Real Analysis Qualifying Exam – May 14th 2016

Written by Prof. S. Lee and Prof. B. Shekhtman

Solve 8 out of 12 problems.(1) Prove the Banach contraction principle:

Let T be a mapping from a complete metric space X into itself such that

d(Tx, Ty) qd(x, y)

for all x, y 2 X and for some q < 1. For arbitrary x0 2 X let xn = Txn�1. Prove that xn ! x such that

Tx = x.(x is a fixed point for T ). Also prove that this is the unique fixed point for T .

(2) Does the following limit exist

lim

Z 1

0

dx

(1 +xn )

nx1n

?

If it does, find the limit.

(3) Let f be a continuous linear functional on a Hilbert space H and M := {x 2 H : f(x) = 0}. Prove that

dimM? 1.

(4) Let T be a surjective linear map from a Banach space X onto a Banach space Y such that

kTxk � 1

2016kxk

for all x 2 X. Show that T is bounded.

(5) Let A := span{xn(1� x) : n � 1} ⇢ C([0, 1]). Describe its closure in the uniform norm.

(6) Let µ be a finite positive measure on the measurable space (⌦,⌅, µ) and let F 2 Lp(µ)⇤for some

1 p < 1. Prove that there exists g 2 L1(µ) such that

Z

Agdµ = F (�A)

for all A 2 ⌅. (Here �A is the characteristic function of A).

(7) Show that

f 2 L1(R) () g 2 L

1(R⇥ R)

where g(x, y) = f(x+ y)f(x).

(8) Show that, for 1 p < 1, a bounded sequence {fn} in Lp(R) such that {fn} ! f pointwise a.e.

converges weakly to f in Lp(R).

(9) Let �, µ be finite measures on X. Let

F = {f 2 L1(X,µ) :

Z

Efdµ �(E)}

for all measurable sets E. Show that there exists f0 2 F such that

Z

Xf0dµ = sup

f2F

Z

Xfdµ.

(10) Find a bounded sequence in L1([0, 1]) such that

limn!1

Z x

0fn =

Z x

0f for all x 2 [0, 1]

and {fn} does not converge weakly to f in L1([0, 1]).

(11) Let f be Lipschitz on R and g be absolutely continuous on [0, 1]. Show that the composition f � g is

absolutely continuous.

(12) Describe the Caratheodory construction of a measure from a set function µ : S ! [0,1] where S is a

collection of subsets in X. Show that µ is countably monotone if and only if the outer measure induced by

µ is an extension of µ.

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Real Analysis Qualifying Exam – September 24th 2016

Written by S. Kouchekian and S. Lee

INSTRUCTIONS : Do at least 7 problems. Justify your reasoning. State the theorems you

use so that all the hypothesis are checked.

1. Suppose that f is a measurable function in R. Prove that there exists a sequence of step functionsthat converges pointwise to f(x) for almost every x.

2. Let F : R ! R be a function satisfying

F (x) =

Z x

af(y)dy

for an integrable function. Prove that F is absolutely continuous.

3. Compute the following limit and justify the calculation.

1X

n=0

Z ⇡/4

0(1�

psinx)n cosx dx.

4. Let (X,M, µ) be a �-finite positive measure space and {⌫n} be a sequence of �-finite positivemeasures on M with ⌫n ⌧ µ for all n 2 N such that

⌫n(E) ⌫n+1(E) for all n 2 N, E 2 M.

Define ⌫(E) = limn!1 ⌫n(E) for each E 2 M. Show that ⌫ defines a measure on M and ⌫ satisfies⌫ ⌧ µ. And express d⌫

dµ in terms of ⌫n.

5. Let fn, f 2 Lp[0, 1] with 1 p < 1. Suppose that fn ! f pointwise. Prove that kfn � fkp ! 0 ifand only if kfnkp ! kfkp as n ! 1.

6. For each polynomial f on [0, 1], letkfk = kfk1 + kf 0k1.

Let X be the normed linear space of polynomials on [0, 1] with the norm k · k1 and Y be the normedlinear space of polynomials on [0, 1] with the norm k · k. Let T : X ! Y be the linear operatordefined by T (f) = f 0 for f 2 X. Show that T is unbounded.

