CALIFORNIA STATE UNIVERSITY, NORTHRIDGE

MEASURE THEORY

AND

APPLICATIONS

A thesis submitted in partial fulfillment of the requirements for the degree of

Master of Science in

Mathematics

by

Keith A. Barker

June, 1993

The Thesis of Keith A. Barker is approved:

Peter Collas

Phillip Emig I

David M. Klein, Chair

California State University, Northridge

11

ACKNOWLEDGMENTS

I would like to express my gratitude for the continued support and

encouragement of several people who helped make this paper a reality. First, I

would like to thank Dr. Norman Herr, Dr. Tung Po Lin, Dr. Barnabus Hughes,

and Dr. Elena Marchisotto for their stimulation and guidance in my graduate

studies. I would like to thank the members of my committee, Dr. Phillip Emig

and Dr. Peter Collas for their insights which made the paper more logical and

coherent. This project would not have come into existence without the patience,

inspiration, and the constant prodding given to me by the chair of my committee,

Dr. David Klein. Thank you, David. I needed that.

I would also like to thank my brother, Chris, and his wife Sue, for their

constant support and encouragement throughout the pursuit of my studies.

And, of course, I want to dedicate this effort to my wife, Kay, and my beautiful

daughter, Christy, without whose patience, understanding, and reinforcement

this project would never have been completed.

iii

TABLE OF CONTENTS

Acknowledgements ......................................................................................................... iii

Abstract ............................................................................................................................... v

1. Introduction .................................................................................................................. 1

Historical Perspective

Elementary Measures

2. The Area Under a Curve ........................................................................................... 5

The Riemann Integral

The Lebesgue Integral

·3. Measure Spaces ...................................................................................................... 11

Definitions

Outer Measure

Measurable Sets

Measure Spaces

4. Integration .................................................................................................................. 30

Riemann Integration

Lebesgue Integration

The Fundamental Theorem of Calculus

5. Applications ............................................................................................................... 42

Fourier Series

Probability

Bibliography .................................................................................................................... 51

IV

ABSTRACT

MEASURE THEORY AND APPLICATIONS

by

Keith A. Barker

Master of Science in Mathematics

This graduate paper is an excursion into the field of measure theory. It

begins by focusing on the historical development of the measures of lengths,

areas, and volumes, followed by the introduction of various methods to

calculate these quantities. The area under a curve is discussed descriptively in

terms of both the Riemann integral and the Lebesgue integral. A series of

fundamental definitions lead to the development of the outer measure set

function. An example of a family of sets on which outer measure is not a

countably additive set function is then formulated. Measurable sets are defined

and a series of theorems are proved to establish that outer measure, restricted

to measurable sets, is a countably additive set function.

After a discussion of measure spaces, Riemann integration is developed

in terms of step functions, followed by a rigorous development of Lebesgue

integration in terms of simple functions. The usefulness of Lebesgue integration

is then illustrated by applying the Lebesgue Dominated Convergence Theorem

to a sequence of functions that converge pointwise, but do not converge

uniformly. The Fundamental Theorem of Calculus is presented for the Riemann

integral and is then generalized to the Lebesgue Integral. The thesis concludes

with an application indicating how Lebesgue integration is used in Fourier

series and with an application of measure theory into the study of probability.

v

Chapter 1

Introduction

HISTORICAL PERSPECTIVE

Historians generally agree that mathematics arose from the necessity to

solve the practical problems of counting and recording numbers. The annual

flooding of the Nile River plain forced the Egyptians to develop a method for the

relocation of land markings. Since, according to Herodotus (c. 485- c. 430 s.c.),

taxes were paid on the basis of land area, Egyptian surveyors developed, and

used, various mensuration formulas. To convert the land along the Tigris and

Euphrates rivers into a rich agricultural region, the Babylonians engineered

structures which accomplished marsh drainage, irrigation and flood control.

The mathematics needed by the Babylonians was extensive. The word

geometry is a compound of the two Greek words meaning "earth" and

"measure," indicating that the subject arose from the necessity of land measure.

(Eves, 1)

As primitive counting evolved into mensuration and practical arithmetic, it

became necessary to develop methods to instruct, and to study, this developing

science. The study of the empirical reasoning inherent in mensuration and

practical arithmetic led to the abstraction needed for the beginnings of

theoretical geometry and elementary algebra.

When theoretical geometry and elementary algebra are studied in the

abstraction, formulas for both areas and volumes of various geometric shapes

arise. Although many of the Egyptian and Babylonian formulas contained

errors (Eves, 2), it was their groundwork that led to the generalized concepts of

length of a segment, the area of a plane figure, and the volume of a figure in

1

space. These notions are the intuitive values that are associated with the

"measure" of some particular object.

Since the time of the ancient Egyptians and Babylonians, the problem of

finding areas and volumes has captured the imagination of mathematicians.

Archimedes (287- 212 B.c.), through an ingenious method called "exhaustion,"

verified calculations of areas and volumes (Burton, 221). In this method, formal

logic, not infinitesimal quantities, was used. Nicole Oresme (c.1323 - 1382)

anticipated an aspect of analytic geometry by graphing the de pendent variable

against the independent variable as the independent variable took on small

increments. Using this method, he was able to approximate areas under curves

(Eves, 93).

With the advent of the scientific revolution in the seventeenth century,

interest in the calculation of area and volume reappeared. Johannes Kepler

(1571 - 1630), in his second law of celestial mechanics, stated that the

displacement vector from the sun to a planet sweeps out equal areas in equal

time intervals (Burton, 341). Kepler treated the area swept out by the planet as

a collection of an infinite number of small triangles with one vertex at the sun

and the other two vertices at points infinitely close together along the planet's

elliptical orbit. Using this unrefined form of calculus, Kepler was able to find the

sum of the areas of these triangles, and verify his law (Burton, 337).

Throughout the sixteenth and seventeenth centuries, mathematicians

and scientists developed their own peculiar methods for finding areas and

volumes. Isaac Newton (1642- 1727) and Gottfried Leibniz (1646- 1716) were

the first to apply integral calculus, in a systematic way, for the determination of

areas and volumes.

2

ELEMENTARY MEASURES

Area, as a measure of a plane figure, has evolved from the land measure

of the Egyptians and Babylonians to the abstract Banach spaces of modern

analysis. To develop area as a measure, it is requisite to start with a unit length.

By establishing an arbitrary length as unity, a unit square follows immediately.

The measure of any square whose side is a multiple of unity can then be found

by counting the number of unit squares contained in the original square. In a

similar manner, areas of rectangles, whose sides are multiples of unity, can be

determined by totaling the number of unit squares contained in the figure. If the

areas of various squares and rectangles are collected, a general rule can be

ascertained from the data relating area to the lengths of the sides of the

respective figures. The concept of area, as a measure, can be further extended

with the inclusion of rational and irrational numbers as lengths of sides of the

squares and rectangles

The area of a parallelogram is found by removing a triangular region

from one end, and attaching it to the opposite end, to form a rectangle. Areas of

triangular regions, which can be shown to be half of a parallelogram, can now

be determined. The area of any shape which can be broken down into non­

overlapping squares, rectangles, and triangles can be found by summing the

areas of the individual pieces.

By using the definition of 1t (the ratio of the circumference of a circle to the

diameter of the circle), the area of a circular region can now be ascertained.

The first step is to determine the area of an n-sided regular polygon (which can

be inscribed in a circle). In Figure 1, "x" represents the length of a side of the

polygon, and "a" represents the length of the apothem, the distance from the

center of the circumscribed circle to the.side of the polygon. The area of the

3

triangle is (1/2)(x)(a), and, since there are "n" triangles in the n-sided polygon,

the area of the polygon becomes (1/2)(n)(x)(a). As the number of sides of the

polygon become infinitely large, the term "(n)(x)", which is the perimeter of the

polygon, approaches the circumference of the circle, 2rrr, and the length "a"

approaches the radius of the circle, r. The area of the circle now becomes

(1/2)(2rrr)(r), which simplifies to rrr2.

Avoiding rigor, and for the illumination of the reader, the area of a sector

of a circle can be found by using the following ratio:

Area of Sector Arc Length of Sector Area of Circle - Circumference of Circle

If e signifies the angular measure, in radians, of the central angle formed by the

radii of the arc, the formula can now be simplified to:

(re)rrr2 1 _? Area of Sector = 2rrr = 2 re

The measure of any region composed of non-overlapping squares, rectangles,

triangles, circles, or arcs of circles, can now be determined.

As long as a figure is composed of straight line segments, arcs of circles,

or combinations of these, it is relatively easy, although possibly tedious, to find

its measure. If, however, the figure is composed of curves determined by

relations other than circular arcs, the measure must be calculated through the

use of integral calculus.

4

Chapter 2

The Area Under A Curve

Generally, the area under a curve is determined by finding the limit of the

sum of inscribed rectangles, as the number of rectangles increases without

bound. The area is found by two distinct methods. The first technique is called

the Riemann integral, which is the familiar integral of elementary calculus, and

the second is the Lebesgue integral, which is the keystone of modern analysis.

