AVU-PARTNER INSTITUTION
MODULE DEVELOPMENT TEMPLATE
Prepared by Thierry Karsenti, M.A., M.Ed., Ph.D.
With : Salomon Tchaméni Ngamo, M.A. & Toby Harper, M.A.
Version 19.0, Thursday December 21st, 2006
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C.
TEMPLATE STRUCTURE
I.
INTRODUCTION
1. TITLE OF MODULE
Analysis 2
2. PREREQUISITE COURSES OR KNOWLEDGE
Unit 4 Real Analysis
Analysis on the real line (unit 1)
Unit 5 Topology
Real Analysis (unit 3)
Unit 6 Measure Theory
Real Analysis unit 3 and unit 4
3. TIME
The total time for this module is 120 study hours.
4. MATERIAL
Students should have access to the core readings specified later. Also, they will need a
computer to gain full access to the core readings. Additionally, students should be able to
install the computer software wxMaxima and use it to practice algebraic concepts.
5. MODULE RATIONALE
The rationale of teaching analysis is to set the minimum content of Pure Mathematics
required at undergraduate level for student of mathematics. It is important to note that skill
in proving mathematical statements is one aspect that learners of Mathematics should
acquire. The ability to give a complete and clear proof of a theorem is essential for the
learner so that he or she can finally get to full details and rigor of analyzing mathematical
concepts. Indeed it is in Analysis that the learner is given the exposition of subject matter
as well as the techniques of proof equally. We also note here that if a course like calculus
with its wide applications in Mathematical sciences is an end in itself then Analysis is the
means by which we get to that end.
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II.
CONTENT
6. Overview
Overview
This module consists of three units which are as follows:
Unit 4 – Real Analysis
In this unit we define and give examples of general metric spaces which show that they
occur abundantly in mathematics. We then look at the structure of a general metric space
a long the lines of unit 1. In addition we introduce the concept of compactness and its
effects on continuity of functions.
Unit 5 – Topology
The structures of topological spaces are studied along side those of metric spaces.
However the main difference here is that the axioms that define a metric space are
dependent on the concept of distance whereas in the axioms that define a topological
space the concept of distance is absent.
In particular the study of the twin concepts of convergence and continuity brings out this
difference very well. Finally a look at different topologies like product or quotient topology
endowed on a set is essential in this unit.
Unit 6/7 – Measure Theory
In this unit we start with the study of both the Lebesgue outer measure and the real line
before we look at the Lebesgue measurable subsets of the real line. A look at the sigma
algebra of subsets of a given underlying set gives rise to measurable space on which we
can also study a class of functions called measurable functions. A part from the Lebesgue
measure we also study an abstract measure leading to an abstract measure space on
which we introduce an abstract integral. Finally a brief comparison of the Lebesgue
integral and the well known Riemann integral is also essential in this unit.
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Outline: Syllabus
Unit 4 - Real Analysis (30 hours)
Level priority Unit 1 is the Pre-requisite

Definition and examples of metric spaces

Nieghbourhoods, interior points and open sets

Limit points and closed sets

Dense and compact subsets

Compactness and continuity
Unit 5 – Topology (30 hours)
Level priority Unit 3 is the Pre-requisite

Review of metric spaces

Topological spaces, neighbourhoods, interior and open sets

Limit points closed sets, closure and boundary

Bases, relative and product topologies

Continuity and homeomorphisms

Convergence and Hansdorff axiom.
Unit 6 - Measure Theory (40 hours)
Level priority Unit 3 and Unit 4 are the Pre-requisite

Lebesgue outer measure and Lebesgue measure on real line

Lebesgue measurable subsets of 

Measurable spaces and measurable functions

Abstract measure spaces and abstract integral

Monotone convergence theorem, Fatou’s lemma and Lebesgue dominated
convergence theorem

Relation between Riemann Integral and Lebesgue Integral
Unit 7 - The Lebesgue Integral (20 hours)
Level priority Unit 6 is the Pre-requisite
 Define the Lebesgue integral
 Give examples of Lebesgue integrals defined on different sets.
 State some properties of Lebesgue integral
 Verify some properties of the Lebesgue integral
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Graphic Organiser
Space
Structure of a
metric space
Structure of a
topological
space
Functions on
metric spaces
Functions on
topological
spaces
Measurable
space
Measurable
functions
Measure
Space
Abstract
Integral
Continuity
and
compactness
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Continuity and
Homeomorphisms
5
8. Specific Learning Objectives (Instructional Objectives)
You should be able to:
1. Demonstrate understanding of basic concepts and principles of mathematical
analysis.
2. Develop a logical framework for stating and proving theorems.
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III.
TEACHING AND LEARNING ACTIVITIES
9. PRE-ASSESSMENT
1. Which of these sets is countable?
(a)
(b)
where
C
(c)
0,1 
(d) A 
  a, b 
is the set of rational numbers.
2. Which of these sets has both supremum and infimum
(a) A   x : 0  x  1
(b)
B   x :    x  1
(c) C   x : x  1
(d) D   x : x  1
1

3. Given the set A   , n  J   how many limit points does A have?
n

(a) 2
(b) 0
(c) 1
(d) Infinite
4. Given the sequence of real numbers
1 if n is odd
xn  
0 if n is even
which of the following statements is true about the sequence  xn  ?
(a) xn converges to 1
(c) xn is bounded
(b) xn converges to 0
(d) xn is Cauchy
5. Which of the following subsets of
(a) A   x : 1  x  2 
is an open set?
(b) B   x : 0  x  1
(c) C   x : 0  x  2 
(d) D   x : 1  x  2 
6. Which of the following subsets of
(a) A   x : 1  x  5 
is a closed set?
(b) B   x : 20  x  30 
(c) C   x : 0  x  1
(d) D   x : 6  x  0 
7. State which of the following subsets of
(a) A   x : 2  x  5 
(d)
(b)
is neither open nor closed.
B   x : 0  x  3
(c) C   x : 0  x  4 
D   x : x  0
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8. Which of the following statements about a function of bounded variation is true?
(a) It is continuous
(c) It is uniformly continuous
(b) It is discontinuous
(d) It is bounded
9. The following is a Riemann sum:

n
k
 sin n 
n k 1
Use this fact to evaluate
 n
k 
lim   sin  
n  n k 1
n 

(a) 2
(b) 0
(c) -2
 
10. Consider the function f :  ,   
4 
2
f  x   sin x  x

(d) -1
defined by


Using the intermediate value theorem determine the non-zero value of    ,   such
4

that:
2
sin  


(a)

4
(b) 
(c)
3

4
(d)

2
SOLUTIONS
A. Pre-assessment
1. d
7. c
2. a
8. d
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3. c
9. a
4. c
10. d
5. b
6. a
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10. LEARNING ACTIVITIES
ANALYSIS MODULE 2: UNIT 4
TITLE: Real Analysis
SPECIFIC OBJECTIVES
At the end of this activity the learner will be able to:




Define the concept of a metric on a given space.
Give examples of metric spaces
Demonstrate how continuity of a function is dependent on the structure of the space
on which it is defined.
Define and distinguish among various concepts that constitute the structure of a
metric space such as neighbourhoods, interior and limit points, open and closed sets
etc.
SUMMARY
It is of great significance to note that proper analysis as a branch of Pure Mathematics
starts with a critical look at the structure of a space. The classical theory was
developed heavily on the real line
whose structure is determined by the absolute
value function and the well known inequalities that hold true on it. Thus on
we can
define a map
:
  by
 x, x  0
x 
  x, x  0
which satisfies the following axioms:
(i)
a b  0
 a, b 
(ii)
a b  0
(iii)
a b  a  b
(iv)
a  b,  a  b
iff a  b
 a, b 
iff a  b
These axioms constitute a geometric structure on
since it refers to distance which
over the years has been used in Analysis without paying special attention to the
structure itself. In modern analysis this is where we start. Thus for a non-empty set X
we define a map d : X x X   called distance. Such that a number of axioms are
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satisfied (see the section on key words below). In this case we say that the non-empty
set X has been endowed with the geometric structure d (also called a metric) and the
order pair  X , d  is called a metric space. We therefore note that all the definitions of
the concepts or components of the structure of the space  X , d  are given in terms of
the metric d. it is also of great interest to note that even concepts like continuity of a
function defined on such a space is given in terms of the metric d.
KEY WORDS AND CONCEPTS


Space: This is a non-empty set which is furnished with either geometric or
algebraic structure.
Metric: This is a map X x X   defined on a non-empty set X such that:
d  x, y   0
(i)
 x, y  X
(ii) d  x, y   0 iff x  y
(iii) d  x, y   d  y, x 
 x, y  X
 x, y  X
(iv) d  x, z   d  x, y   d  y, z 
 x, y z  X
In this case  X , d  is a metric space.