The graph of T is the set {(x, T (x)) 2 X ⇥ Y |x 2 X}. Is the graph of T closed in the producttopology of X ⇥ Y ?

7. Let (X,M, µ) be a �-finite positive measure space and ⌫ be a finite measure on M. Show that ⌫ ? µif and only if there is no nonzero measure ⇢ on M such that ⇢ ⌧ µ and ⇢ ⌫ on M.

8. Given an example of an increasing function on R whose set of discontinuity is precisely Q.

9. Let f 2 Lp(R) \ Lq(R) with 1 p < q < 1. Prove that f 2 Lr(R) for any r with p < r < q.

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Real Analysis Qualifying Exam – January 28th 2017

Written by B. Shekhtman and S. Lee

INSTRUCTIONS : Do at least 7 problems. Justify your reasoning. State the theorems you

use so that all the hypothesis are checked.

1. Suppose that f and g are continuous function on [0, 1] and

Z 1

0f(x)x

ndx =

Z 1

0g(x)x

ndx

for all n � 0. Prove that f(x) = g(x).

2. Let X be a Banach space and Y be a proper closed subspace of X. Prove that for every 0 < " < 1

there exists x 2 X with kxk = 1 such that

kx� yk > "

for all y 2 Y .

3. Let T be a bounded linear operator on a Hilbert space H. Show that kTk = kT ⇤k and kT ⇤Tk = kTk2.

4. Let f be a non-negative function in L1([0, 1]). Show that

limn!1

Z 1

0

npf(x)dx = m{x 2 [0, 1] : f(x) > 0}.

(Here m stands for Lebegues measure on [0, 1]).

5. Let f be a continuous function on [0, 1]. Show that limn!1R 10 (n+ 1)x

nf(x)dx = f(1).

6. Let (X,B, µ) be a measure space and f 2 Lp(µ) \ L

1(µ). Show that

limp!1

kfkp = kfk1

7. Suppose {fn}1n=1 is a sequence of nonnegative measurable functions on [0, 1] with

Z 1

0fndx 1

n2for all n � 1.

1

Prove that fn ! 0 a.e. on [0, 1].

8. Let f 2 L1(R). Show that g(x) =R1�1 cos(xt)f(t)dt is continuous and bounded.

9. Let {En} be a sequence of Lebesgue measurable subsets of R with

1X

n=1

µ(En) < 1. (1)

a) Show that

µ(lim supn!1

En) = 0,

where lim supn!1 En =T1

m=1

S1n=m En

b) Is the conclusion in a) still true if (1) is replaced byP1

n=1(µ(En))2< 1?

10. Let 0 < a < b. Show that the function f : R2 ! R given by

f(x, y) =

(e�xy

(x, y) 2 [0,1)⇥ [a, b],

0 otherwise

is integrable with respect to the Lebesgue measure on R2. Compute the integral

Z 1

0

e�ax � e

�bx

xdx.

11. Prove f : R ! R is absolutely continuous over any compact set if and only f · g is absoletely

continuous for all smooth g : R ! R supported on a compact set.

12. Let E ⇢ [0, 1] be measurable. Define a function f : [0, 1] ! R by

f(x) =

Z x

0(1� 2�E(t))dt.

Prove that f is of bounded variation and determine the total variation of f on [0, 1].

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QUALIFYING EXAM: REAL ANALYSISMay 13, 2017

Prof. C. Beneteau

Prof. S.-Y. Lee

Answer 4 questions from Part A and 3 questions from Part B.

Part A.

1. Let m be Lebesgue measure on R, and suppose that E ⇢ [0, 1] is a set ofreal numbers with the property that for any x and y in E with x 6= y, x�yis not equal to a rational number. Prove that the collection {E + r : r 2Q \ [4, 5]} is a countable collection of mutually disjoint sets. Prove thateither m(E) = 0 or E is not measurable.

2. Let X = [0, 1] and let m be Lebesgue measure on X. Define what it meansfor a sequence of measurable functions fn to converge to a function f inmeasure. Prove or disprove the following statement: If fn converges to fin measure, then fn converges to f pointwise a.e.

3. Consider the function f(x) = x cos(1/px) defined on the interval [0, 1],

where we set f(0) = 0. Is f of bounded variation? Prove or disprove.

4. (a) Given a measure space (X,M, µ) and a sequence of integrable functions{fn}1n=1, define what it means for the sequence to be uniformly integrable.

