HOMEWORK 3 FOR 18.100B/C, FALL 2010
SOLUTIONS
As usual, physical homework due in 2-108 by 11AM. Electronic submission (to
rbm at math dot mit dot edu) up to 5PM.
HW3.1 Let X be a set with the discrete metric
(
0 if x = y
d(x, y) =
1 otherwise.
Which subse

Math 563 - Fall 15 - take home final solutions
1. Let An be a sequence of independent events such that limn P (An )
exists. Prove that
1 X
[1A P (Ak )]
n k=1 k
n
converges in distribution to a normal random variable and find the mean and
variance of this

Math 563 - Fall 15 - Homework 6 Solutions
1. (from Durrett) Let Xn be a sequence of integer valued random variables,
X another integer valued random variable. Prove that Xn converge to X in
distribution if and only if
lim P (Xn = m) = P (X = m)
n
for all

Math 563 - Fall 15 - Homework 4
5. (Resnick) Let X be a random variable. A number m is called a median of
X if P (X m) 1/2 and P (X m) 1/2. You should convince yourself
that there is always at least one median, but it need not be unique. Recall
that the m

Math 563 - Fall 15 - Homework 2 - Solution
4. (from Durrett) Let X be a real valued function on and let (X) be
the -field generated by the sets X 1 (B) where B is a Borel set in R. (This
is the smallest -field with respect to which X is measurable.) Let Y

Math 563 - Fall 15 - Homework 1
6. (from Resnick) Let P be a probability measure on B(R), the Borel sets in
R. Prove that for any E B(R) and any > 0 there exits a finite union of
disjoint intervals A such that P (EA) < .
Hint: Define F to be the collectio

Math 563 - Fall 15 - Homework 3 - selected solutions
4. Prove that a random variable X is independent of itself if and only if there
is a constant c such that P (X = c) = 1. Hint: What can you say about the
distribution function of X?
Solution Consider th

MATH31011/MATH41011/MATH61011:
FOURIER SERIES AND LEBESGUE INTEGRATION
Chapter 3: Measure and Lebesgue Integration
Revision of Riemann Integration
One of the most important operations in mathematics is integration. The rst rigorous
treatment of integratio

MATH5011 Suggested Solution to
Exercise 2
(1) Let f be a non-negative measurable function.
(a) Prove Markovs inequality:
Z
n
o
1
x X : f (x) M
f d,
M X
for all M > 0.
(b) Deduce that every integrable function is finite a.e.
R
(c) Deduce that f = 0 a.e.

REAL ANALYSIS II HOMEWORK 2
CIHAN
BAHRAN
Stein & Shakarchi, Chapter 4
2. In the case of equality in Cauchy-Schwarz inequality we have the following. If
| hf, gi | = kf k kgk and g 6= 0, then f = cg for some scalar c.
Writing e =
g
,
kgk
we have |hf, ei| =

Measure zero and the characterization of Riemann integrable
functions
Jeffrey Schenker
July 25, 2007
Let us define the length of an interval I (open or closed) with endpoints a < b to be
`(I) = b a.
(1)
The extension of the notion of length to sets other

MATH414 FUNCTIONAL ANALYSIS
HOMEWORK 5
Hw 5: 5/140 8,9/141 1,2/150
Problem (5/140)
Show that for a sequence (xn ) in an inner product space the conditions |xn |
|x| and hxn , xi hx, xi imply convergence xn x.
Solution
|xn x|2 = hxn x, xn xi
= hxn , xn i

MATH 4010 (2014-15)
Functional Analysis
CUHK
Suggested Solution to Homework 5
Yu Mei
P140, 7. Show that in an inner product space, x y if and only if we have kx + yk = kx yk for all scalars
.
Proof. It follows from the definition of inner product that
kx

REAL ANALYSIS I BASIC MEASURE AND INTEGRATION
This course will follow a near minimal path through those aspects of the theory
of Lebesgue measure and integral used in virtually all applications in analysis and
probability theory.
We begin, not with abstra

MATH 1010
FINAL EXAM
MAY 10, 2012
NAME: _ A NSW E l2 KE\l
IMPORTANT: When you start, tear out the last two pages, which state full
version of the exam problems, including necessary denitions and lemmas, and also
hints. For space, exam pages only give abbr

