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AConcise
Introductionto
MeasureTheory
Satish Shirali

A Concise Introduction to Measure Theory

Satish Shirali
A Concise Introduction
to Measure Theory
123

Satish Shirali
Formerly of Panjab University
Chandigarh, India
and
The University of Bahrain
Zallaq, Bahrain
ISBN 978-3-030-03240-1 ISBN 978-3-030-03241-8 (eBook)
https://doi.org/10.1007/978-3-030-03241-8
Library of Congress Control Number: 2018960220
Mathematics Subject Classification (2010): 28-01, 28A05, 28A10, 28A12, 28A20, 28A25, 28A35
© Springer Nature Switzerland AG 2018
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part
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This Springer imprint is published by the registered company Springer Nature Switzerland AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface
The concept of measure in an abstract space, and of the integral with respect to it,
has played a fundamental role in analysis and probability in the decades after the
pathbreaking work of H. Lebesgue in 1902. It has therefore been an essential
component of a mathematics curriculum, accessible to any student who has
acquired a facility with basic analysis, including the Heine–Borel theorem, the
theory of Riemann integration, infinite series, and also with the use of sets and
quantifiers (“for all” and “there exists”) in definitions and proofs. This book
assumes such a facility on the part of the reader and also an understanding of the
notions of countability and Cartesian product. Equivalence classes and the axiom of
choice are needed only once.
It is well known that the abstract integral can be interpreted as an improper
Riemann-type integral of a related function on the real line, thereby rendering it
relatively more concrete. The related function is one that closely resembles what
is called the cumulative distribution in probability and statistics and may reason-
ably be called simply the distribution. This book introduces the concept of the
integral of a nonnegative measurable function on a measure space via the improper
integral of the related distribution function, without the intermediacy of simple
functions. Doing so brings out more vividly how the Lebesgue approach begins
by partitioning the range rather than the domain of the integrand. Also, the concept
is then amenable to a quick extension to “fuzzy” measures. It also helps to
appreciate how the monotone convergence property is unrelated to the additivity of
measure.
v

It is natural to begin by recasting the Riemann integral of a step function as the
integral of its distribution function, which is essentially the Abel summation for-
mula. This is what Chap. 1 starts with. For other kinds of functions, it is not always
clear that a distribution function exists at all. A countably additive measure as the
“size” of a set is then presented as motivated by the need for a limit of functions
having distributions again to have a distribution.
The concept of a countably additive measure on a r-algebra and measurability of
functions are introduced in Chap. 2 against the backdrop of the motivation
described earlier. The integral of a nonnegative measurable function is defined as
the improper integral of its distribution function, and the monotone convergence
theorem is proved. The integral of a function that may take negative values is
introduced, but its properties are dealt with in the next chapter.
Simple functions are defined in Chap. 3 and used to establish the additivity of the
integral. The dominated convergence theorem, for which the additivity of the
integral is used, is also treated in this chapter. There is a discussion of the extension
to subadditive fuzzy measures, but this material is optional.
Chapter 4 is about constructing Lebesgue measure. After defining Lebesgue outer
measure, Carathéodory’s ideas are applied to an abstract outer measure in the usual
manner. It is shown that the classical Riemann integral agrees with the Lebesgue
integral for Riemann integrable functions, and the existence of a nonmeasurable
subset of the reals is discussed. The chapter ends with induced measures, which find
application in connection with product measures and in identifying certain improper
integrals as being Lebesgue integrals.
The counting measure and interchanging the order of summation in a repeated
sum as amounting to interchanging the order of integration are treated at length in
Chap. 5. In this connection, the unconditional sum is identified with the integral
respect to the counting measure.
Product measures and the theorems of Tonelli and Fubini are taken up in Chap. 6.
In Chap. 7, the relation to differentiation is discussed in some detail. There are
many approaches to the topic, the one adopted in this book being via the Vitali
covering theorem and Tonelli’s theorem. While the concept of total variation is
presented ad hoc in most texts, here it is presented as a natural outcome of attempts
to decompose a function as the sum of an increasing and a decreasing function.
The relation between differentiation and Lebesgue integration is not as
straightforward as in the case of Riemann integration, and no discussion of the
matter can be complete without the Cantor set and function. Their essentials are
discussed in Chap. 8.
The symbols R, Q, Z and N are commonly understood these days, and no
explanation of them has been given.
For matters that are treated differently or not treated at all in most other works
covering measure and integration, a prospective reader having prior experience
of the subject may wish to check out the following:
vi Preface