THE RIEMANN INTEGRAL

Given a function, f, continuous on the interval [a, b], we begin by

partitioning the interval into n subintervals by inserting points x 1, x2, ... , Xk-1, Xk,

... , Xn-1 into the interval as shown in Figures 2 and 3. For convenience, the

figures shown are of positive valued functions, but the procedure is valid for any

continuous function on the closed interval. y y

____,-;:::::!q'"""- Y = f(x)

--f----L...L...-...1.....1..--"'-'~1...-.--....___.L---7}{

a ~ }{2 XK-1 XK xn -1 b ~~.L-~-~~-~1...-.-.L-~}{

a ~ x2 xk-1 xk Xn-1 b

Figure 2 Figure 3

These points subdivide [a, b] into n subintervals, each of length ffi<1 = x1 -a, Ax2

= x2- x1, ... , Axn = b- Xn-1, which need not be of equal length. Since f is

5

continuous, it has a minimum value, m k· and a maximum value, Mk in each

subinterval (for a proof of this consequence of continuity, see Cronin-Scanlon,

pages 69 -70). The sum of the areas of the rectangles in figure 2 is called the

Lower Sum:

L = m1Llx1 + m2Llx2 + ... + mn.ill<n = _Lmk.ill<k,

and the sum of the areas of the rectangles in Figure 3, comprise the Upper

Sum:

U = M1Llx1 + M2Llx2 + ... + Mn.ill<n = _LMk.ill<k.

The difference in the areas, U - L, is of particular importance. By

increasing the number of subintervals, the closer the tops of the rectangles will

fit the curve, thus decreasing the difference, U - L. First, define the "norm" of

the subdivision as the largest subinterval width and then, as this norm

approaches zero, all of the subintervals must both decrease in width and

become more numerous at the same time. As the number of rectangles

increases, the difference U - L can be made as small as any given positive

number. This means that the

lim (U - L) = 0, which implies that lim U = lim L. norm~ 0 norm~ 0 norm~ 0

The existence of these limits is a consequence of a special property that

continuous functions have on a closed and bounded interval, called "uniform

continuity." (For a discussion of uniform continuity, see Cronin-Scanlon, pages

71 - 74).

In each interval [Xk-1, xk], a point Ck is arbitrarily chosen and the sum 5 =

_Lf(ck).ill<k is formed. The sum 5, like L and U, is the sum of the product of

functional values and interval widths, and is therefore the sum of areas of

rectangles. Since each Ck is randomly chosen, we know that m k:::;; f(ck):::;; Mk

6

which implies that L ~ 5 ~ U. By the Sandwich Theorem, (Thomas/Finney, 65),

it is concluded that

lim L = lim 5 =lim U, nonn~ 0 nonn~ 0 nonn~ 0

which means that, no matter how the point Ck is chosen, the sum 5 = _Lf(ck)~k.

always has the same limit as the norm of the subdivisions approaches zero.

This conclusion, without uniform continuity, was first reached in 1823 by

Augustin-Louis Cauchy ( 1789 - 1857), and was then put on a solid logical

foundation, with uniform continuity, by other nineteenth century mathematicians.

This limit, called the Riemann Integral of function f over the interval [a, b], is b

denoted by: J f(x) dx. This integral was named for Georg Friedrich Bernhard a

Riemann (1826- 1866). It was Riemann's idea to trap the limit between the

upper and lower sums. The sum 5 = IJ(ck)~k is called the approximating

sum, or the Riemann sum for the integral (Thomas/Finney, 267).

THE LEBESGUE INTEGRAL

The usual derivation of the Riemann integral is based on dividing the

domain of the function f into smaller and smaller subintervals. Henri Lebesgue

(1875- 1941), in his doctoral thesis published in 1902, made a simple

modification of the Riemann theory. Instead of dividing the domain, he divided

the range into finer and finer pieces. This method, illustrated in Figure 4,

depends more on the function rather than the domain, so it has the possibility of

working for more types of functions than does Riemann integration.

The first step in Lebesgue integration is to define a measure function 1J

which will assign a non-negative real number to sets that are deemed

measurable. This function must have the property that the measure of an

7

interval on the real line, IJ([a, b]), must be equal to (b- a). This requirement

suggests that the measure function on the real line is actually the formula for the

length of the interval between pairs of points on the line as defined in

elementary algebra.

The next step is to formulate the "inverse image" of the set [yo, Ynl It is

denoted as: F1([yo. Yn]) = {x: f(x) E [yo, Yn]}. The inverse image is the set of all

points of the domain of the function which are mapped into [yo, Yn] by f. As an

example, let f be a function such that:

f 1 if XE [2, 3) f(x) = ~

l 0 if X e; (2, 3)

then r\1) = [2, 3) and r\o) = (- oo, 2) u [3, oo).

Using the concepts of measure and inverse image, the Lebesgue

Integral can now be developed. Referring to Figure 4 below, the range of f(x)

must be partitioned into n subintervals, each of width ~, and note that portions

of the graph of the function are contained within the "horizontal rectangle" m m + 1

bounded by n and n

a Figure 4

8

m+1 tr m n

1 n

b

Let Ln (f)= L ~ IJ(f1 [[~ , m : 1

) ]) . Since ~is a y-coordinate, and

IJ(f1 [[~ , m : 1

) ]) represents the measure, or the "length" of an interval on

the horizontal axis, the sum. Ln (f), therefore denotes the aggregate of the

areas of rectangles. As n, the number of horizontal subintervals, increases

without bound, the limit of the sum will be the area under the curve. The

Lebesgue Integral can now be defined in the following manner:

J f(x) dx = lim Ln(f)

(Reed-Simon, 13-14). The existence of this limit can be proven using, for

example, the Lebesgue Dominated Convergence Theorem.

To illustrate the Lebesgue Integral, define the function f(x), on the

interval [0, 1], in the following manner:

r1 f(x) = i

lo

if x is irrational

if x is rational

Attempting to integrate this function by the Riemann integral, the limit of

the lower sum, 5, is found to be equal to zero, and the limit of the upper sum, U,

is determined to be one, which means that the function is not Riemann

integrable. The alternative is to now try the Lebesgue integration technique.

By using the horizontal rectangles as defined by the Lebesgue integral, 1

and, as shown in Figure 5, it is concluded that: J f(x) dx = (1){1J[f-1(1)]}. 0

1 ::::::::::::

0 Figure 5

9

An analysis of the measure of the interval [0, 1] is necessary to find IJ[f\1)]. The

measure of the interval [0, 1] is equal to the length of the interval, which is one.

Since the union of the disjoint sets, {Rationals n [0, 1]} and {Irrationals n [0, 1]},

is equal to the interval [0, 1], 1-J[O, 1] = 1-J{Rationals n [0, 1]} + 1-J{Irrationals n [0,

1]}. The set of rational numbers is countably infinite, so the set, {Rationals n [0,

1]}, can be written as the set {r1, r2, r3, ... }, where each rk represents a unique

rational number. The set {r1, r2, r3, ... } can be represented as the union of {r 1},

{r2}, {r3}, .... Since {ri} n {rj} = 0, fori* j, 1-J{Rationals n [0, 1]} = IJ{r1, r2, r3, ... }

= IJ{r1} + IJ{r2} + 1.1{r3} + ... , which is the sum of the measures of sets whose only

element is a single point. A set consisting of a single point is an interval with

both endpoints being the same point. Since the measure of an interval is the

length of the interval, IJ{rk} = rk- rk = 0, which implies ~J{Rationals n [0, 1]}

must be 0. Therefore,

1-J{Irrationals n [0, 1]} = 1-J[O, 1] - 1-J{Rationals n [0, 1]}

= 1 - 0

= 1.

But, f 1(1) is equal to {Irrationals n [0, 1]}, which means ~-J[r\1 )] =

1-J{Irrationals n [0, 1]} = 1, and 1

f f(x) dx = (1){1-J[f-1(1)]} = (1)(1) = 1. 0

To understand this kind of integration, it is necessary to comprehend the

notion of length, i.e. the "measure," of a set. Do all sets have measure? If not,

what are the necessary and sufficient conditions for a set to have a measure?

What are the properties of a set that will make it measurable? The answers to

these, and other questions, will be found as Lebesgue measure and measure

theory are developed in the remainder of this paper.

10

Chapter 3

Measure Spaces

DEFINITIONS

To introduce the study of measure spaces, it is necessary to establish a

few working definitions. To begin with, the real number system is not sufficient

for this investigation and must be enlarged to include the two elements,- oo and

+ oo. This set will be referred to as the extended real numbers. It should be

noted that, for any real number x,- oo < x < oo and the following properties hold:

1) x + oo = oo and x- oo = - oo

2) x(oo) = oo and x(- oo) =- oo, for all x > 0

3) oo + oo = oo and - oo - oo = - oo

4) oo(± oo) = ± oo, - oo(+ oo) =- oo, and - oo(- oo) = + 00

5) O(oo) = 0 and (oo- oo) will be left undefined. (Royden, 36)

Let A be a collection of sets such that if A and B are elements of A, then

(Au B) eA. and (-A) e A, where (-A) is the complement of set A. When these

two conditions are met, then the set A is called an algebra of sets. Using

DeMorgan's laws, it follows that (A n B) is also an element of A. If the above

two conditions are satisfied, and if any union of a countable collection of sets in

A is again in A, then A is called a a-algebra. It again follows that the

intersection of a countable collection of sets in A is also an element of A.

An interval I, on the real number line, can be expressed as open, closed,

or half-open, and can be denoted in the following forms: (a, b), [a, b], (a, b], or

[a, b). The length of I, I(I), is defined by (b- a). I(I) is an example of a set

function. A set function is a function that assigns an extended real number to

11

each set in some collection of sets. The domain of this length function would be

the collection of all intervals contained in the real number line.

If S is a set of real numbers, d is an upper bound for set S if, for each x e

S, d ~ x. A number c is called the least upper bound of set S if c is an upper

bound of setS and c ::;; d for each upper bound d of set S. The least upper

bound of set S is also referred to as the supremum of set S and denoted as sup

S. The number a is a lower bound of setS if, for each x e S, a ::;; x. If b is a

lower bound for set S and a ::;; b for each lower bound a of set S, then b is

defined as the greatest lower bound or the infimum of set S, and denoted as inf

S. The Completeness Axiom states that every setS, of real numbers, which has

an upper bound, has a least upper bound. From the this axiom, it follows that

every set of real numbers with a lower bound has a greatest lower bound.