Neighbourhood: In any metric space  X , d  if x0  X is any fixed point in X
then the neighbourhood (in brief neighbourhood) of x0 is given by

 


N x0 , r  x  X : d x0 , x  r

 for some real number r  0.
Interior point: In any metric space  X , d  if A  X then a point p  A is said to
be an interior point of A if  a neighbourhood say N  p, r  of p such that
N  p, r   A.

Interior of a set:
For any subset A of
X, d
interior of A is given by
A0   x : x is an interior point of A}.




Open set: Any subset A of
X, d
is said to be open if every element of A is an
interior point of A.
Limit point: In any metric space  X , d  if A  X then a point p  X is said to
be a limit point or cluster point or accumulation point of A if every neighbourhood
of p contains at least one point of A (different from p if p  A ).
Closed set: Let A be a subset of  X , d  . Then A is said to be closed if every
limit point of A belongs to A.
Closure of a set: For any subset A of  X , d  closure of A is given by
A  A  x : x is a limit point of A}.
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
Bounded set: Any set A in  X , d  is bounded if for a fixed p  X  M  0 such
that d  p, x   M

 x A.
Open cover: Let A be a subset of  X , d  and let  E    be a collection of
open subsets of
X, d
such that A 

 
E . Then the family
 E   
is
called an open cover for A. If also for i , i  1,..................., n we have that
n
n
A   E , then E
is called a finite open sub cover for A.
i 1
i
 
i
i 1

Compact Set: Let A be a subset of  X , d  . Then A is said to be compact in X if

every open cover for A has a finite open sub cover for A.
Pointwise continuity of a function: A function f :  X , d x   Y , dY

metric space  X , d x  into another metric space Y , dY

 from a
 is said to be continuous
at a point x0  X if for each   0    0 such that



  

d X x0 , x    dY f x0 f  x   
Where d for all x  X and d denote metrics on X and Y respectively.
Y
X
Uniform continuity of a function: A function f :  X , d x   Y , d y is said to be


uniform continuous on X if for each   0    0 (depending only on  ) such
x, y  X
that
for
any
pair
of
points
we
have:
d X  x , y     dY
 f  x  f  y 
 
4. Learning Activity: Real Analysis: On the Geometric Structure of a Space.
The Story of Building a House
A story has been told of a young man called Mr. Waweru from one of the villages in Kiambu
district in Kenya. He one day constructred a house with rectangular walls but he had no
idea that he had to ensure that parallel walls were to be of same lengths and that adjacent
walls were to be at right angles to each other. He ignored these factors and after he had
finished thatching the house he tried viewing it from different corners and found out that it
was not as rectangular as he would have wished it to be. He also found out that even
putting furniture in the house there was a problem how different furniture could be
arranged.
Picture showing a man roofing a house.
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Question
Write a short passage to explain the mathematics that Mr. Waweru should have considered
when he was building his house?
Introduction
The most important message we get from the story above is that measurements of both
length and angles is essential and it is what constitutes the structure that comes out of
geometry. Indeed the geometric structure will create symmetry in any arrangement. In the
story above geometry was lucking this is why even the arrangement of furniture was a
problem.
In a metric space as seen earlier we have a non-empty set endowed with a metric which in
itself is a concept of distance and hence gives rise to a geometric structure. Thus, every
component of the structure must be defined in terms of distance. If you look back at all the
definitions of the key words and concepts are given sequentially and it terms of distance.
In fact we can summarise them in form of the following diagram.
neighbou
rhood
Interior
point
Interior of a
set
Open set
Limit point
Closed set
Closure of a
set
Examples 4.1
(i)
The set
of real numbers is one of the most common examples of metric
spaces. Here we define the metric as:
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d  x , y  x  y
(ii)
The set
 x, y  .
of complex numbers is also a metric space with the metric defined as


d z1, z2  z1, z2
(iii)
 z1 , z2 
The n-dimensional Euclidean space
defined as
d  x, y 
n

i 1
xi  yi
n
is also a metric space with the metric d
 x, y  n


y   y , y ,............, yn 
1 2
Where x  x , x ,............, xn
1 2
Remarks 4.2
(i)
(ii)
Note that on a given set we can define more than one metric.
Note also that every definition has its negation. In general if the demand of the
definition is that “for every ……..”. then its negation is that “there exists at
one……”. For instance we have already seen that a set is open if every member
of the set is an interior point. In this case its negation is that “a set is not open if
there exists at least one member of the set which is not an interior point”.
Exercise 4.3
(i)
(ii)
Try to give the negation of all definitions in this activity and any other related
definitions you have come across elsewhere.
Use definitions and their negations to show that the set A given below is neither
open nor closed
A   x  : 0  x  1
Also state the closure of A denoted by A .
(iii)
Show that any set A is open iff A  Int A and that A is closed iff A  A .
Remark 4.4
Note that definitions alone cannot tell how all the concepts are related to each other. We
also need theorems to do this. The following theorem gives us the relationship between an
open set and a closed set and also the relationship between a compact set and a closed
set.
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Theorem 4.5
(i)
(ii)
In any metric space  X , d  a set A is closed iff Ac is open.
A is compact implies A is closed and bounded. The converse is true if the space
is n .
Remark 4.6
Note that the structure of a space such as a metric space can also be examined through
mappings on the space, in particular the continuous ones. To this end we have the
following theorem.
Theorem 4.7
Let X and Y be metric spaces and f : X  Y be a function. Then we have
(i)
(ii)
If X is compact and f is continuous then f is uniformly continuous.
If X is compact and f is continuous then the range f  X  is also compact.
Exercise 4.8
(i)
(ii)
Look for a counter example in your readings to show that the converse of
theorem 4.5 part (ii) is not true in general.
Let f :  0,1  
be given by f  x   x 2  x  . Show that f is uniformly
continuous on  0,1 .
(iii)
Let X be a metric space which is compact and f : X 
function.
Define the numbers m and M by m  x  X f  x  and
be a continuous
M  lub f  x  .
x X
Show that  p, q  X such that f  p   m and f  q   M
4.9
Readings
1. The student should read Mathematical Analysis 1 by Elias Zakon, The Trillia Group.
Read the sections fully and complete the exercises.
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ANALYSIS 2 MODULE: UNIT 5
LEARNING ACTIVITY 3
TITLE: Topology
SPECIFIC OBJECTIVES
At the end of this activity the learner will be able to:





Define a topology on a set
Give examples of topological spaces
Recast the definitions of the components of the structure like neighbourhoods,
interior point, limit point etc without reference to the concept of distance
Recast definition of continuity of a function without reference to the concept of
distance
State and prove properties of continuity purely in terms of subsets of the space
SUMMARY
We summarise this activity along the same lines as in activity 2 in which we saw that
structure of a metric space is heavily depended on the concept of distance. In the case
of a topological space (see definition in the next section) the structure depends on both
open and closed sets.
Thus our main task in this activity is to see how we can recast those definitions in the
metric space without making any reference to the concept of distance. Since we note
that in a set based structure the concept of distance is generally absent. It is also
important to note that any definition in a metric space which does not refer to distance
can still carry through in a general topological space.
We also note that the concept of continuity of functions in particular acquires
considerable generalization when its definition and properties are stated in terms of
open sets. This is significant since the concept of continuity has exceptional properties
which carry a lot of applications in mathematics.
KEY WORDS AND CONCEPTS

Topology: Let X be any non-empty set and  be a collection of subsets of X. Then
 is called a topology on X if the following properties are satisfied
 
(i)
(ii) X 
(iii)
n
 Oi  whenever Oi   i  1,......., n .
i 1
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 O  whenever O  for each    .
 