1.1.3, 1.1.6 (motivation), 1.2.4, 3.1.9, 3.1.10, 3.1.P1, 3.1.P7, 3.1.P8, 3.1.P9, Figures in
Sect. 3.2, Sect. 3.3, 4.1.2(b), 4.2.P6, 4.2.P8, 4.2.P9, 4.4.1, 5.1.2, 5.1.4, 5.1.10, 5.2.5, 5.2.7,
5.2.8, 6.2.P4, 6.6.4, 7.1.3(b), 7.1.P3, beginning of Sect. 7.6 (motivation), 7.8.2, Sect. 8.2
(manner of constructing the Cantor function), 8.2.P3, 8.2.P4, 8.2.P5.
The author is grateful to Dr. H. L. Vasudeva for his valuable comments on the
presentation, which contributed signiﬁcantly to the improvement in readability.
Gurugram, India Satish Shirali
September 2018
Preface vii

Contents
1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Riemann Integral Revisited. . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Measure Space and Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1 Measure and Measurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Measurability and the Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 The Monotone Convergence Theorem . . . . . . . . . . . . . . . . . . . . . 35
3 Properties of the Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 Simple Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Other Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Subadditive Fuzzy Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 Construction of a Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1 Lebesgue Outer Measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Measure from Outer Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3 Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4 Induced Measure and an Application . . . . . . . . . . . . . . . . . . . . . . 85
5 The Counting Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1 Interchanging the Order of Summation . . . . . . . . . . . . . . . . . . . . 91
5.2 Integration with the Counting Measure . . . . . . . . . . . . . . . . . . . . 98
6 Product Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.1 Algebras and Monotone Classes of Sets . . . . . . . . . . . . . . . . . . . . 103
6.2 Deﬁning a Product r-Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3 Sections of a Subset and of a Function . . . . . . . . . . . . . . . . . . . . 108
6.4 Deﬁning a Product Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.5 The Tonelli and Fubini Theorems . . . . . . . . . . . . . . . . . . . . . . . . 116
6.6 Borel Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
ix

7 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.1 Integrability and Step Functions . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.2 The Vitali Covering Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.3 Dini Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.4 The Lebesgue Differentiability Theorem. . . . . . . . . . . . . . . . . . . . 142
7.5 The Derivative of the Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.6 Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.7 Absolute Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.8 The Integral of the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8 The Cantor Set and Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
8.1 A Surjection from the Cantor Set to [0, 1] . . . . . . . . . . . . . . . . . . 169
8.2 A Bijection from the Cantor Set to [0, 1] and the Cantor
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
x Contents

Chapter 1
Preliminaries
1.1 The Riemann Integral Revisited
It is elementary from the basic theory of the Riemann integral, for example as in
Rudin [7] or Shirali and Vasudeva [9], that if a function on a closed bounded interval
[α, β] is constant on (α, β), its integral is the product of that constant with the length
of the interval. The values of the function at the endpoints make no difference. Each
term in a lower or upper sum of a bounded function f : [a, b] → R is a product
of this form. To put it more precisely, denote the lower sum of a bounded function
f : [a, b] → R over a partition P: a x0 x1 · · · xn b of [a, b] as usual by
L( f, P)
n

j1
m j (xj − xj−1)
where mj means the infimum of f over the subinterval [xj−1, xj] of P. The term
mj(xj −xj−1) is the Riemann integral over [xj−1, xj] of the function that equals mj on
the subinterval (xj−1, xj), regardless of what value the function takes at the endpoints.
Taking into account that the sum of the integrals of a function on contiguous intervals
[α, β] and [β, γ] is the integral of that function over [α, γ], it is seen that the lower
sum is equal to the integral
b
a m of any function m: [a, b] → R which, for each
j, equals the constant mj on the interval (xj−1, xj). The values of m at the points xj
make no difference; but in order to be specific, we shall choose m to be mj on the left
closed subinterval [xj−1, xj) of P and m(b) to be 0. The argument why refining the
partition increases the lower sum can then be interpreted in terms of such a function
m as proving that refining increases the function m.
Similar observations apply to the upper sum, which will be denoted as usual by
U( f, P)
n

j1
Mj (xj − xj−1)
S. Shirali, A Concise Introduction to Measure Theory,
https://doi.org/10.1007/978-3-030-03241-8_1
1