The concept of measure of a set is a natural generalization of the ideas of

length, area, or volume. Other measures can be created, for example, by

considering the increment cj>{b)- cj>{a) of a non-decreasing function cj>{x) on the

interval [a, b), or the integral of a non-negative function over some line, plane

surface or space region. In all cases, the notion of measure is a set function that

assigns a non-negative extended real number to a given set.

Let n be a collection of sets of real numbers, and let "m" be a set function

that assigns non-negative extended real numbers to elements of n. If set E is

an element ofn, then the extended real number assigned toE by m, noted as

"mE," is called the measure of E. In the ideal situation, the set function m should

have the following properties:

1. mE should be defined for each set of real numbers

2. For an interval I, on the real number line, ml = 1{1)

3. If {En} is a sequence of disjoint sets, for which m is defined, then

m(UEn) = L{mEn). This means that "m" is a countably additive

12

set function.

4. m is translation invariant. If E is a set for which m is defined and if

{E + y} is the set {x + y: x e E} obtained by replacing each point in

x in E by the point (x + y), then m(E + y) =mE (Royden, 54).

As will be illustrated later, it is impossible to construct a set function

having all of these properties and it is not known if a set function satisfying the

first three properties exists. Consequently, one of the properties must be

weakened and generally it is most useful to weaken the first property and retain

the remaining three properties. Therefore, mE need not be defined for all sets E

of real numbers, however it is desired that mE be defined for as many sets as

possible. It is convenient to require that the family rt. of sets for which m is

defined be a a-algebra. The set function "m" is said to be a countably additive

measure if it is a non-negative extended real valued function whose domain is a

a-algebra rt.of sets of real numbers and m(UEn) = L{mEn) for each sequence

{En} of disjoint sets in M. It is now desired to construct a countably additive

measure which is translation invariant and has the property that m I = I{I) for

each interval I (Royden, 55).

OUTER MEASURE

Consider a set of real numbers, A, and the countable collection {In} of

open intervals that cover A. Note that A c Uin and that the length of each

interval is a positive number. Since the sum of the lengths of the intervals is

uniquely defined independent of the order of terms, the outer measure, m*A, of

A can be defined as the greatest lower bound or the infimum of all such sums of

the lengths of the intervals that cover A. In symbols, m*A = inf ~)(In) for A c

Uin. The following three propositions arise from this definition.

13

Proposition 1: If A c B, then m*A::; m*B.

1. Suppose A c B.

2. Let LA = {_LI{In): In is an open interval and A c Uin}.

Let LB = {_LI(In): In is an open interval and B c Uin}.

3. As an example, let set A= (0, 1), which implies 1 E LA· and

let set B = (-1, 2), which implies 3 E LB·

-1 0 1 2

....

4. Every element of LB is also an element of LA.which implies that

LB CLA·

5. Therefore, inf LA::; inf "Ls which implies m*A::; m*B.

Proposition 2: Each set consisting of a single point has outer measure zero.

1.

2.

3.

4.

5.

6.

Suppose set A contains only the single point x.

m*A = inf {_LI(In): In is an open interval and A c Uin}. 1 1

Let It = (x- 2 E, x + 2 E), for some E > 0.

Since It is an open covering of set A, A c (x- ~ E, x +~E). 1 1

Therefore, m*A::; l(x- 2 E, x + 2 E)= E.

Since E is arbitrarily small, m*A must be zero.

Proposition 3: m*0 = 0.

1. Since 0 c {x} for any x E R, m*0::; m*{x}, by Proposition 1.

2. By Proposition 2, m*{x} = 0, which implies that m*0::; 0.

3. By the definition of outer measure, 0 ::; m*0.

4. By combining steps 2 and 3, 0::; m*0::; 0, which implies m*0 = 0.

The following three properties concerning outer measure are direct con­

sequences of the definitions presented here. Elegant proofs of these properties

are provided in Royden, pages 56- 58.

14

Property 1: The outer measure of an interval is its length.

Property 2: Let {An} be a countable collection of sets of real numbers. Then 00

m*(U~=1An) =::;; I,(m*An). This is called countable subadditivity. n=1

Property 3: Outer measure is translation invariant.

As a consequence of Property 2 above and Proposition 2, if {x1, x2, ... } is

a countable collection of real numbers, then 00 00

m*{x1, x2, ... } :::;; I,m*{xi} = L,.o = 0. i=1 i=1

This establishes that the outer measure of any countable set of real numbers is

equal to zero.

Outer measure is defined for gil sets of real numbers. However, as

indicated in Property 2 above, it is not necessarily a countably additive set

function. To provide an example of a family of sets of real numbers on which

outer measure is not a countably additive set function, it is first necessary to

define the sum modulo one of two numbers. If x and yare real numbers in the

interval [0, 1), then the sum modulo one of x andy, denoted by (x 63 y), is

defined as follows:

f X + y if X + y < 1 x63y = i

l X + y - 1 if X + y ~ 1

The operation, 63, is a commutative and associative operation that maps pairs of

numbers in [0, 1) into numbers in [0, 1 ). As an example of this operation,

consider the unit circle, and assign to each x e [0, 1 ), the angle 27tx. The sum

modulo one now corresponds to angle addition. For example, let x = 0.5 and let

y = 0.75, then (x 63 y) = 0.5 + 0.75-1 = 0.25 and would correspond to 21t(0.5) +

21t(0.75) = 21t(1.25) = 2.51t which is equivalent to (0.5)1t or 21t(0.25).

15

Suppose set E be a subset of [0, 1 ), then the translation modulo one of

set E is defined as (E $ y) = {z: z = (x $ y), for all x E E}. If the example of sum

modulo one is angle addition, then translation modulo one by y would

correspond to rotation of set E through the angle 21ty. The following lemma

establishes that outer measure is invariant under translation modulo one.

Lemma 1: If E is an outer measurable subset of [0, 1), then, for each y E [0, 1),

(y $ E) is outer measurable and m*(y $ E)= m*E.

An e~cellent example of the proof of this lemma is provided by Munroe

on page 142.

If x andy are real numbers in [0, 1) and if (x- y) is a rational number,

then x andy are defined to be equivalent and will be written as (x = y). This is

an equivalence relation since

i) (x = x) : x- x = 0 E Q, the set of rational numbers.

ii) If (x = y), then (y = x) : (x- y) E Q implies (y- x) E Q.

iii) If (x = y), (y = z), then (x = z) : if (x- y) E Q, (y- z) E Q, then (x- y) + (y- z)

= (x- z) E Q.

This equivalence relation partitions the interval [0, 1) into equivalence

classes, that is, classes such that any two elements of one class differ by a

rational number. The reader should note that a partition separates a set into a

collection of disjoint subsets whose union is the original set. The equivalence

class for the element x is the set of all y such that (x = y). The following are three

examples of equivalence classes:

i) Q, the set of rational numbers

ii) V2 +Q

iii) 1t + Q

The Axiom of Choice states that, given any collection of non-empty sets, it

is possible to choose an element from each of the sets. Using this axiom, let P

16

be the set consisting one element from each equivalence class as defined

above. If {ro, r1, r2, ... } is an enumeration of the rational numbers in [0, 1 ), with

ro = 0, then define Pi= ri ffi P, where Po= ro ffi P = 0 ffi P = P. The following

lemma establishes that {Pi} is a pairwise disjoint sequence of sets.

Lemma 2: Pin Pj = 0, fori"# j.

1. Suppose x e Pin Pj. then xis in both Pi and Pj.

2. Then x = Pi ffi ri = Pj ffi rj. where both Pi and Pj e P.

3. But Pi- Pj e Q.

4. Therefore, Pi= Pj·

5. But P contains exactly one element from each equivalence class which

implies Pi= Pj and ri = rj. which means Pi= Pj. Therefore, if the

intersection is non-empty, the sets are equal, and, if they are different, the

intersection of Pi and Pj must be empty.

1. Let x e [0, 1), then xis contained in some equivalence class. Suppose

x= peP.

2. Therefore x = p ffi ri, for some ri e Q n [0, 1), which implies x e Pi.

3. Thus, [0, 1) = U;:'0Pi.

From lemmas 2 and 3, m*[O, 1) = m*(U;:'0Pi) = 1. If outer measure is

00

countably additive on {Pi}, then m*(U;:'0Pi) = Im*(Pi) = 1. Since each Pi is a i=O

translation modulo one of P, then, by lemma 1, m*(Pi) = m*(P) fori= 1, 2, 3, ... 00 00 (X) 00 00

and Im*(Pi) =Im*(P) = 1. Suppose, m*(P) = 0, then Im*(Pi) = Im*(P) = :Lo i=O i=O i=O i=O i=O

. 00 00

= 0 "# 1. Suppose m*(P) > 0, for example, let m*(P) = ~,then Im*(Pi) = I,m*(P) i=O i=O

17

00

= L~ = oo * 1. It can now be concluded that Im*(P) is either 0 or oo, i=O i=O

depending on whether m*(P) is zero or a positive number. In either case, outer

measure is not a countably additive set function on the pairwise disjoint

sequence of sets, {Pi} (Royden, 64-66). It is clear from this example that in order

to guarantee countable additivity of the outer measure set function, it is

necessary to, at the minimum, restrict outer measure to a family of subsets of :R

which does not contain set P or its translates {Pi}- Set P will then be a "non­

measurable" subset of :R. The notion of measurability is defined in the next

section.