(iv)

Open set: Let A be a subset of a topological space  X ,  , then A is said to be open

if A  .
Closed set: Let A be a subset of a topological space
 X ,  , then A is said to be
closed if its complement denoted by C  A   .

Neigbourhood: Let  X ,  be a topological space and p  X . Then a subset N of
X is called a neighbourhood of p if N contains an open set O which contains p. i.e.
p  O  N for O  . In this case p is also called an interior point of the subset N.

Interior of a set: Let A be a subset of a topological space  X ,  , then interior of A

denoted by A0 or Int A is the set of all interior points of A. clearly Int A  A.
Limit Point: Let  X ,  be a topological space and A be a subset of X. Then a point
x  X is said to be a limit point of A if for each neighbourhood N of x we have that
NA   .

Closure of a set: For any set A in  X ,  closure of A denoted by A is given by
A  A  x : x is a limit poitn of A . Clearly A  A

Boundary of a set: For any subset A of a topological space X, boundary of A
denoted is given by B ' dary  A  A  C  A  .

Continuity of a function: Let

 

 X ,1 

and Y , 2

be two topological spaces and
f : X ,1  Y , 2 be a function. Then f is said to be continuous on X if f 1  0  is
open in X for every open set O in Y. Where f 1  0    x  X : f  x O . Thus f is

continuous if f 1  O   1 whenever O   2 .
Homeomorphism: Let X and Y be topological spaces. If both f : X  Y and its
inverse f 1 : Y  X are continuous then f is called a homeomorphism.
5. Learning Activity: On Set Based Structure of a Space.
Story of a designer
One day while walking in the streets of Nairobi, the Capital City of Kenya I came
across a designer who was using a drawing package (OpenOffice Draw) to create
some symmetrical images.
She was doing this in stages whereby she first drew a star then produced about a
hundred copies of them. She then assembled the copies into a symmetrical image.
Stage 1
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Stage 2
Question
Can you make a suggestion of any other shape whose copies can be assembled to create
a symmetrical pattern?
Introduction
We note that in our learning activity 2 we referred to some geometric structure which
accounts for symmetry or beauty. However, in our story above we have a case where
beauty or symmetrical images can be created without necessarily using the concept of
distance. The structure here depends on some arrangement of collections of items in the
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space. This is a situation that is analogous to the structure of a general topological space
which is dependent on sets and their collections. Indeed the definitions in a topological
space as we have already seen depend on components like neighbourhoods, open sets or
closed sets or their collections. It is also important to note that the definitions in metric
spaces which do not refer to the concept of distances will remain unchanged in general
topological spaces. For instance definitions like, open cover, compact set etc are intact
whether in metric space or topological space.
In this connection the set based structure on a topological space re-organises the
sequence of concepts in such a way that can be summarized by the following diagram.
Open set
Neighbourhoods
Closed set
Interior of
a set
Limit point
Closure of
a set
Boundary
of a set
Example 5.1
The following are examples of general topological spaces.
(i)
Let X  a, b, c. Define
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1   , X , a, b , b , a , b, c 
 2   , X 
 3  All subsets of X 
Then 1 , 2 , and  3 are all topologies on X. Where  3 is called the finest topology
on X and  2 is called the coarsest topology on X.
Thus
 X ,1 ,  X , 2 , and  X , 3  are topological spaces.
(ii)
Consider X 
with its standard metric an let
  All open intervals in
(iii)
.
Then
Let X be any metric space and let
  All open subsets of X 
is a topological space.
Then  X ,  is a topological space.
In order for the learner to understand example 5.1 part (iii) better we now work
out the following exercise together.
Exercise 5.2
In any metric space X show that:
(i)
(ii)
(iii)
(iv)
The null set  is open
The whole space X is open
n
 Oi is open whenever Oi is open  i  1,..........., n
i 1
 O is open whenever O is open for each
 
  .
Solution
(ii)
If  is not open then  x   which is not an interior point. But there is no such x
in  since  is empty. This is a contradiction. Hence  is open.
For any point p  X  r  0 such that the neighbourhood N  p, r   X . Thus any
(iii)
point p  X is an interior point. Hence X is open.
Assume O1 , O2 ,..........., On are open.
(i)
Let
n
O =  Oi
i 1
and
p
be
any
point
in
O.
Thus
p O
implies
p  Oi  i  1, ..........., n.
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Thus  nbhs




N p , ri such that N p , ri  Oi  i  1,........., n.
Let r  min ri . Then the nbh N  p , r  is such that:
1 i  n

n
p  N  p , r   O =  Oi
i 1
n
Hence O =  Oi is open.
i 1
Let O be open for each   and let O =  O .
 
Now p  O  p  O for some   . But O is open. Hence  a nbh say
(iv)
N  p , r  of p such that
p  N  p,r   O   O i.e. p is an interior point of O.
 
Hence O   O is open.
 
Remark 5.3
We note from the solution of the exercise above that if  is the collection of all open
subsets of a metric space then we have that:
(i)   
(ii) x  
n
(iii)  Oi  , whenever Oi   i  1,......., n.
i 1
 O  , whenever O  for each   .
 
This is why every metric space is an example of a topological space
(iv)
Now try the following exercise.
Exercise 5.4
(a) In any metric space X show that:
(i)  is closed
(ii) X is closed
(iii)
(iv)
n
 Fi is closed whenever Fi is closed  i  1,......., n.
i 1
 F is closed whenever F is closed for each   .
i 1 
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(b) Give reasons why we can not use the collection of closed sets to define a topology on a
metric space X.
We now state some theorems which give us deeper relationship among some concepts in
a topological space.
Theorem 5.5
Let X be a topological space. Then we have:
(i)
(ii)
(iii)
(iv)
For each x  X ,  a neighbourhood N of x in X.
If N is a neighbourhood of x  X then x  N .
If for each, neighbourhood N of x we have N   N then N  is also a
neighbourhood of x.
If for each x  X both N and M are neighbourhoods of x then N  M is also a
neighbourhood of x.
Theorem 5.6
Let X be a topological space. Then a subset O of X is open iff O is a neighbourhood of
each of its points.
Theorem 5.7
Given a subset A of a topological space X we have:
(i)
(ii)
If F is a closed set such that A  F , then A  F .
If  F   is a family of closed sets with each one of them containing A, then
A   F .
 
(iii)
If O 

is a family of open subsets of X with each one of them contained in A
then Int A   O .
 