(b) If X = [0,1), M is all Lebesgue measurable sets, and µ is Lebesguemeasure, is the sequence fn(x) =

pn�[0,1/n)(x) uniformly integrable? Prove

or disprove.

5. (a) State the Radon-Nikodym Theorem.

(b) Give an example that shows the necessity of the requirement that thespace X be � finite in the Radon-Nikodym theorem.

(c) Suppose X = [0, 1], M is the �-algebra of Lebesgue measurable sets,µ is Lebesgue measure on X, and ⌫ is counting measure on X. Prove ordisprove each of the following statements: (1) µ << ⌫; (2) ⌫ << µ; (3) µand ⌫ are mutually singular.

6. Suppose {fn}1n=1 is a sequence of nonnegative measurable functions on [0, 1]with Z 1

0

fn dx an for all n � 1,

whereP1

n=1 an < 1. Prove that fn ! 0 a.e. on [0, 1].

7. Let F : R ! R be a function satisfying

F (x) =

Z x

a

f(y)dy

for an integrable function f . Prove (without referring to any theorems)that F is absolutely continuous.

Part B.

8. Let fn(✓) = ein✓ for 0 ✓ 2⇡. Show that fn(✓) ! 0 weakly in Lp[0, 2⇡]for any 1 < p < 1.

9. Let µ be a complex measure µ on a measure space (X,M), i.e., µ is acountably additive set function from M to C satisfying µ(;) = 0. Define anew set function |µ| on M as follows:

|µ|(E) := sup

( 1X

k=1

|µ(Ek)| : Ek 2 M are pairwise disjoint , E = [1k=1Ek

).

Prove that |µ| is a positive measure. (|µ| is sometimes called the totalvariation measure.)

10. Let (X,M, µ) be a finite measure space. Show that Lp(X,µ) ( L1(X,µ)for any p > 1. Is it true if we remove the hypothesis that µ(X) < 1?

11. Let X = Y = [0, 1] and for each positive integer n, let 'n be the charac-teristic function of the interval [1 � 1/2n�1, 1 � 1/2n), and let gn = 2n 'n.Define

f(x, y) =1X

n=1

[gn(x)� gn+1(x)] gn(y).

Prove that f is measurable as a function on the product space X ⇥ Y(with respect to Lebesgue measure), and calculate each of the iteratedintegrals

RX

RY f(x, y) dy dx and

RY

RX f(x, y) dx dy. Is the function f(x, y)

integrable on [0, 1]⇥ [0, 1]? Justify fully.

12. Let f 2 L1(R). Show that g(x) =R1�1 sin(xt)f(t)dt is continuous and

bounded.

13. Let T be a surjective linear map from a Banach space X onto a Banachspace Y such that

kTxk � 1

2017kxk

for all x 2 X. Show that T is bounded.

14. Find a linear functional L on C[�2, 2] such that there does not exist afunction g 2 C[�2, 2] with kgk1 = 1 and |L(g)| = kLk.

QUALIFYING EXAM: REAL ANALYSISSeptember 30, 2017

Prof. C. Beneteau

Prof. S.-Y. Lee

Prof. B. Shekhtman

Answer 4 questions from Part A and 3 questions from Part B.

Part A.

1. Let X be a set and B a �-algebra of subsets on X. Let {µn}1n=1 be a

sequence of (positive) measures on (X,B) such that µn+1(E) � µn(E) for

every E 2 B. Let µ(E) := limn!1 µn(E). Prove that µ is a measure on B.Is this conclusion still true if we assume that µn+1(E) µn(E)?

2. Consider the function f(x) = x2sin(1/x) defined on the interval [0, 1],

where we set f(0) = 0. Is f of bounded variation? Prove or disprove.

3. State and prove Fatou’s lemma, and give an example where strict inequality

occurs.

4. (a) Given a measure space (X,M, µ) and a sequence of integrable functions

{fn}1n=1, define what it means for the sequence to be uniformly integrable.

(b) If X = [0,1), M is all Lebesgue measurable sets, and µ is Lebesgue

measure, is the sequence fn(x) = n2�[0,1/n3)(x) uniformly integrable? Prove

or disprove.