The construction of the "non-measurable" family of sets {Pi}. required the

use of the Axiom of Choice. Robert M. Solovay, in 1970 showed that " ... the

existence of a non-Lebesgue measurable set cannot be proved in Zemelo­

Frankel set theory (ZF) if use of the axiom of choice is disallowed. In fact, even

adjoining an axiom DC (principle of dependent choice) to ZF, which allows

many consecutive choices, does not create a theory strong enough to construct

a non-measurable set" (Solovay, 1). In this paper, Solovay concluded that the

axiom of choice implies the existence of non-measurable sets and the existence

of non-measurable sets implies the axiom of choice.

MEASURABLE 5 ETS

Since all subsets of the real number line have outer measure, to meet the

condition of countable additivity, it will be necessary to decrease the number of

sets on which outer measure is defined. To suitably reduce the set of outer

measurable sets to a collection of measurable sets forming a a-algebra on

which outer measure will be countably additive, the definition first provided by

18

Caratheodory (page 161 ), and used by Halmos (page 44), and Royden (page

58) will be ery1ployed.

Definition: A set E is said to be measurable if, for each set A, we have

m*A = m*(A n E)+ m*(A n -E),

where the symbol -E represents the complement of set E. A member of the a­

algebra formed by this definition will be called a measurable set.

It is difficult to get an intuitive understanding of this definition except

through familiarity with its implications as will be done in the proofs of the

succeeding theorems. Since outer measure is not a countably additive set

function on the collection of all subsets of R, this definition creates a class of

sets on which outer measure is, at the minimum, additive. Given any two sets of

real numbers, A and E, set A is equal to (A n E) u (A n -E) and applying the

definition of measurability, set E, therefore, splits set A in such a manner that (A

n E) and (A n -E) will have outer measures that add to the outer measure of set

A. Since set E splits set A in such a way that outer measure is additive on set A,

set E will be regarded as measurable. In other words, this definition achieves

the rudimentary additive notion for outer measure.

Finally, it should be noted that the definition of measurability of set E

starts with the outer measure of set A, not with the outer measure of set E. Set A

is an arbitrary "test set." The measurability of set E has nothing to do with the

outer measure of set E itself, but it depends on what set E does to the outer

measure of other sets. A measurable set is one which separates no set in such

a way as to abrogate additivity for outer measure (Munroe, 86).

From this foundation of the countable additivity of outer measure on

certain sets, the concept of countable additivity can now be expanded to pairs of

disjoint sets, providing one of the sets is measurable.

19

Proposition 4: If E1 is measurable, and E1 n E2 = 0, then m*(E1 u E2) = m*(E1) + m*(E2)-

1. Since E 1 and E2 are disjoint,

[(E1 u E2) n E1] = E1, and [(E1 u E2) n -E1] = E2.

2. Since E 1 is measurable,

m*(E1 u E2) = m*[(E1 u E2) n E1] + m*[(E1 u E2) n -E1].

3. By substitution, m*(E1 u E2) = m*(E1) + m*(E2).

This proposition is a special case of Theorem 5, which is proved later in this

chapter. Using the definition of measurability, two other sets can now be

classified as measurable.

Theorem 1: The empty set, 0, and the set of real numbers, R, are measurable.

1. Let A be any set of real numbers.

2. By proposition 1, m*0 = 0, therefore, m*A = m*0 + m*A.

3. Since 0 = (An 0), and set A = (An R) = (An -0), by substituting into

statement 2, it can be inferred that m*A = m*(A n 0)+ m*(A n -0).

4. Therefore, the empty set, 0, is measurable.

5. It should be noted that the definition of measurability is symmetric in E

and -E which means that -E is measurable whenever E is measurable.

6. Since -0 =Rand -R = 0, the set of real numbers is also measurable.

Combining the notion that, for any sets of real numbers, A and E, set A is

equal to (An E)+ (An -E) with the proposition that outer measure is countably

subadditive, it can be concluded that m*A:;:;; m*(A n E) + m*(A n -E) is true for

all sets of real numbers, A and E. It can now be inferred that, in order to prove

that a set E is measurable, it is only required to demonstrate that

m*A ~ m*(A n E)+ m*(A n -E).

20

The following theorems establish that outer measure, restricted to a class

of measurable sets, is a translation invariant, countably additive set function,

and that the measure of an interval is the length of the interval.

Theorem 2: If Eisa set of real numbers and m*E = 0, then E is measurable

(Royden, 58).

1. Let A be any set of real numbers, then (A n E) c E.

2. Therefore, m*(A n E)::; m*E and, since m*E = 0, it can be inferred that

m*(A n E)= 0, since out~r measure is never negative.

3. Since (A n -E) c A, m*(A n -E)::; m*A.

4. Reversing this inequality and adding 0, m*A 2 m*(A n -E)+ 0, and sub-

stituting from statement 2, m*A 2 m*(A n -E)+ m*(A n E).

5. Therefore set E is measurable.

Theorem 3: If E1 and E2 are measurable sets of real numbers, so is (E 1 u E2)

(Royden, 59}.

1. Let A be any set of real numbers.

2. Since E2 is measurable,

m*(A n -E1) = m*(A n -E1 n E2) + m*(A n -E1 n -E2).

3. Since An (E1 u E2) =(An E1) u (An E2 n -E1), then

m*(A n (E1 u E2))::; m*(A n E1) + m*(A n E2 n -E1).

4. Adding m*(A n -E1 n -E2) to both sides of statement 3:

m*(A n (E1 u E2)) + m*(A n -E1 n -E2)::; m*(A n E1) +

m*(A n E2 n -E1) + m*(A n -E1 n -E2).

5. Substituting statement 2 into statement 4,

m*(A n (E1 u E2)) + m*(A n -E1 n -E2)::; m*(A n E1) + m*(A n -E1).

6. By definition of the measurability of E 1,

m*(A n (E1 u E2)) + m*(A n -E1 n -E2)::; m*A.

7. Since (-E 1 n -E2) = -(E1 u E2), statement 6 becomes

21

m*(A n (E1 u E2)) + m*(A n -(E1 u E2)::; m*A.

8. By the definition of measurability, (E 1 u E2) is measurable.

Theorem 4: The family of measurable sets, n, is an algebra of sets (Royden,

59).

1. Let the sets E1 and E2 be members of the set X of measurable sets.

2. Set E 1 being measurable, implies that -E 1 is also measurable and is

therefore contained in n:.

3. As demonstrated by Theorem 3, (E 1 u E2) is measurable and is also a

member ofn.

4. By definition, the family of measurable sets n is an algebra of sets.

Theorem 5: If A is any set, and if E 1, ... , En is a finite sequence of disjoint n

measurable sets, then m*(A n [U~1 Eil) = L m *(A n E i) (Royden, 59).

This theorem will be proven by induction. 1

i=1

1. lfn=1,m*(AnE1)=L m*(A n Ei)=m*(AnE1). i=1

n-1

2. Assume the induction hypothesis: m*(A n [U~-11 Ei]) = L m*(A n Ei). i=1

3. Since En is measurable,

m*{A n [U~1 Ei]) = m*(A n [U~1 Ei] n En)+ m*(A n [U~1 Ei] n -En)

n-1

4. m*(A n [U~1 Ei]) = m*(A n En)+ L m*(A n Ei). by statement 2. i=1

n

5. m*(An [U~1EiD=I m*(A n Ei). i=1

Theorem 6: The collection n of measurable sets is a cr-algebra (Royden, 60).

1. Theorem 4 demonstrated that n is an algebra of sets. Thus, to prove that

n is a cr-algebra, all that must be verified is that any union of a countable

22

collection of measurable sets is measurable. In other words, it must be

shown that if A1, A2, ... e :M., then A1 u A2 u ... e :M., or E = Ui:1 A;e :M..

2. Let E1 = A1_, E2 = A2 \ A1, E3 = A3 \ (A1 u A2), E4 = A4 \ (A1 u A2 u A3), ... ,

En= An\ (A1 u A2 u ... u An-1).

3. For each E; e :M., E; n Ej = 0 fori :t j, U~1 E; = U~1A;, for any~. and

00 00

Ui=1 E; = Ui=1 A; .

n 4. Let A be any subset of real numbers and Fn be defined as Ui=1 E;. (Note

that Fn e :M., Fn c E which implies that -E c -Fn).

5. Since F n is measurable, m*A = m*(A n Fn) + m*(A n -Fn).

6. Since -E c -Fn, m*A ~(An Fn) + m*(A n -E), n

m*A ~ m*( An Ui=1 E;) + m*(A n -E).

n 7. Using Theorem 5, m*A ~ L m*(A n E ;) + m*(A n -E).

i=1 n

8. As n ~ oo, lim [m*A] ~lim [ L m*(A n E ;) + m*(A n -E)] which i=1

00

becomes m*A~ L m*(A n E;)+m*(An-E). i=1

00 00

9. Using statements 1 and 3, E = Ui=1 A;= Ui=1 E;, it is concluded that

00 00

An E = (Ui=1 E;) n A, which implies that m*(A n E)= m*((Ui=1 E;) n A).

10. Since outer measure is countably subadditive, 00

m*{A n E)= m*((Ui:1 E;} n A)::;; L m*(E; n A), therefore, i=1

00

m*(A n E)::;; I, m*(E; n A). i=1

11. By substitution into statement 8, m*A ~ m*(A n E)+ m*{A n -E) which

23

makes set E measurable, and it is concluded that 1't is a a-algebra.

Theorem 7: The interval (a, oo) is measurable (Royden, 60).

1. Let A be any set and define A1 =An (a, oo) and A2 =An (- oo, a].

As an example, let a= 1, and A= [-10, 10]. Then A1 = [-10, 10] n (1, oo)

= (1, 10] and A2 = [-10, 10] n (- oo, 1] = [-10, 1].