Remark 5.8
Note that part (i) of the theorem above simply asserts that “every closed set F that contains
A also contains closure of A” while part (ii) simply says that: “Closure of A is the smallest
closed set that contains A”. We also note that part (iii) asserts that interior of a set is the
largest open subset of the set.
Exercise 5.9
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Using the readings try to give the proof of theorem 5.6 above.
Remark 5.10
Note that the definition of continuity on a topological space can also be stated in terms of
closed sets. Thus f : X  Y is continuous if f 1  A  is closed in X whenever A is closed
in Y.
It is in view of this equivalent definition that the theorem stated below can be proved.
Theorem 5.11
Let X and Y be topological spaces and f : X  Y be a function. Then f is continuous iff for
any subsets A of X we have
f A   f A 
Exercise 5.12
(a) For any two subsets A and B of a topological space X show that
(i)
A B  A B
(ii) A  B  A  B
(b) By expressing closure of a set A as:
A  A  B dary  A ,
show that:
A is closed iff B dary  A   A
(c)
Make a brief comparison between continuity of a function in metric spaces and
continuity of a function in general topological spaces.
(d) Consider the open interval (0,1) with any other open interval say
X   0,1 and Y   a, b  in
 a, b 
in
. Let
be endowed with the standard topology.
Let also f : X  Y be defined by f  t   a 1  t   bt  t   0,1
Show that f is a homeomorphism from X onto Y.
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ANALYSIS 2 MODULE: UNIT 6
TITLE: On Set Based Structure with Measures.
SPECIFIC OBJECTIVES
At the end of this activity the learner will be able to:
 Give definition of the Lebesgue outer measure and state its properties.
 Give definition and examples of Lebesgue measurable sets.
 Define and give examples of   algebras.
 Define and give examples of measurable spaces.
 Define and give examples of measurable functions.
 Define and give examples of measure spaces.
SUMMARY
We have already seen in the last two activities that a space is a non-empty set endowed
with either geometric or algebraic structure. Indeed, in the case of a metric space  X , d 
we have a geometric structure since d is distance while in the case of a topological space
 X ,  we have an algebraic structure since the analysis in X depends on the collection 
of sets and their combinations. In this activity we look at a measurable space along the
same lines and then extend this to the concept of a measure space.
We also define a class of functions called measurable functions and study their properties
in order to understand the structure in a deeper sense. We note here that like the case of a
topological space, the structure of a measurable space also depends on collections of sets.
However, these collections of sets are such that we can introduce a measure on them.
This in itself suggests that the structure of a measure space is both geometric and
algebraic since in a measure we have the concept of length.
KEY WORDS AND CONCEPTS

Lebesgue Outer Measure: Let   I  denote the length of an interval I and
  r  
  I 
I r
where r is a countable collection of open subintervals of . For any subset E of .
we denote the class of all countable collections of open subintervals of
which
cover E by C  E  . Then the number given by
  E  
inf
r C  E 
  r 
is called Lebesgue outer measure of the Set E.
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23

Lebesgue measurable set: Any subset E of
if for any other subset y of
we have that:

   y     Y  E     Y  E C

is said to be Lebesgue measurable

  algebra: Let X be a non-empty set and  be a collection of subsets of X such
that:
(i)   
(ii) If   x then AC  



(iii) If  An  n1 is a sequence of elements of  , then  An   .
n1
Measurable space: Let X be a non-empty set and  denote the   algebra of all
subsets of X. Then the ordered pair  X ,   is called a measurable space. Thus

any member of  is called  -measurable set.
Measurable function: Let  X ,   be a measurable space and f : X  Re be an
extended real valued function. Then f is said to be  -measurable if the set:
x  X : f  x   r 
r .

Measure: Let  X ,   be a measurable space. The restriction of the outer measure

to only members of  is called a measure.
Measure Space: Let  X ,   be a measurable space and  be a measure
on  X ,   . Then the ordered triple  X ,  ,   is called a measure space.
6. Learning Activity: On Set Based Structure with Measures.
Story of Designing Symmetrical Patterns
In our activity 3 we had a story of a designer who could create some symmetrical
images by simply generating patterns without reference to distance or any
measurements. In this story we draw your attention to the fact that there are times
when
designers
also
require
measurements in order to create
symmetrical images. While walking on
the streets of Nairobi one day I also
came across such a designer who would
start with say a number of squares on a
line then measure some distance and
angle before putting other figures like
equilateral triangles on a line e.t.c. until a
complete figure like a hexagon is
created.
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Question
Apart from generating a hexagon whose sides has different shapes of figures can you
generate other patterns?
6.1 Introduction
We note that in the story above the designer is not just concerned with producing
copies of a given figure like a star but has also to use measurements of length and
angles to generate a pattern as seen in the diagram above.
This is a case which best relates to a structure that combines both geometry and
algebra in a space. In Measure Theory we deal with a space called Measure Space (as
already defined in previous section) where we have the ordered triple  X ,  ,   . We
note that apart from the underlying set X itself, the   algebra  is about
combinations of subsets of X which is the algebraic aspect of the space. However the
measure  is about measurements on the subsets of X. For example if I is an interval
in
, then   I  is simply the length of the interval I.
Thus  constitutes the
geometric aspect of the structure of a measure space. It now follows that the
definitions of concepts in a measure space have to depend on these basic components
of the structure in the space. In fact we can summarise the sequence of concepts in
this space as follows.
Underlying set
X
Outer Measure
  algebra


Measurable set
Measurable
space
X,  
Measure

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Measure space
 X,  ,
Measurable
function
25
Example 6.2
Let X  , the class of all Lesbegue measurable subsets of is denoted by  m
and we have that the null set  ,
and belong to  m .
Also  m is a  - algebra of subsets of .
(ii) Let    and    denote the class of all subsets of and the class of the Borel
subsets of .
Then    and    are also  - algebra and we have that:
(i)
    m     .
Note that we also have that:
   *  m is the Lebesgue measure on .
(iii) Let X  and      or    m or    

(iv)
,
 ,  ,  m
and

,

,
then
we
have
that
are measurable spaces.
,    m and    *  m be the Lebesgue measure on
ordered triple  ,  m,   is a measure space.
Let X =
. Then the
Example 6.3
Let E be a measurable subset of X.
Consider the function;  E defined by:
1 if
0 if
E  x  
xE
x  EC
We show that  E is measurable, where  E is called the characteristics function of the
set E.
Indeed, for r  1, the set:
 x  X : E
 x

r  X   .
For 0  r  1, the set:
 x  X :  E  x   r   EC   .
Also for r  0, the set:
 x  X : E  x   r     .
Hence for all r 
we have that the set:
 x  X : E  x  r 
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
26
Thus  E is measurable.
Note that the class of all measurable functions on a measurable functions on a
measurable space  X ,   is denoted by M  X ,   .
In this case  E  M  X ,   .
and    m then we say that  E is a Lebesgue Measurable function.
If X 
Exercise 6.4
and    m sketch the graph of  E and use it to
verify that  E is indeed Lebesgue measurable.
In the example above letting X 
We now state some theorems which give us properties and some relationship among
some concepts in measurable spaces and measure spaces.
Theorem 6.5
Let   be the Lebesgue outer measure on the subsets of
(i)
. Then we have:
     0
(ii)  x  0  x 
(iii)   is a monotone i.e.    A     B  for A  B
(iv)   is translation invariant. i.e.     r  E       E 
where  r  x   x  r
x 
(v)   is countably sub-additive i.e. if


 En 

n 1
is any sequence of subsets of
then
    En       En 
n1

 n1
Remarks 6.6
(i)
Note that apart from the properties stated above we have from definition of
 :  
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
 that    0.
e
27
(ii)
Note also that the main difference between the properties of the Lebesgue outer
measure   on
and the Lebesgue measure  on
is that  is countably

additive. Thus if  En n1 is a sequence of subsets of


   En      En .
 n1
 n1
then
Theorem 6.7
Let  X ,   be a measurable space and f , g : X 
constant. Then we have:
(i)
(ii)
(iii)
e
be  measurable. Let c be any
f  g is  measurable
c  f is  measurable
cf is  measurable
Exercise 6.8
(a) Let E be a countable set in the sense that we can arrange it as:
E   x1 , x1 ,.................
(i)
Show that    E   0
(ii)
Show that for the set
of rational numbers   
  0.
(b) From the definition of a   algebra  , show that
(i) X  

