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 +

CONTINUOUS ALMOST EVERYWHERE
Definition 1. Let be a subset of R. Say that has
S measureP0 if, for each > 0, there is a
sequence of balls (Bj )
with
radii
r
,
such
that
j
j=1
j Bj and
j rj < .
Given a function f : [a, b] R, we say it is continuous almost e

Problem Set #10
Math 471 Real Analysis
Assignment: Chapter 8 #2, 3, 6, 8
Clayton J. Lungstrum
December 1, 2012
Exercise 8.2
Prove the converse of Holders inequality for p = 1 and p = . Show also that for
0
/ L1 (E).
real-valued f
/ Lp (E), there exists a

POWER SERIES ARE CONTINUOUS
Let (an ) be a sequence in C. Let = lim sup |an |1/n , and assume that < . Then, setting
R = 1/ (or R = + in the case = 0), we know that the series
f (z) =
X
an z n
(1)
n=0
converges absolutely for |z| < R (this is Theorem 3.39

THE WEIERSTRASS PATHOLOGICAL FUNCTION
Until Weierstrass published his shocking paper in 1872, most of the mathematical world
(including luminaries like Gauss) believed that a continuous function could only fail to
be differentiable at some collection of i

Two Propositions about Subsequences
Let (an ) be a bounded sequence in R. In this note, we prove two propositions mentioned
in the 18.100B lecture on March 9: that there is a subsequence tending to the lim sup, and
that there is a monotone subsequence.
Pr