2. For (a, oo) to be measurable, it must be shown that

m*A = m*(A n (a, oo)) + m*(A n -(a, oo)) which is equivalent to

m*A = m*(A n (a, oo)) + m*(A n (- oo, a]), or m*A = m*A1 + m*A2.

Since m*A::; m*A1 + m*A2, all that needs to be shown is that

3. If m*A = oo, then the theorem is proven.

4. If m*A < oo, then there is a countable collection, {In}. of open intervals

which cover A and for which :LI(In)::; m*A + c:.

5. Let In' =Inn (a, oo) and In II =Inn (- oo, a]. Then In' and In II are intervals

(possibly empty sets) and I(In) = I(In ') + I(In 11

) = m*(In ') + m*(In 11

).

6. Since A 1 c U {In'), m*A1 ::; m*(U {In')) ::; .L,m*{In '). This inequality

follows from the notion that outer measure is countably subadditive.

7. By a similar argument, A2 c U {In 11

), m* A2 ::; m*(U On")) ::; .L,m*{In ").

8. Combining statements 6 and 7, .L,[m*(In ') + m*(In ")] ~ m*A1 + m*A2.

9. Substituting from statement 5, .L,I(In} ~ m*A1 + m*A2 and from statement

4, m*A + c: ~ m*A1 + m*A2 which makes m*A ~ m*A1 + m*A2, since c: is

arbitrary.

10. Therefore, the interval (a, oo) is measurable.

At this point in the development of measure theory, it is necessary to

consider more general types of sets than the open and closed sets. This is

required because the intersection of any collection of closed sets is closed, and

the union of any finite collection of closed sets is closed, however, the union of a

24

countable collection of closed sets is not necessarily closed. As an example,

consider the set of rational numbers. The set of rational numbers is the union of

a countable collection of closed sets, each of which contains exactly one

number. But the union of these closed sets is not closed. Thus, if a-algebras of

sets that contain all of the closed sets are to be considered, a more general type

of set must be specified (Royden, 52).

Definition: The Borel sets, :8, of Rare the smallest family of subsets of R with

the following three properties:

(i) The family is closed under complements. If Be :8, then-Be :8.

(ii) The family is closed under countable unions. If B1, B2, ... e :8,

then u;:1 Bi E :8.

(iii) The family contains each open interval.

Since the family of all subsets of R is a a-algebra that satisfies (i), (ii),

and (iii) above, the collection of all such a-algebras is non-empty. To see that

the smallest family exists, consider that if {:Sa} is a non-empty collection of

families satisfying the three conditions stated above, then so does rl{:Ba}, and

this intersection is the smallest family that satisfies all the conditions (Reed and

Simon, 14- 15). The reader should note that, from statements (i) and (ii) of the

definition, the Borel sets form a a-algebra. The only properties of Borel sets

needed in this paper follow from the fact that they form the smallest a-algebra

containing the open and closed sets (Royden, 53).

Theorem 8: Every Borel set is measurable, specifically each open set and each

closed set is measurable (Royden, 61 ).

1. By Theorem 6, the collection n, of measurable sets, is a a-algebra. By

Theorem 7, (a, oo) En ,which means -(a, oo) = (- oo, a] E n.

25

2. Since 1't is a a-algebra, the union, as well as the intersection, of a

countable collection of sets in 1't is also an element of n. Each open interval of

the form (- oo, b), can be represented as U~=1 (- oo, b- ~],which makes(- oo, b)

an element of n. Every open interval of the form (a, b) can be represented as

(- oo, b) n (a, oo). This implies that all open intervals are contained inn.

3. Since :M., contains all of the open intervals, and since 1't is a a-algebra,

1't contains the smallest a-algebra containing all the open intervals, therefore n contains the Borel sets.

The following theorem states that if outer measure is defined on a

sequence of disjoint subsets of the set n, of measurable sets, then outer

measure is a countably additive set function.

Theorem 9: If {Ei} is a sequence of measurable sets and Ei n Ej = 0, fori-:~:- j, 00

then m*(U~1 Ei) = :L<m*Ei). (Royden, 62) i=1

1. Proposition 2 on page 14 demonstrated that outer measure is a 00

subadditive set function, that is, m*(U~1 Ei)::; :L<m*Ei). Therefore, to complete i=1

00

the proof of this theorem, it only necessary to prove that m*(U~1 Ei);::: :L<m*Ei). i=1

n

2. Theorem 6 proved that, for any set A, m*(A n [U~1 E~) = :Lm*(A n Ei). i=1

n

If set A is the set of real numbers, R, then m*(U~1 Ei) = :Lm*(Ei) and it can be i=1

inferred that outer measure, on set :M., is a finitely additive set function.

3. If {Ei} is an infinite sequence of disjoint measurable sets, then U~1 Ei ::>

26

n 4. By substitution from statement 2, m*(U;1Ei) ~ :Lm*(Ei).

i=1

5. The left side of the inequality in statement 4 is independent of n, and the n ~ ~

limit, as n ~ oo, of :Lm*(Ei) is L(m*Ei) which implies m*(U;1Ei) ~ :L<m*Ei). i=1 i=1 i=1

~

6. Combining statements 1 and 5, m*(U;1Ei) = :L<m*Ei). i=1

It can now be concluded that if outer measure is restricted to the set n of

measurable sets, it will be a countably additive set function which is translation

invariant (Royden, 58}, and has the property that the measure of an interval will

be the length of the interval (Royden, 56). If set E is a measurable set, the

Lebesgue measure, mE, is defined to be the outer measure of E. Lebesgue

measure is, therefore, outer measure, restricted to the collection of measurable

sets.

MEASURE SPACES

A measurable space is usually represented as a couple, {'X, :'8),

consisting of a set 'X and a a-algebra :'8 of subsets of 'X. Since :'8 is a a­

algebra, :'8 is a family of subsets of 'X which contain the empty set, and is closed

with respect to complements and with respect to countable unions. A subset .A,

of 'X, is defined to be measurable if A e :'8. A measure, ~. on the measurable

space, {'X, :'8), is a nonnegative set function defined for all sets of :'8, and

satisfying the following conditions:

(i) ~(0) = 0 ~

(ii) ~(U~1 Ai) = :L~Ai, for any sequence Ai, of disjoint measurable i=1

subsets of :'8.

27

A measure space, represented by the triple (X, :8, ~). is a measurable space

with a measure~ defined on :8 (Royden, 254).

An example of a measure space is (:R., n, m) where :R. is the set of real

numbers, Xis the set of measurable sets of real numbers, and m is Lebe$gue

measure. Another example would be (R, :8, m) where R is again the set of real

numbers, :8 is the a-algebra of Borel sets, and m is again Lebesgue measure.

A measure space, {X, :8, ~) is defined to be complete if :8 contains all

subsets of sets of measure zero. This means that if B e :8, ~B = 0, and A c B,

then A e :8. The measure space defined by (R, n, m) is complete, since n contains subsets of all sets of measure 0, while the measure space {:R., :8, m),

where :8 is the a-algebra of Borel sets, is not. To see this, the reader should

note that, by Theorem 8, every Borel set on the real line is Lebesgue

measurable and that there are Lebesgue measurable sets of measure zero

which are not Borel sets (Munroe, 148- 149). It can also be shown that any

Lebesgue measurable set of measure zero is contained in a Borel set of

measure zero (Munroe, 97- 98). Thus, (:R., :8, m) does not contain all subsets

of sets of measure zero, and is therefore, not complete. It can be proven that

each measure space can be completed by the addition of subsets of sets of

measure zero. This is called the completion of the measure space.

Formally, the completion of measure space (X, :8, ~) is the measure

space (X, :Bo. ~0} where

(i) :8 c :8o

(ii) If E e :8, then !JE = !JoE

(iii) E e :8o if and only if E = A u B, where B e :8 and A c C, C e :8, !JC = 0.

Thus, the completion of measure space (R, :8, m), where :8 is the a­

algebra of Borel sets, is the measure space (R, n, m), where set n, is the set of

28

Lebesgue measurable sets (Royden, 257). This provides an alternative

description of the Lebesgue measurable sets. A set is Lebesgue measurable if,

and only if, it is the union of a Borel set and a subset of a Borel set with

Lebesgue measure zero.

With the foundations of measure theory now complete, it is time to return

for a closer look at the Riemann integral, its shortcomings, and develop a more

general form of integration as defined by the Lebesgue integral.

29

Chapter 4

Integration

Riemann Integration

An elegant formulation of the Riemann integral can be obtained by the

use of step functions and, to introduce this formulation, it is necessary to

establish a few working definitions. To begin with, let f(x) be a real valued,

bounded function on the interval [a, b] and let xo, ... , xn be a partition of [a, b]

such that xo =a, and xn =b. Let Mi be defined as the sup {f(x): Xi-1 ~ x ~Xi} and

let mi be defined as the inf {f(x): Xi-1 ~ x ~Xi}, then the upper sum is designated n n

asS= 'LMi(Xi- Xi-1), and the lower sum denoted ass= 'Lmi(Xi- Xi-1). The i=1 i=1

expressions "s" and "S" are called Darboux Sums, and from their definitions,

it can be seen that s ~ S (Cronin-Scanlon, 127- 128). The Upper Riemann

integral of f(x) is defined as

b

R+ J f(x) dx = inf S, a

and the Lower Riemann integral of f(x) is defined as

b

R- J f(x) dx =sups, a

with the infimum and supremum taken over all possible subdivisions of [a, b]. If

the Upper and Lower Riemann integrals are equal, then f(x) is defined to be b

Riemann integrable, and is denoted as R J f(x) dx, (Royden,76). a

If xo .... , Xn is a partition of [a, b] such that xo =a and xn = b, then the step

function, 'I', has the form 'l'i(X) = Ci, where Xi-1 ~ x ~Xi and Ci is a constant. For

30

example, suppose n = 3, c1 = 2, c2 = 3, c3 = 1, then the graph of the step

function formed would be represented as shown in Figure 6.