5. State the Hahn Decomposition Theorem. Consider the setX = [�⇡/2, ⇡/2],

and let M be the �-algebra of Lebesgue measurable sets on X. Let f(t) =

sin(2t). Define, for any measurable set E,

µ(E) =

Z

E

f(t) dt,

where dt is the usual Lebesgue measure. Find the Hahn Decomposition for

this measure, and compute µ+(X), µ

�(X), and the total variation |µ|(X).

6. Let f and g be continuous functions on [0, 1] such that f(x) < g(x) for all

x 2 [0, 1]. Prove that there exists a sequence of polynomials pn such that

f(x) < pn(x) < g(x)

for all n, and pn ! g uniformly on [0, 1].

7. Let A = span{xn(1 � x), n = 1, 2, . . .} ⇢ C([0, 1]). Determine the closure

of A in the uniform norm.

Part B.

8. Define what a complex measure µ on a measure space (X,M) is, and

define what the total variation |µ| is. Prove that the total variation |µ| isa (positive) measure.

9. Let (X,M, µ) be a measure space. For 1 p < 1, define what Lp(X) is,

and state and prove Holder’s inequality.

10. Show that ZZ

[0,1]⇥[0,1]

1

1� xydxdy < 1.

Justify fully.

11. Let H be a Hilbert space, and let T be a continuous linear operator on H.

Let T⇤be the operator defined by

hT ⇤x, yi = hx, Tyi for all x, y 2 H.

Prove that T is invertible if and only if T⇤is.

12. Let X be a Banach space, and define the mapping J : X ! X⇤⇤

by

J(x)(f) = f(x)

for each f 2 X⇤and each x 2 X. The space X is called reflexive if J is

surjective. Prove that on a reflexive Banach space, every functional attains

its norm, that is, for every f 2 X⇤, there exists x 2 X with kxk = 1 such

that f(x) = kfk.

13. Let f : R ! R be a smooth function with a compact support. Show that,

for each positive integer N ,

����Z

Rsin(tx)f(t)dt

���� Cx�N

, x > 0

for some constant C > 0.

Real Analysis Qualifying Exam – January 27th 2018

Written by S.-Y. Lee and B. Shekhtman

Solve 8 out of 12 problems.

1. Assume that the function f is of bounded variation on [0, 1]. For each x 2 [0, 1], define v(x) to be thetotal variation of the restriction of f to [0, x]. Prove that if f is absolutely continuous then v isabsolutely continuous.

2. Let {fn} be a sequence of nonnegative Lebesgue measurable functions on [0, 1]. Show that {fn}converges to zero in measure if and only if

limn!1

Z 1

0

fn(x)

1 + fn(x)dx = 0.

3. Assume that the functions f and g are both integrable and bounded on R. Define the function h onR by

h(x) =

Z 1

�1f(x+ y)g(y)dy.

Prove that h is a continuous function on R, and that lim|x|!0 h(x) = 0.

4. Let f and g be real valued measurable functions on [0, 1] with the property that for every x 2 [0, 1], gis di↵erentiable at x and

g0(x) = f(x)2.

Prove that f 2 L1[0, 1].

5. Let f : R ! R be a C1 function with a compact support. Show that, for each positive integer N ,����Z

Rsin(tx)f(t)dt

���� Cx�N , x > 0

for some constant C > 0.

6. Show that ZZ

[0,1]⇥[0,1]

1

1� xydxdy < 1.

Justify your answer.

1

7. Let fn be a sequence of functions in L1([0, 1]) with respect to Lebegues measure such that, as n tendsto infinity, Z

|fn(x)| dx ! 0.

Does it follow that fn ! 0 a.e.. Give a proof or a counterexample.

8. Describe all intervals [a, b] ⇢ R such that span{x2n, n = 0, 1, 2, ...} is dence in C[a, b].

9. Show that for every continuous function f on [0,1)

limn!0

Zf(x)e�nxdx = 0.

10. Let T be a linear map from a Banach space X onto a Banach space Y such that

kTxk � 1

10kxk

for all x 2 X. Show that T is a bounded linear operator.

11. Let Y be a closed subspace of C([0, 1) and g /2 Y . Prove that there exists a signed Borel measure µon [0, 1] such that Z

fdµ = 0

for all f 2 Y andRgdµ = 1.

12. Show that l1 is not a separable space.

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