(ii)  An  
n1
(c) (i) Let  X ,   , Y , Y  be two measurable spaces and f : X  Y be a function. Give
definition of measurability of f along the lines of definition of continuity in topological
spaces.
(ii) Show that if f :

is continuous then f is Lebesgue measurable.
(d) (i) Show that if a function f is measurable then f is also measurable.
(ii) Let f : A 
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where A  m be given by
28
 x A
 1
f  x  
 x  AC
1
Show that f is Lebesgue measurable but f is not.
Readings 6.9
1. Theory of functions of a real variable by – Shlomo Stenberg (2005) pp 95 – 116
and pp 133 – 156.
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ANALYSIS 2 MODULE: UNIT 7
Title: The Lebesgue Integral
Specific Objectives
At the end of this activity the learner will be able to:




Define the Lebesgue integral
Give examples of Lebesgue integrals defined on different sets.
State some properties of Lebesgue integral
Verify some properties of the Lebesgue integral
Summary
We have already seen in unit one of Analysis 1 Module how the Riemann integral is
developed and the restrictions on the nature of the functions that are Riemann integrable.
Indeed Riemann integrable functions are subject to rather stringent continuity conditions.
However the Lebesgue theory enables us to integrate a much larger class of functions. Its
greatest advantage lies perhaps in the ease with which many limit operations can be
handled and from this point of view the Lebesgue convergence theorems may well be
regarded as the core of the Lebesgue theory.
One of the difficulties which is encountered in the Riemann theory is that limits of Riemann
integrable functions (or even of continuous functions) may fail to be Riemann integrable.
This difficulty is eliminated in Lebesgue theory since limits of measurable functions are
always measurable.
Key Words and Concepts

Simple function: Let f : X  be a function whose range is finite. Then f is called
a simple function. For example a constant function is a simple function.

Canonical representation of a simple function: Suppose the range of a simple
function f consists of the distinct non zero real numbers a1 , a2 ,.......an . Let
f 1  ai   Ei  i  1,................., n, so that f  x   ai  x  Ei .
Ei  E j   for i  j
representation as:
Then
f 
n
 ai
i 1
and the simple function f has the canonical
E
i
where  E is the characteristic function on Ei  i  1,................., n.
i
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
Integral of a simple function: Let  X ,  ,   be a measure space and let Q be a
simple function in M   X ,   . Thus
is non-negative  measurable simple
function. Let Q have the canonical representation as

n

k 1
ak  E .
k
Then the extended real number given by
n
 ak   Ek 
k 1
is called the integral of the simple function Q with respect to  and is denoted by


Q d
X
Abstract integral: Let  X ,  ,   be a measure space and f : X  e be any
function in M   X ,   . Then the integral of f denoted by  f d  is given by
X


f d   sup   S d  
X
 X

where the supremum is taken over all simple functions S satisfying the condition

S  x  f  x  x  X .
If E   , then the integral
If sup  s d     then
s f
E f
d 
X f
E d 
 f d    .
If  is the Lebesgue measure then
f
d  is called the Lebesgue integral of the
function f.

General case of abstract integral: Let  X ,  ,   be a measure space and
f : X  e be  measurable. Let
f x   max  f  x  , 0
x X

f  x   max  f  x  , 0
x X



where f  f  f
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
and f  f   f 
31
Then if
f

f
d  and

d  are finite we say that f is integrable and the integral of f
with respect to  is given by:
 f d   f

d   f  d
we then write f  L  X ,  ,   . If E   then we have
E f
If

f

d 
E f

f

d    and
d 
E f

d
d    then
f
d    and in this case f  L  X ,  ,   .
Property of  -almost everywhere  .a.e  : Let  X ,  ,   be a measure space. A
property P is said to hold true .a.e on X if there is a subset N of X with   N   0
such that P fails on N, but P is true on X  N.
7
Learning Activity The Lebesque Integral
Story of felling a tree
It is a story told in a village called Luyia of Transzoia district in Kenya
that about fifty years ago a man called Mulunda wanted to build a
house. He took only a panga to try and fell a big tree for this
purpose. One passerby just laughed at him wondering why Mulunda
had embarked on an almost impossible mission. The second
passerby sympathised with Mulunda so much that he went and
borrowed an axe for him. While using an axe Mulunda realized that
it took him a short time to fell the tree. He also admitted his ignorance of some neighbours
owning tools like axes which can make work easier.
Question
Apart from an axe can you think of other tools that can be used to fell a big tree?
7.1 Introduction
We note that the main message from the story above is that Mr. Mulunda had owned a
panga for a long time over trusting it with every work without being innovative enough to
think of other tools. Indeed he only needed to consult with some of his neighbours to know
that some jobs are beyond certain tools. It is significant to note that mathematical tools
also behave in a similar manner. There are concepts which are beyond certain techniques.
Indeed the Lesbegue theory enables us to evaluate integrals of a large class of functions
than the Riemann theory would do. It is a well known fact that the limit of a Riemann
integrable function need not be Riemann integrable. However, in the case of the Lebesgue
integral the limit of a Lebesgue integrable function is still integrable.
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We now state without proofs some of the most important theorems on monotonicity and
convergence of the Lebesgue integral.
Theorem 7.1
Let  X ,  ,   be a measure space. Then we have:
(i)
If f , g  M   X ,   such that f  g then
(ii)
If E , f   with E  F then
E f
d 
 f d   g d
F f
d
Remark 7.2
Note that the result above shows that the Lebesgue integral is monotonic with respect to
functions and sets.
Theorem 7.3 (Monotonic convergence theorem)
Let  X ,  ,   be a measure space and
that
 fn 
be a sequence of functions in M   X ,   such
 fn  is monotonic increasing and converges to a function f
lim
n 
.a.e on X. Then we have:
f d
 fn d     nlim
 n 
  f d
Theorem 7.4 (Fatou’s Lemma)
Let
 X ,  ,   be
a measure space and
 fn 
be a sequence of functions in M   X ,   .
Then we have


f n  d   lim  f n d  .
  nlim

n 

where lim denotes limit infimum.
Corallary 7.5
If  fn  in theorem 7.4 above converges to f then Fatou’s Lemma gives
f
d   lim
n 
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d
33
Remark 7.6
Note that in corollary 7.5 above since f n converges to f we have that
lim f n  f
Hence
f
d    lim f n d   lim
 f n d .
Remark 7.7
Let  X ,  ,   be a measure space and f : X  e be  - measurable. Then we have
f  L  X ,  ,   iff f  L  X ,  ,  
Furthermore,
 f d 

f d .
Theorem 7.8 (Lebesgue dominated convergence theorem)
Let  X ,  ,   be a measure space and
which converges .a.e to f.
 fn  be a sequence of measurable functions on X.
Let g  L  X ,  ,   such that f n  x   g  x  .a.e on X.
Then
 fn d     nlim

lim
n 

fn d 
  f d
Remark 7.9
Note that in the Monotone convergence theorem there is no analogue for a monotonic
decreasing sequence of functions  fn  as the example below shows.
Example 7.10
Consider the measure space

, B,  B
 where  B
is the Borel measure on
.
Let  fn  be a sequence of functions given by
fn 
1

n  n,  
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where  n,  is the characteristic function of the interval  n,   . We first note that
 
lim f
n  n
 0.
In fact f n  0 function, uniformly on
.
1
Indeed, f n  x   0 

 0
n  x
 n,  

1
1
   n     1 and  x  .
n
 
However
f
d    0 d   0;
whereas
1
 fn d    n 
d
 n,  
1

d     n J 
n
 n,  

Thus
lim
n 
 fn d     0   f d  .
Exercise 7.11
Let
 X,  ,
be a measure space and f : X 
be such that g  L  X , x ,   and f  g .
e be
measureable. Let also f : X 
e
Use the Lebesgue Dominated convergence theorem to show that
f  L  X ,  ,  .
7.12 Readings
1. Theory of functions of a real variable by – Shilomo Stenberg (2005) pp 134- 140.
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11. COMPILED LIST OF ALL KEY CONCEPTS (GLOSSARY)


Space: This is a non-empty set which is furnished with either geometric or
algebraic structure.
Metric: This is a map X x X   defined on a non-empty set X such that:
(i) d  x, y   0
 x, y  X
(ii) d  x, y   0 iff x  y
(iii) d  x, y   d  y, x 
 x, y  X
 x, y  X
(iv) d  x, z   d  x, y   d  y, z 
 x, y z  X
In this case  X , d  is a metric space.