3 2 1

I

b

From the definition of the step function, it can be seen that the I 'JI(X) dx a

n

is equal to Ici(Xi - Xi-1), which implies that i=1

b b

R+ I f(x) dx = inf I 'JI(X) dx, for all step functions 'JI(X) ~ f(x), and a a

b b

R- I f(x) dx = sup I 'Jf(X) dx, for all step functions 'JI(X)::; f(x). a a

(Royden, 76). This means that if f(x) is Riemann integrable, its integral can be

expressed in terms of the integral of step functions and this formulation

represents the intuitive notion of the Riemann integral as presented in

elementary calculus.

As demonstrated in chapter 2, the function defined by:

r 1 if X iS irrational f(x) = ~

l 0 if x is rational for x e [0, 1]

is not a Riemann integrable function. This example illustrates the shortcomings

of Riemann integration and shows the need do develop a more general form of

integration. The required general form was first introduced by Henri Lebesgue,

in his doctoral thesis presented at the Sorbonne in 1902, (Gillispie, 110).

31

Lebesgue Integration

Henri Leon Lebesgue was born in Beauvais, France on June 28, 1875.

Lebesgue studied at the Ecole Normale Superieure from 1894 to 1897 and his

first university position was at Rennes, from 1902 to 1906. After his studies at

the Ecole Normale Superieure, Lebesgue became acquainted with the

research of another graduate of the Ecole, Rene Baire. Baire studies included

the theory of discontinuous functions of a real variable which helped focus

Lebesgue on the deficiencies of Riemann integration. In his doctoral thesis,

Lebesgue began the development of a theory of integration which, as he

demonstrated in his presentation at the Sorbonne, included all the bounded,

discontinuous functions introduced by Baire.

By 1922, Lebesgue had produced nearly ninety books and papers.

Much of his work dealt with his theory of integration, but he also presented

significant work in the calculus of variations, the theory of surface area, and in

the structures of sets and functions. In 1922, he was elected to the Academie

des Sciences and, during the last twenty years of his life, he remained active in

his studies of pedagogy, elementary geometry, and history. Lebesgue died in

Paris on July 26, 1941 (Gillispie, 110- 112, and Lebesgue, 1 - 5).

To begin the study of Lebesgue integration, it is necessary to have a

function that has a value of one on a measurable set and zero elsewhere. The

integral of this function should, therefore, be equal to the measure of the set.

The function that satisfies these conditions, on the measurable set E, is called

the characteristic function, XE, and is defined as follows:

r 1 if x e E XE(X) = i

l 0 ifx ~ E

32

As introduced in Chapter 2, the inverse image, J-1(A), is defined as the

set {x: f(x) e A}. Equivalently, the inverse image is the set of points in the

domain of the function which are mapped into A by f.

An extended real valued function, f, is defined to be measurable, or

Lebesgue measurable, if, whenever set A is measurable, J-1(A) is measurable

and f satisfies one of the following conditions:

(i) For each real number a, the set {x: f(x) >a} is measurable.

(ii) For each real number a, the set {x: f(x);:: a} is measurable.

(iii) For each real number a, the set {x: f(x) <a} is measurable.

(iv) For each real number a, the set {x: f(x):::;; a} is measurable.

The reader should note that each of the above conditions are equivalent

(Royden, 66). It should be observed that continuous functions, with measurable

domains, are measurable. All step functions are also measurable. If f is a

measurable function and E is a measurable subset of the domain off, then the

function obtained by restricting f toE is measurable (Royden, 77).

Suppose E 1, ... ,En is a finite class of disjoint, measurable sets, and if

{a1, ... , an} is a set of finite real numbers, then the real valued function <1>,

defined by the relation

n

<I>(x) = IaiXEt(x) i=1

is called a simple function. A simple function is a function that assumes a finite

number of values and assumes each of these values on a measurable set

(Munroe, 155). It should also be noted that a function <1> is simple if, and only if,

it is measurable and assumes only a finite number of values (Royden, 77).

The basic idea of Lebesgue integration is to replace the step functions of

Riemann integration with simple functions. But, before this can be

33

accomplished, a few facts about simple functions need to be established. One

of the more useful portrayals of the simple function is called the canonical

representation and is defined as follows: Suppose <1> is a simple function and n

{a1, ... , an} is the set of nonzero values of <1>, then <I> = :Iai(XAi) where i=1

Ai = {x: <l>{x) = ai}. The canonical representation of <I> is characterized by the

fact that the Ai are disjoint and the ai are distinct and nonzero. If <1> vanishes

outside a set of finite measure, then, by definition

n

I<l>{x) dx = :Iai(mAi). i=1

where (mAi) is the Lebesgue measure of Ai, and <I> is the canonical n

representation, <I> = :Iai(XAi). This integral is usually abbreviated as I<l>. If E is i=1

any measurable set, then I <I> is defined as I <I>Xt (Royden, 77). E E

As an example, suppose f(x) = 3X[o, 11 + 2X[2, 5]· If [0, 1] u [2, 5] c E, then

I <I> dx = I 3X[o, 11 + 2X[2, 5] dx E E

= I 3X[o, 11 dx + I 2X[2, 5] dx E E

= 3m[O, 1] + 2m[2, 5]

= 3(1) + 2(3)

= 9.

The following proposition brings the idea of simple functions a little closer

to the more familiar step functions: Suppose f is a bounded, real valued

function defined on set E, a measurable set of finite measure. If <1> and 'I' are

34

simple functions, then, by analogy with the Riemann integral, it can be

concluded that

inf J\ji(X) dx, for f::::; "', is equal to the sup j<D{x) dx, for f;::: <D,

if, and only if, f is a measurable function. A proof of this conjecture is provided

by Royden, pages 79-81.

The Lebesgue integral for non-negative functions can now be defined as

follows: Iff is a non-negative measurable function defined on a measurable set

E where mE is finite, the Lebesgue integral off over E, denoted as

J f(x) dx, E

is defined as the supremum of J <D{x) dx, where the supremum is over all E

simple functions, <D::::; f, which vanish outside of a set of finite measure. This

integral is sometimes expressed as: J f, or, if E = [a, b], the integral would be E

b

written as J f(x) dx. a

Any non-negative function is said to be "integrable" if J f(x) < oo. Before

defining a Lebesgue integrable function, two further definitions must be

introduced. The positive part, j+(x), of function f(x), is defined as

j+(x) = max {f(x), 0}.

In a similar manner, the negative part, J-(x), of function f(x), is defined as

J-(x) = max {-f(x), 0}.

It should be observed that both j+(x) and J-(x) are non-negative functions and

that f(x) = j+(x) - J-(x). Thus, if f(x) is measurable, then j+(x) and J-(x) are

both measurable and non-negative. This leads to the following definition:

Definition: A measurable function f(x) is defined to be Lebesgue integrable

over set E if j+(x) and J-(x) are both integrable over E. If they

35

are integrable, then J f(x) dx J J-(x) dx, E E

(Royden, 89-90).

In considering both Riemann and Lebesgue integration, what are the

conditions for a function to be integrable? It has been shown that a necessary

and sufficient condition for a function to be Riemann integrable is that the

function must be bounded and continuous almost everywhere. This means that,

for example, a function with a countable number of discontinuities is Riemann

integrable. A necessary and sufficient condition for a function f to be Lebesgue

integrable, is that f be measurable and both J+(x) and J-(x) have finite integrals.

The following proposition establishes that the Lebesgue integral is

actually a generalization of the Riemann integral, (Munroe, 177).

Proposition: If a bounded function f, is Riemann integrable on [a, b], then it is b b

Lebesgue integrable on [a, b], and: R f f(x)dx = f f(x)dx. a a

b b

1. Since f is Riemann integrable, R- J f(x) dx = R+ f f(x) dx.

2.

a

But every step function is also a simple function, and b

a

R- J f(x) dx ~ sup1~~ j<I>(x) dx ~ inf1 ~'1' j'Jf(X) dx ~ R+ a

b

J f(x) dx. a

3. Therefore, sup1~~ j<I>(x) dx = inf1 ~'1' j'Jf(x) dx, and f must be Lebesgue b b

integrable, and R J f(x)dx = J f(x)dx. a a

An alternative proof of this proposition is provided in Cronin-Scanlon, on pages

197 -198. Before illustrating some of the uses of Lebesgue integration, the

following definitions from analysis should be reviewed.

1. A sequence of functions {fn} defined on a set E, is said to converge

pointwise onE, to a function f(x) if, for all x in E, limn~~fn(x) = f(x). This means

36

that, given an x E E, and an arbitrary E > 0, there is an N such that for all n ~ N,

lfn(x)- f(x)l < E, (Royden, 49).

2. A sequence of bounded functions {/n} defined on a set E, is said to

converge uniformly to f(x) on E, if limn~~ suplfn(x)- f(x)l = 0, (Beals, 47) ..

1

0 Figure 7 1

The sequence of functions, fn(x) = xn, on the interval [0, 1), is an example

of a sequence of functions that converge pointwise, but the sequence is not

uniformly continuous on [0, 1 ). f 1 (x) = x 1, /2(x) = x2, and f 10(x) = x 10 are

illustrated in Figure 7.

fn(x) = xn converges to 0 pointwise since limn~~fn(x) = limn~~xn = 0

= f(x), for all x E [0, 1). To converge uniformly, limn~~suplfn(x)- f(x)j must

equal 0. However, limH~suplfn(x)- f(x)l = limH~supjxn- 01 = limn~~supjxnj

= 1 * 0, and it is concluded that the sequence does not converge uniformly.