Some Important Results from Analysis
H. Krieger, Mathematics 156, Harvey Mudd College
Fall, 2008
Denitions: Let cfw_an be a sequence of real numbers. Then we let
lim inf an = lim ( inf a ) = sup( inf a ).
n
n n
n
n
Note that if cfw_an is monotone, then

V
V
fi ( j ) =
ij
V
F
V
F
F
B = cfw_ (1 , , n
1 ,i = j
ij =
0 , i 6= j
V
V
fi
i
B = cfw_f1 , , fn
B
B
0 = c1 f1 ( i ) + + cn fn ( i ) = ci
B
V .
B
V
V
B = cfw_f1 , , fn
f = c 1 f1 + + c n fn = 0
f ( i ) = o( i ) = 0
i = 1n
B
fi (x1 , , xn )B ) = xi
f
f

Misc. Analysis Results
Prakash Balachandran
Department of Mathematics
Duke University
June 16, 2009
1
Analysis
1. Does there exist an infinite -algebra which has only countable many members?
Proof: No. Suppose otherwise. Then, there exists a -algebra F on

3 Measurable Functions
Notation A pair (X, F) where F is a -field of subsets of X is a measurable
space.
If is a measure on F then (X, F, ) is a measure space.
If (X) < 1 then (X, F, ) is a probability space and a probability
measure. The measure can, and

The Lebesgue Integral
Dung Le1
1
Simple Functions
We have seen that there are difficulties in applying the partition process for Riemann
integrals to functions which are badly discontinuous. Such examples include Q , the characteristic function of the rat

APM 504 - PS1 Solutions
1.1) Let P be a probability measure on (, F). We assume that all sets mentioned below are
measurable.
(i) Monotonicity: If A B, then B is the disjoint union of A and B
P(B) = P(A) + P(B A) P(A) since P(B A) 0.
(ii) Subadditivity: S

Austin Mohr
Math 704
Homework 1
Problem 1
Prove that the Cantor set C is totally disconnected and perfect. In other words, given two distinct points
x, y 2 C, there is a point z 2
/ C that lies between x and y, and yet C has no isolated points.
Proof. Let

Anlise
Moiss Toledo
11 de abril de 2012
Soluo de exerccios: 8.1 8.7 - Seo 8
Exerccio 1. O cone C = cfw_(x, y, z) R2 ; z 0, x2 + y 2 z 2 = 0 homeomorfo a
R2 .
Demonstrao.
(i) Seja a funo bijetiva
:
C
R2
(t cos , t sin , t) 7 (t, tan( x2 2 )
(ii) A inversa

Landon Kavlie
Math 711
Homework #4
26-October-2010
1. Show that if f : R R is differentiable everywhere, then f 0 is Borel
measurable.
Proof. First, note that since f is differentiable for all x R, f is continuous.
Therefore, f 1 (U ) is open if and only

.
LGEBRA LINEAR
ISBN 978-85-915683-0-7
ROBERTO DE MARIA NUNES MENDES
Professor do Departamento de Matemtica e Estatstica e do
Programa de Ps-graduao em Engenharia Eltrica da PUCMINAS
Belo Horizonte
Edio do Autor
2013
Sumrio
Prefcio
1
1 Espaos Vetoriais
1.

The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 6 (Week 7)
MATH3969: Measure Theory and Fourier Analysis (Advanced)
Semester 2, 2014
Web Page: http:/www.maths.usyd.edu.au/u/UG/SM/MATH3969/
Lecturer: Daniel Daners
Questi

CONDITIONAL CONVERGENCE
Here is a concrete example of conditional convergence. Recall that the series
X
(1)n+1
S=
n
n=1
converges (by the alternating series test), but not absolutely. (The fact that it converges to
a known limit ln 2 is irrelevent; it is

CONSTRUCTION OF R
1. M OTIVATION
We are used to thinking of real numbers as successive approximations. For example,
we write
= 3.14159 . . .
to mean that is a real number which, accurate to 5 decimal places, equals the above
string. To be precise, this m

1. 20) Construct an example of a sequence of nonnegative measurable
functions from R to R that shows that strict inequality can result in
Fatous Lemma.
Consider the sequence of functions fn (x) = [n,n+1] (x), an indicator
function of the interval [n, n +