Neighbourhood: In any metric space  X , d  if x0  X is any fixed point in X then
the neighbourhood (in brief neighbourhood) of x0 is given by

 


N x0 , r  x  X : d x0 , x  r

 for some real number r  0.
Interior point: In any metric space  X , d  if A  X then a point p  A is said to
be an interior point of A if  a neighbourhood say N  p, r  of p such that
N  p, r   A.

Interior of a set:
For any subset A of
X, d
interior of A is given by
A0   x : x is an interior point of A}.




Open set: Any subset A of
X, d
is said to be open if every element of A is an
interior point of A.
Limit point: In any metric space  X , d  if A  X then a point p  X is said to be
a limit point or cluster point or accumulation point of A if every neighbourhood of p
contains at least one point of A (different from p if p  A ).
Closed set: Let A be a subset of  X , d  . Then A is said to be closed if every limit
point of A belongs to A.
Closure of a set: For any subset A of  X , d  closure of A is given by
A  A  x : x is a limit point of A}.

Bounded set: Any set A in  X , d  is bounded if for a fixed p  X  M  0 such
that d  p, x   M

 x A.
Open cover: Let A be a subset of
open subsets of
X, d
such that
 X , d  and let  E    be a collection
A   E . Then the family  E   
 
of
is
called an open cover for A. If also for i , i  1,..................., n we have that
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n
A

i 1


 
E , then E
i
i
n
is called a finite open sub cover for A.
i 1
Compact Set: Let A be a subset of  X , d  . Then A is said to be compact in X if
every open cover for A has a finite open sub cover for A.
Pointwise continuity of a function: A function f :  X , d x   Y , dY from a

metric space  X , d x  into another metric space Y , dY


 is said to be continuous at
a point x0  X if for each   0    0 such that



  

d X x0 , x    dY f x0 f  x   
Where d for all x  X and d denote metrics on X and Y respectively.
Y
X
Uniform continuity of a function: A function f :  X , d x   Y , d y is said to be


uniform continuous on X if for each   0    0 (depending only on  ) such that
for any pair of points x, y  X we have: d X  x , y     dY  f  x  f  y    

Topology: Let X be any non-empty set and  be a collection of subsets of X. Then
 is called a topology on X if the following properties are satisfied
 
(i)
(ii) X 
(iii)
(iv)
n
 Oi  whenever Oi   i  1,......., n .
i 1
 O  whenever O  for each    .
 

Open set: Let A be a subset of a topological space  X ,  , then A is said to be open

if A  .
Closed set: Let A be a subset of a topological space  X ,  , then A is said to be
closed if its complement denoted by C  A   .

Neigbourhood: Let  X ,  be a topological space and p  X . Then a subset N of
X is called a neighbourhood of p if N contains an open set O which contains p. i.e.
p  O  N for O  . In this case p is also called an interior point of the subset N.

Interior of a set: Let A be a subset of a topological space  X ,  , then interior of A

denoted by A0 or Int A is the set of all interior points of A. clearly Int A  A.
Limit Point: Let  X ,  be a topological space and A be a subset of X. Then a point
x  X is said to be a limit point of A if for each neighbourhood N of x we have that
NA   .
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
Closure of a set: For any set A in  X ,  closure of A denoted by A is given by
A  A  x : x is a limit poitn of A . Clearly A  A

Boundary of a set: For any subset A of a topological space X, boundary of A
denoted is given by B ' dary  A  A  C  A  .

Continuity of a function: Let

 

 X ,1 

and Y , 2

be two topological spaces and
f : X ,1  Y , 2 be a function. Then f is said to be continuous on X if f 1  0 
is open in X for every open set O in Y. Where f 1  0    x  X : f  x O . Thus f

is continuous if f 1  O   1 whenever O   2 .
Homeomorphism: Let X and Y be topological spaces. If both f : X  Y and its
inverse f 1 : Y  X are continuous then f is called a homeomorphism.

Lebesgue Outer Measure: Let   I  denote the length of an interval I and
  r  
  I 
I r
where r is a countable collection of open subintervals of . For any subset E of .
we denote the class of all countable collections of open subintervals of
which
cover E by C  E  . Then the number given by
  E  
inf
r C  E 
  r 
is called Lebesgue outer measure of the Set E.

Lebesgue measurable set: Any subset E of
if for any other subset y of
we have that:

   y     Y  E     Y  E C

is said to be Lebesgue measurable

  algebra: Let X be a non-empty set and  be a collection of subsets of X such
that:
(i)   
(ii) If   x then AC  



(iii) If  An  n1 is a sequence of elements of  , then  An   .
n1
Measurable space: Let X be a non-empty set and  denote the   algebra of all
subsets of X. Then the ordered pair  X ,   is called a measurable space. Thus

any member of  is called  -measurable set.
Measurable function: Let  X ,   be a measurable space and f : X  Re be an
extended real valued function. Then f is said to be  -measurable if the set:
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x  X : f  x   r 
r .

Measure: Let  X ,   be a measurable space. The restriction of the outer measure
to only members of  is called a measure.

Measure Space:
Let
 X ,   be
a measurable space and  be a measure
on  X ,   . Then the ordered triple  X ,  ,   is called a measure space.

Simple function: Let f : X  be a function whose range is finite. Then f is called
a simple function. For example a constant function is a simple function.

Canonical representation of a simple function: Suppose the range of a simple
function f consists of the distinct non zero real numbers a1 , a2 ,.......an . Let
f 1  ai   Ei  i  1,................., n, so that f  x   ai  x  Ei .
Ei  E j   for i  j
representation as:
Then
f 
n
 ai
i 1
and the simple function f has the canonical
E
i
where  E is the characteristic function on Ei  i  1,................., n.
i

Integral of a simple function: Let  X ,  ,   be a measure space and let Q be a
simple function in M   X ,   . Thus
is non-negative  measurable simple
function. Let Q have the canonical representation as

n

k 1
ak  E .
k
Then the extended real number given by
n
 ak   Ek 
k 1
is called the integral of the simple function Q with respect to  and is denoted by


Q d
X
Abstract integral: Let  X ,  ,   be a measure space and f : X  e be any
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function in M   X ,   . Then the integral of f denoted by

f d  is given by
X


f d   sup   S d  
X
 X

where the supremum is taken over all simple functions S satisfying the condition

S  x  f  x  x  X .
If E   , then the integral
If sup  s d     then
E f
X f
d 
E d 
 f d    .
s f
If  is the Lebesgue measure then
f
d  is called the Lebesgue integral of the
function f.