This example will be used to illustrate a use of Lebesgue integration after the

introduction of one more definition and a theorem.

A property is said to hold almost everywhere (a.e.), if the set of points

where it fails to hold is a set of measure zero (Royden, 69). For example,

functions f and g are equal a.e. iff and g have the same domain, and the set of

points where f(x) * g(x) has measure zero.

37

The Lebesgue Dominated Convergence Theorem, first published by

Lebesgue in 1908, states: If fn(x) is a sequence of functions and there exists an

integrable function G(x), such that lfn(x)l :::;; G(x) a.e. and the limit off n(x), as b

n ~ oo, is equal to f(x) a.e., then the limit, as n ~ oo, of J fn(x) dx a

b

= J f(x) dx, (Hawkins, 118, and Reed- Simon, 24). a

Using the previous example, f n(x) = xn, it has been established that fn(x)

converges pointwise to 0, but does not converge uniformly. Therefore, f(x) = 0.

Since lxnl:::;; 1 on [0, 1], let G(x) = 1. By the Lebesgue Dominated Convergence 1 1

Theorem, the limit, as n ~ oo, J x n dx = J 0 d x = 0. 0 0

The Fundamental Theorem of Calculus

Both Isaac Newton (1642- 1727) and Gottfried Leibniz (1646- 1716)

were familiar with the idea of relating integrals as sums with the concept of

integrals as anti-derivatives (Bell, 153). Newton even introduced the notion of

changing the upper bound of an integral to a free variable (Burton, 368).

However, it wasn't until 1823 that the first statement of the Fundamental

Theorem of Calculus was presented by Augustin Cauchy (1789- 1857). Using

the ideas of continuity and uniform continuity, Cauchy defined, and coined the

term "definite integral" of a continuous function on the closed interval [a, b].

In his statement of the fundamental theorem, Cauchy considered a

function f, continuous on the interval [a, b] and defined the function F(x) as X

F(x) = J f. Cauchy also established that F(x) is both continuous and differen-a

tiable. Cauchy's statement of the Fundamental Theorem of Calculus takes the

form of three theorems:

38

Theorem 1: F is a primitive function for f; that is, F' =f. X

Theorem 2: All primitive functions off must be in the form If + C, a

where C denotes a constant; that is, If G is a function with a X

continuous derivative G', then I G' = G(x)- G(a). a

Theorem 3: If G is a function such that G'(x) = 0 for all x in [a, b], then G'(x) re­

mains constant there.

The reader should note that Theorem 3 is an immediate consequence of

the Mean Value Theorem and was used by Cauchy to prove Theorem 2. In

1849, using the idea of limits, Cauchy refined his version of the fundamental

theorem by including discontinuous functions with a finite number of

discontinuities on the interval [a, b], and he extended his definition of the

integral to unbounded functions (Hawkins, 9- 12).

In analyzing the Fundamental Theorem of Calculus with regards to both

Riemann and Lebesgue integration, two basic questions arise. d X

1. When does dx I f(t) dt = f(x)? a

Iff is Riemann integrable on [a, b], then the relation holds at each point x

in [a, b] at which f is continuous. Iff is continuous on [a, b], then the relation

holds for all points of [a, b] (Fulks, 156- 157). If f(x) is Lebesgue integrable,

this relation holds for almost all values of x, (Royden, 97). The reader should

note that, for Lebesgue integration, there is no requirement that f be continuous

at x.

This relation leads to the key observation that whenever f is continuous, X

the integral I f(t) dt is a differentiable function of x. As illustrations of a

functions whose definitions are only in terms of an integral, consider first, the

39

natural logarithm of a positive number. The natural logarithm is defined in the

following form: X

In X = f t dt for all x > 0.

1

As a second example, consider the sine integral: X

si(x) = J si~ t dt 0

which appears in various engineering applications {Thomas, 278).

b

2. When does J f'(x) dx = f(b)- f(a)? a

For Riemann integration, this relation holds as long as f(x) is

differentiable on [a, b] and f'(x) is Riemann integrable on [a, b] (Fulks, 158). For

the Lebesgue integral, the answer to this question depends on the concept of

"absolute continuity."

A real valued function f, defined on the closed interval [a, b], is defined to

be absolutely continuous on [a, b] if, given an E > 0, there is a o > 0 such that n

L lf(xj') - f(xi)l < E i=1

for every finite collection {(xi, Xi')} of non-overlapping intervals with n

L lxi' - x il < o. i=1

An absolutely continuous function is clearly continuous. Since any

continuous function on a closed interval is uniformly continuous, it follows that

an absolutely continuous function is uniformly continuous. However, the

converse in not true. Any absolutely continuous function f, on [a, b], is

differentiable almost everywhere (Royden, 108- 109). The property of absolute

40

continuity was first introduced by Axel Harnack (1851-1888) in 1884 (Hawkins,

77).

The following proposition establishes the Fundamental Theorem of

Calculus as applied to Lebesgue integration: X

Proposition: f(x) = f(a) + j f'(t) dt, for all x E [a, b], if, and only if, f is a

absolutely continuous on [a, b] (Royden, 110, and Munroe, 268).

As an illustration of the power of the Lebesgue integral over the Riemann

integral, consider the function f(x) = .]x on the interval (0, 1 ]. Since f(x) is

unbounded, the Riemann integral does not exist, and, using the method first

introduced by Cauchy, the improper Riemann integral must be evaluated using

limits (Hawkins, 12). Thus, 1

J Jx d x becomes 0

1

lim J _1_ dx asE ~o. Vx

which, because of the limit, becomes a more complex problem than necessary.

It can be shown that f(x) = 2Vx is absolutely continuous, and therefore, it

satisfies the Fundamental Theorem of Calculus for Lebesgue integration, and 1

the integral becomes J .]x d x = 2vxl6 = 2. This integral is now a proper

0

Lebesgue integral, and it can be integrated as indicated.

41

Chapter 5

Applications

Fourier Series

A trigonometric series is an infinite series of the form:

a 00

2° + L (ancos nx + bnsin nx) n=1

where ao, a1, ... and b1, b2, ... are given constants. If the trigonometric series

converges for all x e [0, 21t] then it converges for all real x, since, for all n, both

cos nx and sin nx have periods of 21t (Cronin-Scanlon, 217). In 1822, Joseph

Fourier (1768- 1830) proposed that any bounded function, f(x), can be

expressed in the form of a trigonometric series (Hawkins, 5). In his proofs,

Fourier, a mathematical physicist, demonstrated to the pure mathematicians of

the time, that the notions of "arbitrary" functions, real numbers, and continuity

needed rigorous clarification (Bell, 293).

Fourier's development is based on the following: Suppose the

trigonometric series converges uniformly on [0, 21t] to a function f(x), i.e.,

a 00

f(x) = 2° + L (ancos nx+ bnsin nx), n=1

21t

for all real x (Cronin-Scanlon, 217). If J f(x) dx exists, then the Fourier 0

series of f(x) is the trigonometric series given by

a 00

2° + L (ancos nx + bnsin nx), where n=1 1 21t

an - 1t J f(x)(cos nx) dx 0

42

for n = 0, 1, 2, ... , and

1 21t

bn = - J f(x)(sin nx) dx 1t 0

for n = 1, 2, ....

The numbers ao, an, bn are called the Fourier coefficients of f(x}. The

Fourier series of f(x) is sometimes specified by:

a "" f(x) - 2° + ~ (ancos nx + bnsin nx).

n=1

The symbol"-" is used to indicate that the coefficients an and bn are

obtained from the functions defined above. In general, the series to the right of

the "-" may not converge to f(x) for every x. However, if the Fourier series does

converge to f(x), then the "-" symbol can be replaced by the equal sign

(Cronin-Scanlon, 219- 20).

The Fourier series is, of course, a trigonometric series. However, if given

a trigonometric series, can it be determined if there is a function, f(x), such that

the Fourier series of f(x) is the given trigonometric series? Solving this problem

led to the discovery, and proof, of the well-known Riesz-Fischer Theorem.

Before introducing this theorem, it is necessary that the reader be familiar with

the properties of the LP spaces.

Suppose "p" is a positive real number and E is a measurable set of real

numbers. Then LP(E) is defined as the set of measurable functions on E such

that lfiP is integrable. Thus L1[0, 1] would consist of the set of Lebesgue

integrable functions on [0, 1] (Royden, 119). As another example, consider

f(x) = x(-1/4) on [0, 16]. It should be recognized that f(x) e L1[0, 16] since 16

I [x (-1/4)]1 dx = ~ x(3/4) l1g = 3{ . However, f(x) ~ L4[0, 16] since 0

16

I [x (-1/4)]4 dx 0

16

= J ~dx 0

is not integrable.

43

Since any two functions that are equal almost everywhere on a

measurable set E, have the same integral on E, the distinctions between the

functions are insignificant. The relation "equal almost everywhere" is reflexive,

symmetric, and transitive, thus it forms an equivalence relation. The Lebesgue

space, denoted as LP(E), is the collection of equivalence classes from the

associated equivalence relation on LP(E). Even though the elements of LP(E)

are usually thought of as individual functions, they are actually equivalence

classes of functions, such that, in each class, any two functions are equal almost

everywhere.

Suppose [f] denotes an equivalence class in LP = L P(E) containing

function f. Define addition in LP by: [f] + [g) = [f + g), and scalar

multiplication in LP by: a[f] = [af]. Using the relation If+ giP ~ 2P(IfiP + lgiP), it

can be shown that the Lebesgue space, LP, is a linear space, or a vector space.

