterm
Answer at most four questions.
1. Let f be strictly increasing and continuous on [0, a] and dierentable on
(0, a) with f (0) = 0.
(i) Prove that if
u
g (u) :=
f (u)
f 1 (t)dt uf (u)
f (t)dt +
0
0
then g (u) = 0 for all u [0, a].

Modern Analysis 2
Test 2
Answer (1 or 2) and (3 or 4).
1. Decide (with proof) on which intervals [a, b] an arbitrary continuous function may be uniformly approximated by functions of the form:
(i) c0 + c1 t1/3 + + cn tn/3 ;
(ii) c0 + c1 t4 + + cn t4n .
2.

Modern Analysis 2
Test 3
Answer TWO questions.
1. Let (fn ) be a sequence of measurable functions. Prove that each of the
n=0
following sets is measurable:
(e) E = cfw_x : fn (x) > en ;
(f) F = cfw_x : fn (x) < 0 for nitely many n;
1
(h) H = cfw_x : n > 0

Modern Analysis 2
Test 4
Answer TWO questions.
1. Let f L() and dene : M R by
A M (A) =
f d.
A
Show that if > 0 is given then there exists > 0 such that
(A) < |(A)| < .
2. Let f : R R be Lebesgue integrable. For t R dene
t
f
F (t) =
0
with respect to Lebe

Modern Analysis 2
Homework 01
1. Let (an : n > 0) be a strictly increasing sequence in (0, 1) converging to
1 and let A = cfw_an : n > 0. Dene f : [0, 1] R by f (t) = 1 if t A and
f (t) = 0 if t A.
/
1
Is f Riemann-integrable over [0, 1]? If so, what is 0

Modern Analysis 2
Homework 03
1. Let cfw_fn : n N be a sequence of functions, each continuous on [a, b];
assume that each is continuously dierentiable on (a, b). Assume further that
there exist constants M0 and M1 such that |fn (t)| < M0 and |fn (t)| < M1

Modern Analysis 2
Homework 04
1. Let the dierentiable function f : (a, a) R satisfy f (0) = 0 and
f = 1 + f 2 . By considering the function g := 2/f (or otherwise) show that
a is at most /2.
Note: The cosine as the solution to a second-order IVP may be as

Modern Analysis 2
Homework 05
1. Let (X, M, ) be a measure space with (X ) < . Let (fn ) be a
n=1
sequence of measurable real-valued functions on X converging pointwise to
f . Prove that if > 0 and > 0 then there exist A M and N > 0 such
that (A) < and
n

Modern Analysis 2
Homework 06
1. Declare that fn f (and say fn converges to f in measure) i
( > 0) cfw_|f fn |
0.
When (X ) < dene
d(f, g ) =
Show that
|f g |
d.
1 + |f g |
fn f d(f, fn ) 0.
1

Modern Analysis 2
Homework 07
1. Let (An ) be a sequence in M. For m a positive integer, let Bm be the
n=1
set comprising all elements that lie in An for at least m values of n. Prove
that Bm M and
m (Bm )
(An ).
n=1
Remark: It follows that if
n=1
(An ) <

Modern Analysis 2
Test 1
Answer (1 or 2) and (3 or 4) and 5.
1. Let f : [a, b] R be continuous and nonnegative, with maximum value
M . Prove that
b
fn
lim
n
1/n
= M.
a
2. Let f : [a, b] R be continuous. Prove that there exists p between a and
b such that