General case of abstract integral: Let  X ,  ,   be a measure space and
f : X  e be  measurable. Let
f x   max  f  x  , 0
x X

f  x   max  f  x  , 0
x X



and f  f   f 
where f  f  f
Then if
f


f
d  and

d  are finite we say that f is integrable and the integral of f
with respect to  is given by:
 f d   f

d   f  d
we then write f  L  X ,  ,   . If E   then we have
E f
If

f

d 
E f

f

d    and
d 
E f

d
d    then
f
d    and in this case f  L  X ,  ,   .
Property of  -almost everywhere  .a.e  : Let  X ,  ,   be a measure space. A
property P is said to hold true .a.e on X if there is a subset N of X with   N   0
such that P fails on N, but P is true on X  N.
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12. COMPILED LIST OF COMPULSORY READINGS (Complete reference +
abstract/rationale)
1.
Mathematical Analysis 1 by Elias Zakon, The Trillia Group.
2.
Theory of functions of a real variable, 1990, Lynn Loomis and Shlomo Stenberg, pp 95
– 116 and pp 133 – 156.
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13. COMPILED LIST OF MULTIMEDIA RESOURCES and USEFUL LINKS (Complete
reference + abstract/rationale)
Reading # 1: Wolfram MathWorld (visited 03.11.06)
Complete reference : http://mathworld.wolfram.com
Abstract : Wolfram MathWorld is a specialised on-line mathematical encyclopedia.
Rationale: It provides the most detailed references to any mathematical topic. Students
should start by using the search facility for the module title. This will find a major article. At
any point students should search for key words that they need to understand. The entry
should be studied carefully and thoroughly.
Reading # 2: Wikipedia (visited 03.11.06)
Complete reference : http://en.wikipedia.org/wiki
Abstract : Wikipedia is an on-line encyclopedia. It is written by its own readers. It is
extremely up-to-date as entries are contunally revised. Also, it has proved to be extremely
accurate. The mathematics entries are very detailed.
Rationale: Students should use wikipedia in the same way as MathWorld. However, the
entries may be shorter and a little easier to use in the first instance. Thy will, however, not
be so detailed.
Reading # 3: MacTutor History of Mathematics (visited 03.11.06)
Complete reference :
http://www-history.mcs.standrews.ac.uk/Indexes
Abstract : The MacTutor Archive is the most comprehensive history of mathematics on the
internet. The resources are organsied by historical characters and by historical themes.
Rationale: Students should search the MacTutor archive for key words in the topics they
are studying (or by the module title itself). It is important to get an overview of where the
mathematics being studied fits in to the hostory of mathematics. When the student
completes the course and is teaching high school mathematics, the characters in the
history of mathematics will bring the subject to life for their students. Particularly, the role of
women in the history of mathematics should be studied to help students understand the
difficulties women have faced while still making an important contribution.. Equally, the role
of the African continent should be studied to share with students in schools: notably the
earliest number counting devices (e.g. the Ishango bone) and the role of Egyptian
mathematics should be studied.
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15. SYNTHESIS OF THE MODULE
We note that having gone through this module you should now be fully equipped with the
knoweldege of the concepts involved in the following contents:
In unit 4 the main task has been to understand the structure of a general metric space. The
grasping of cencepts such as interior points, limit points or open sets and closed sets
together with their combinations, is essential, properties of functions defined on metric
spaces are well exposed in this unit. Leading to some of the most important results in
analysis like those covering the concepts of continuity and compactness.
In unit 5 the main aspect of understanding is that of removing the geometric structure
dealing with the concept of dictance from a metric space and replacing it with a set based
structure which constituts a topological space. The analysis of the concepts is then carried
out in a similar manner up to the level of continuity and homeomorphisms.
Finally, in units 6 and 7 we have dealt with the structure that combines both geometric and
algebraic components. Thus in concepts like Lebesque Outer Measure and Lebesque
Measure on the real line have both algebraic combination of sets like union and
measurements like length of an interval. A good grasp of the properties of the Outer
Measure leads to easy understanding of Lebesque measureable subsets of the real line. A
study of measurerable functions opn a measureable space is analagous to that of
continuous functions on topological spaces. This leads to the definition of an abstract
integral on an abstract measure space.
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16. SUMMATIVE EVALUATION
FINAL ASSESSMENT Questions
1. Let  X , d  be a metric space and A  X . Give the definition of interior of A denoted by
AO
Prove that
(a) AO  A iff A is open.
(b) AO is the largest open subset of A.
2. Let  X , d  be a metric space and A  X . Give the definition of closure of A denoted by
A.
(a) For the set A  x : 0  x 1 state A .
(b) Prove that A  A iff A is closed.
3. Give the definition of a Cauchy sequence in a metric space  X , d  .
(a) Prove that a convergent sequence is Cauchy.
(b) Use an example to show that a Cauchy sequence need not be convergent.
4. Show that a real-valued continuous function on a closed interval attains its minimum
and maximum values on the interval. Use an example to show that this fails when the
interval is not closed.
5. Let  X ,  be a topological space and A  X .
Given a collection  A such that:
 A  0  A : 0  A  U , U   , show that  A is a topology on A.
5.
Let X be a topological space and A and B be subsets of X.
Give the definition of boundary of A denoted by B’dary (A).
Prove that:
(a) A is closed iff B’dary (A)  A.
6.
(b)
A B  A  B
(c)
A B  A  B
(a) Give the definition of a Hausdorff topological space.
Let X  a, b, c  and  be a collection of subsets of X such that
   , X , a, c 
Show that  is a topology on X and that  X ,  is not Hausdorff.
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(b) Let S be a subspace of a Hausdorff topological space X. Show that S is also
Huasdorff.
7.
Let   be the Lebesgue outer measure on .
(a) Show that if A is a countable subset of
(b) For any two subsets A and B of
  A  B     B  .
8.
then   A  0.
with A countable show that:
Let  m denote the  -algebra of all Lebesgue measurable subsets of
(a) Show that if A is a subset of
(b) Let E be a subset of
.
with   A  0 then A   m .
with    E   . If there is a subset A of E such that
A   M and   A     E  , show that E   M .
9.
(a) Let f :

be a continuous function. Show that f is Lebesgue measurable.
(b) Show that if a function f is a measurable then f 2 is also measurable.
Use an example to show that the converse is not true in general.
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Final Assessment SOLUTIONS
1.
For a subset A of a metric space
Ao  x : x is an interior point of A
 X,d
interior of A denoted by Ao is given by
(a) We show that Ao  A iff A is open.
Assume A is open. Then every point of A is an interior point of A. Thus we have:
x  A  x is an interior point of A
 x  A0
i.e. A  Ao
……
(1)
But the inclusion
Ao  A
……
(2) is obvious
From (1) and (2)
Ao  A .
Conversely let A  Ao . Then A is open since Ao is open.
(b) We show that Ao is the largest open subset of A. We prove this by contradiction.
Suppose there is an open subset of A such that:
Ao  B  A
Then we also have:
Ao  Bo  Ao
i.e. Ao  Bo
But B is open  B  Bo .
Thus Ao  B.
Hence Ao is the largest open set contained in A.
2. For a subset A of a metric space  X , d  the closure of A denoted by A is given by
A  A  x : x is a limit point of A
(a) Given A  x : 0  x  1
A  A  0 
(b) We show that A  A iff A is closed.
First assume that A is closed and let A denote the set of all limit points of A.
Then we have that A  A
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Thus A  A  A  A .
Conversely assume that.
A  A. Then A is closed since A is always a closed set.
3.
Let
 X,d
be a metric space and  xn  be a sequence of elements of X. Then
 xn  is called Cauchy if for each   0
d  xn , xm     n, m  N
 N    J  such that:
(a) We show that a convergent sequence is Cauchy
Let xn  x. Then for each   0  N    J  such that d  xn , xm  
Now take m  n , then we also have that:

d  xm , x  
m  n
2
 
Thus d  xn , xm   d  xn , x   d  xm , x     
2 2
Hence  xn  is Cauchy.