A norm, II• 11. on a vector space V over real numbers, is a function, such

that II• II: V ~ R, which satisfies the following:

(i)

(ii)

(iii)

(iv)

II f II ~ 0, for all f e V.

II f II = 0, if and only iff = 0.

II af II = (lal)(ll f II), for all a e R, and f e V.

II f + g II ~ II f II+ II g 11. for all f, g e V, (Beals, 70).

Define II If] II as II I II = [ ~III P }/p, thus II I II " 0, and condition (i)

is met. Condition (ii) is satisfied since LP consists of equivalence classes of

functions. For any constant a e R, condition (iii) is fulfilled, and condition (iv),

called subadditivity, can be shown to hold providing p ~ 1. Thus, for p ~ 1, the

LP spaces are normed linear spaces, (Royden, 119).

Any normed linear space is defined as a metric space with the metric:

p(f, g) = II f- g II. (Reed and Simon, 9). A Banach space is a normed linear

44

space (over the real or complex numbers) which is a complete metric space. A

normed linear space is complete if every Cauchy sequence in the space

converges. A sequence, {/n}. in a normed linear space, is a Cauchy sequence

if, given any E > 0, there is anN, such that, for all n, m;;:: N, II fn- fm II< E.

A sequence Un} in a normed linear space is said to converge to an

element f in the space if, given an E > 0, there is an N such that, for all n > N,

II f- fn II< E. Convergence in the LP spaces is referred to as convergence in

the mean of order p, (Royden, 123- 4).

The Riesz-Fischer Theorem simply states that the LP spaces are

complete, thus, they are Banach spaces. The theorem, first introduced in 1907,

was proved by F. Riesz (1880 - 1956) and Ernst Fischer (1875 - 1959)

independent of each other. Part of the proof employed by Riesz involved the

special case for the interval [0, 21t] in the L2 (Hilbert) space. He deduced the

convergence of trigonometric series as long as an and b n are Fourier

coefficients, (Hawkins, 174 - 175). An elegant proof of the Riesz-Fischer

Theorem is presented in Royden, on pages 124- 125. From the Riesz-Fischer

Theorem, it has been established that if the sequences {an} and {bn} are such

that the series

converges, then 1 00

2ao +L (ancos nx + bnsin nx) n=1

is the Fourier series of a function f e L2[0, 21t]. It has also been shown that the

partial sums of the Fourier series converge in the mean to f, (Cronin-Scanlon,

235). It can also be shown that any f e L2[a, b], has a Fourier series which

converges to f in the mean. Furthermore, it can be shown that any Fourier

45

00

series, such that L (a n2 + b n2) < oo, converges in the mean to some f E

n=1

L2[a, b].

It should be noted that the integration required by the Riesz-Fischer

Theorem is Lebesgue integration. One reason that the Riesz-Fischer Theorem

uses Lebesgue integration is that the convergence properties of the Lebesgue

integral are essential in the proof of the theorem (Cronin-Scanlon, 234-235).

Another reason that Lebesgue integration is used is that the class of Lebesgue

integrable functions is larger than the class of Riemann integrable functions.

The search for this theorem was one of the chief motivating forces behind the

development of the Lebesgue integral (Munroe, 258).

One crucial property of the LP spaces not shared by the corresponding

spaces of Riemann integrable functions is that of completeness in the metric

given above. This was pointed out in a note by Fischer in 1907 (Hawkins, 177).

As an example, consider the set of continuous functions on [a, b] with the

corresponding metric b

II f- g II = J lf(x) - g(x)l dx. a

Let C[O, 1] denote the set of continuous functions on the interval [0, 1].

Since each function in C[O, 1] is bounded and continuous, each is Riemann

integrable. To illustrate that C[O, 1] is not complete, it is necessary to find a

Cauchy sequence of functions on C[O, 1] that does not converge to a function in

C[O, 1]. Let fn(x) be defined as in Figure 8.

46

1 3 4 4

Figure 8

fn(x) is a Cauchy sequence of functions since, given an E > 0, there 1

exists anN, such that, if n, m > N, J If n(x) - f m(x)l dx < E. But fn(x) does 0

not converge to a function in C[O, 1]. It converges to the characteristic function

of [ 1, ~]which is not contained in C[O, 1]. Thus, the set of continuous functions

in [0, 1] do not form a complete metric space. The reader should note that the

completion of C[a, b] is L 1 [a, b] (Reed and Simon, 13).

Probability

The purpose of this section is provide the reader with an intuitive

justification for the treatment of probability in terms of a measure space. This

development is excerpted from Halmos, pages 184- 191.

Using a common example, consider the rolling of an ordinary six-sided

die with the number "x" showing on the top face of the die. The number x may,

therefore, take on values of 1, 2, 3, 4, 5, or 6. This means that x e {1, 2, 3, 4, 5,

6}. Possible outcomes of this experiment could be

(i) the number xis even, which is equivalent to x e {2, 4, 6}

(ii) x is less than 4, which is equivalent to x e {1, 2, 3}

(iii) x is equal to 6, which is equivalent to x e {6}.

47

There are as many events associated with this experiment as there are

combinations of the first six positive integers taken any number at a time. If the

impossible event, x not being equal to any of the first six positive integers, is

considered, then there are 26 possible events. The impossible event wiU be

symbolized as 0, while the certain event, x e {1, 2, 3, 4, 5, 6}, will be denoted

as X.

If E is an event, then the complementary event of E will be written as -E.

An experiment, one of whose outcomes lies in E, will be said to result in -E if,

and only if, the outcome does not lie in E. Thus, if E = {2, 4, 6}, then -E would

equal {1, 3, 5}. Given two events, E and F, the union, E u F, occurs if, and only

if, at least one of the two events occur, and the intersection, E n F, occurs if, and

only if, both E and F occur. Suppose E = {2, 4, 6} and F = {1, 2, 3}, the E u F =

{1, 2, 3, 4, 6} and E n F = {2}.

The preceding examples, with generalizations to more complex

experiments, justify the conclusion that probability theory consists of the study of

an algebra of sets. The reader should recall that a collection A. of subsets of X

is an algebra of sets if

(i) Au B is in A. whenever A and B are, and

(ii) -A is in A. whenever A is.

An event is, therefore, a set, and its opposite event is the complementary

set. Mutually exclusive events are disjoint sets and an event consisting of

simultaneous occurrences of two other events, is a set obtained by the

intersection of two sets. For the study of probability where the total number of

possible events is finite, the treatment of the class of possible events as an

algebra of sets is adequate. For situations arising in modern probability theory

and practice, it is necessary to make the additional assumption that the system

48

given in Halmos, Chapter IX. For example, a "random variable" in probability is

just a measurable function.

It seems appropriate to finish this introductory account of measure theory

and Lebesgue integration with a quote given by Lebesgue himself:

"And now, gentlemen, I pause to thank you for your kind attention and to

offer a final word of conclusion, if you will permit. It is that a generalization

made not for the vain pleasure of generalizing but in order to solve previously

existing problems is always a fruitful generalization. This is proved abundantly

by the variety of applications of the ideas that we have just examined."

(Lebesgue, 194).

50

BIBLIOGRAPHY

Beals, Richard. Advanced Mathematical Analysis. New York: Springer-Verlag, 1973.

Bell, Eric T. The Development of Mathematics. 2d ed. New York: McGraw-Hill Book Company, Inc., 1945.

Burton, David M. The History of Mathematics, An Introduction. Dubuque: William C. Brown Publishers, 1985.

Caratheodory, Constantin. Algebraic Theory of Measure and Integration. Reprint, New York: Chelsea Publishing Co., 1963.

Cronin-Scanlon, Jane. Advanced Calculus. Rev. ed. Lexington: D. C. Heath and Company, 1969.

Eves, Howard. Foundations and Fundamental Concepts of Mathematics. 3d ed. Boston: PWS-Kent Publishing Company, 1990.

Fulks, Watson. Advanced Calculus, An Introduction to Analysis. 3d ed. New York: John Wiley & Sons, 1978.

Gillispie, Charles C. (ed.). "Henri Lebesgue," Dictionary of Scientific Biography. vol. VIII. New York: Charles Scribner's Sons, 1973.

Halmos, Paul R. Measure Theory. New York: D. Van Nostrand Company, Inc. 1951.

Hawkins, Thomas. Lebesgue's Theory of Integration, Its Origins and Development. Madison: The University of Wisconsin Press, 1970.

Kolmogorov, A. N. and S. V. Fomin. Measure, Lebesgue Integrals, and Hilbert Space. translated by Natascha Artin Brunswick and Allan Jeffrey. New York: Academic Press, 1961.

Lebesgue, Henri. Measure and the Integral. edited by Kenneth 0. May. San Francisco: Holden-Day, Inc., 1966.

Munroe, M. E. Introduction to Measure and Integration. Reading: Addison­Wesley Publishing Company, Inc., 1953.

Reed, Michael and Barry Simon. Methods of Modem Mathematical Physics. New York: Academic Press, 1972.

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Royden, H. L. Real Analysis. 3d ed. New York: Macmillan Publishing Company, 1988.

Solovay, Robert M. "A model of set-theory in which every set of reals is Lebesgue measurable," Annals of Mathematics. 2d ser., vol. 92, 1970, 1-56.

Temple, G. The Structure of Lebesgue Integration Theory. London: Oxford University Press, 1971.

Thomas, George B. and Ross L. Finney. Elements of Calculus and Analytic Geometry. Menlo Park: Addison-Wesley Publishing Company, 1989.

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