2
n  N
 n, m  N
(b) Example
Let X  (0, 1 with the standard metric on
elements of X given by
xn 
1
,
n
. Consider the sequence
 xn 
of
 n J  .
Then  xn  is Cauchy. Indeed for a given   0 we can find
Now taking m  n we also have
1 

 m  n  N.
m 2
 

Thus d  xn , xm         m  n  N
2 2

Hence  xn  is Cauchy.
1 

n 2
 n N .
1
 0  X.
n
n n
Thus  xn  is Cauchy but not convergent.
However, lim xn  lim
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4. Given f :  a, b 
is continuous. Let X   a, b.
Then X is compact. Since f is continuous f  X  is also compact. Hence f  X  is
closed and bounded.
In particular f  X  is a bounded subset of . Thus by the completeness axiom
the numbers m and M given by
m  glb f  x  and M  lub f  x  do exist.
xX
xX
Since f  X  is closed m, M  f  X  .
Thus  p, q  X such that:
f  p   m and f  q   M .
Hence f attains its minimum and maximum values on  a, b which are m and M
respectively.
Example
Let f :  0, 1 
be given by
f  x   x2  1
Then f has no minimum nor maximum value on  0,1 because the interval (0, 1) is
not closed. Thus f  0  and f 1 are not defined.
5. Given a topological space  X ,  and A  X , we show that:
 A  0  A : 0  A  U : U   is a topology on A.
(i)   A :   A   ,        A
(ii) A  A : A  A  X , X    A   A
(iii) Let O1 , O1 , ............, On   A . Then
O1  A : O1  U1  A, U1  
O2  A : O2  U 2  A, U 2  
On  A : On  U n  A, U n   .
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
n
n
n

 O  A :  Oi   U i  A
i 1 i
i 1
i 1

n
n
  U i  A,:  U i  
i 1
i 1
n
Hence
(iv)
 O   A.
i 1 i
Let O   A for each  
Then O  A : U  A, U 
i.e.
U O  A : U O  U U  A
  
 
 


  U U   A, U U  
 
  

Hence U O  A
 
Hence  A is a topology on A.
6. Let A be a subset of a topological space X. Then boundary of A denoted by B’dary
A is given by
B’dary A  A  C  A  .
(a) We show that A is closed iff B’dary A  A
We first note that
A  A  B’dary A
Assume A is closed. Then A  A. Thus
A  A  A  B’dary A 
B’dary A  A.
Conversely assume
B’dary A  A. Then A  A  B’dary A  A .
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Hence A is closed.
(b) We show that A  B  A  B .
We first note that:
A  A and B  B
A B  A B.
Since A  B is closed we also have that:
A B  A B
……. (1)
Also A  A  B and B  A  B.
A 
A  B and B  A  B
i.e. A  B  A  B  A  B  A  B
i.e. A  B  A  B
……. (2)
From (1) and (2) A  B  A  B .
(c) We show that:
A B  A B
We first note that:
A  A and B  B
A B
A  B.
Since A  B closed. We also have that
A B  A B
6.
A topological space X is said to be Hausdorff if for any pair of points x, y  X  nbhs
say U and V for x and y respectively such that
U V   .
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Given X  a, b, c and    , X , a, c 
We have:
(i)   
(ii)
X 
(iii) X  a, c  a, c 
a, c     
X     
(iv) For A   or X or a, c we have that U A  a, c or X  .
Hence  is a topology on X. We now show that  X ,  is not Hausdorff.
Consider the points a and b. The open sets in  containing a are X and
a, c and X is the only open set in  that contains b. Thus for any nbh U of a
and any nbh V of b we have that.
U  V a, c  X  a, c   or U  V  X  X  X   .
Hence  X ,  is not Hausdorff.
(b) Given S a subspace of a Hausdorff topological space X. We show that S is also
Hausdorff.
Let x any y be any pair of distinct points in S. Then x and y are also members of
X. Since X is Hausdorff  nbhs U and V of x and y respectively such that
U  V  .
Consequently U  S and V  S are disjoint nbhs of x and y in S.
Hence S is also Hausdorff.
7.
(a) Let A be a countable subset of

. Then we can enumerate A as:

A  x1 , x2 , x3 ,................ .

Thus A  U
n1
xn 
i.e. By countable subadditivity of   we have:
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
   U

xn   

 n1


   xn 
n 1
 
x1    x2   ..................
 0  0  ........  0  .....
0
 

i.e.    U  xn    0
 n1

But    0 .
 

Thus    U  xn    0
 n1

Hence   A  0.
(b) We first note that, by countable subadditivity of  
   A  B      A     B 
Now A is countable implies    A  0.
Hence we have that:
  A  B     B 
…….. (1)
We also have by monotone property of   that since B  A  B
  B     A U B 
……
(2)
From (1) and (2) we have:
  A  B     B  .
8.
(a) Given that    A   0, we show that A  m .
Let X be any other subset of
. Then we have that.
A X  A
Thus by monotone property of   we have:
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   A  X      A   0.
i.e.    A  X   0.
Also X  Ac  X . Thus   X     Ac  X 
In this case we have that:
  X   0    Ac  X 
i.e.   X     A  X     Ac  X  .
Hence A   m .
(b) Given that A  m means for any subset X of
we have:
  X     A  X     Ac  X 
In particular put X  E  A.
Then we have:
   E      A  E      Ac  E 
    A     Ac  E 
   A     E  A .
i.e.   E A    A    E   0
i.e.    E  A   0  E  A   m
Now A, E  A   m 
E  A   E  A   m
Hence the result.
9.

(a) Given f :
a continuous function, we show that f is Lebesgue measurable.
Now f is continuous implies for the open set  , r  in
open in
or f 1   , r   is also
.
Thus f 1   , r     
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53
i.e. f 1   , r     x 
: f  x   r   m.
Hence f is Lebesgue measurable.
(b) Given that a function f is measurable we show that f 2 is also measurable.
Now f is measurable means  r 
the set x  X : f  x   r   .
Thus for r  0, the set:
 x  X : f 2  x   r   x  X : f  x    r  U  x  X : f  x    r .
But
x  X : f  x    r   
and
x  X : f  x    r   .
Hence there union
also belongs to  .


Thus for r  0, the set x  X : f 2  x   r   .
For r  0 the set:
 x  X : f 2  x   r   x  X : f  x    r  U  x  X : f  x    r   
Since
r does not exist in
But    . Hence  r 
.
the set:
 x  X : f 2  x   r   .
Hence f 2 is measurable whenever f is measurable.
However the converse of this result is not necessarily true as the following
counter example shows:
Counter example
Let A 
be such that A   m. Thus A is not Lebesgue measurable.
Define a function
f :

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by
54
 1 x  A
f  x  
c
1  x A
Then f 2  x   1  x  .
Thus f 2 is a constant function which is measurable. But f is not measurable
since for r = 0 we have that the set:
x 
: f  x  r    x 
Module Development Template
: f  x   0   A m .
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17. REFERENCES
1. Mathematical Analysis 1, Elias Zakon, 1973, The Trillia Group, Indiana, USA
2. Theory of functions of a real variable, 1990, Lynn Loomis and Shlomo Stenberg,
Jones and Bartlett, Boston, USA
18. MAIN AUTHOR OF THE MODULE
The author of this module on Basic Mathematics was born in 1953 and went through the
full formal education in Kenya. In particular he went to University of Nairobi from 1974
where he obtained Bachelor of Science (B.Sc) degree in 1977. A Master of Science
(M.Sc.) degree in Pure Mathematics in 1979 and a Doctor of Philosophy (P.HD) degree in
1983. He specialized in the branch of Analysis and has been teaching at the University of
Nairobi since 1980 where he rose through the ranks up to an Associate Professor of Pure
Mathematics.
He has over the years participated in workshop on development of study materials for
Open and Distance learning program for both science and arts students in which he has
written books on Real Analysis, Topology and Measure Theory.
Address:
Prof. Jairus M. Khalagai
School of Mathematics
University of Nairobi
P. O. Box 30197 – 00100
Nairobi – KENYA
Email: khalagai@uonbi.ac.ke
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