2 1 Preliminaries
where Mj means the supremum of f over the interval [xj−1, xj].
The notation L(f , P) for the lower sum and U(f , P) for the upper sum will be used
in the sequel without explanation.
Functions that remain constant on the open intervals of a partition are lurking
within the definition of upper and lower sums, although one may not make explicit
mention of them. As we intend to consider them at some length now, we formally
introduce the standard name by which such functions are known.
1.1.1. Definition. A function f : [a, b] → R is called a step function if there is some
partition P: a x0 x1 · · · xn b of [a, b] such that f is constant on each
of the open intervals (xj−1, xj). It is said to be constant on the subintervals of P.
It should be borne in mind that such a function is constant only on the open subin-
tervals of P, although we shall usually suppress the word “open” for convenience.
Evidently, the function is also constant on the subintervals of any refinement of P;
thus the partition P mentioned in the definition is far from being unique.
Although we shall generally denote the Riemann integral over [a, b] of a function
φ by
b
a φ, it will be convenient to use the more customary notation
b
a φ(x)dx when
the function is known by its expression φ(x) without any symbol such as φ having
been introduced for it.
The Riemann integral
b
a f of a step function f is easy to compute: It is simply
n
j1 cj (xj − xj−1), where cj denotes the value on (xj−1, xj). In the language of
elementary calculus, the “area under the graph” of a nonnegative step function is the
sum of the areas of rectangles stretching downward from the graph all the way to
the horizontal axis. The terms in the sum represent the areas of the rectangles. If the
graph is drawn with the axes interchanged, then the rectangles stretch leftward from
the graph all the way to the vertical axis.
1.1.2. Exercise. For each of the functions below, determine whether it is a step
function, and describe in terms of intervals the set S mentioned along with it:
(a) f : [0, 4] → R, where f (x) 6 if 0 ≤ x 1, f (x) 2 if 1 ≤ x ≤ 3,
f (x) 6 if 3 x ≤ 4; S {x ∈ [0, 4]: f (x) 2}.
(b) f : [0, 2] → R, where f (x)x2
; S {x ∈ [0, 2]: f (x) 1
4
}.
(c) f : [0, π] → R, where f (x)sin x; S {x ∈ [0, π]: f (x) t}, where 0≤t.
(d) f : [0, π] → R, where f (x) 1 + sin x; S {x ∈ [0, π]: f (x) t}, where
0≤t.
Solutions: (a) Consider the partition P: 0134. On the (open) subin-
tervals (0, 1), (1, 3) and (3, 4), the function has the constant values
6, 2 and 6 respectively. Therefore f is a step function. S [0, 1)∪(3, 4].
(b) The function takes infinitely many different values and therefore cannot be a
step function. S (1
2
, 2].
(c) The function takes infinitely many different values and therefore cannot be a
step function. If t 1, we have S (sin−1
t, π − sin−1
t), as a graph would
reveal; we dispense with the formal argument. When t ≥1, the set S is empty
because of the inequality sin x ≤1.

1.1 The Riemann Integral Revisited 3
(d) The function takes infinitely many different values and therefore cannot be a step
function. S [0, π] if 0 ≤ t 1 and S (sin−1
(t−1), π−sin−1
(t−1)) if 1 ≤
t 2. For t ≥2, the set S is empty.
In view of what has been noted in parts (c) and (d) of the foregoing Exercise,
we shall henceforth consider the empty set as an interval having length 0. With this
convention, we can say that the set S has turned out to be a finite union of disjoint
intervals in each case. In part (b), if the number 1
4
is replaced by an arbitrary t ≥0,
then S is the interval (
√
t, 2] if 0≤t 4 and ∅ if 4≤t.
Since the length of the interval ∅ is finite, we shall count it among bounded
intervals; moreover, just as in the case of R, it will be considered as closed and
open at the same time. This changes the meaning of “interval” slightly, but all that
is needed is to exercise some care in using results about intervals.
In part (a) of the exercise, one can ask for S {x ∈ [0, 4]: f (x)t} for an arbitrary
t ≥0. The reader will find it helpful to draw a graph of the step function f to see that
S varies with the value of t in the following manner:
S
⎧
⎪
⎨
⎪
⎩
[0, 4] if 0 ≤ t 2
[0, 1) ∪ (3, 4] if 2 ≤ t 6
∅ if 6 ≤ t
Thus the subset of the domain on which the function exceeds any given number t
≥0 turns out to be a finite union of intervals in each of the instances in the exercise.
There are many other functions for which this happens, for example, those that are
monotone. When it does happen, the function has a distribution, to be denoted herein
by ˜
f , which is defined to be the function whose value at any t ≥0 is the sum of the
lengths of all the finitely many disjoint intervals whose union is the subset S of the
domain on which the function exceeds t. If the subset is empty, we take ˜
f (t) to be
0. In particular, when t ≥sup f , the set S is empty and ˜
f (t) 0. A significant point
to note is that the distribution, if it exists, is a decreasing function (which means
nonincreasing in this book), and therefore Riemann integrable (see Theorem 9.3.3.
of Shirali and Vasudeva [9] or 6.9 of Rudin [7]). It is decreasing because t1 t2
implies {x: f (x) t1} ⊆ {x: f (x) t2}.
The reader may note however that disjoint intervals comprising the subset need
not be unique—Example: [0, 3] is the union of the disjoint intervals [0, 1), [1, 3] as
well as [0, 1], (1, 2], (2, 3]—and therefore one could legitimately ask whether their
total length is unique, and consequently whether the concept of the distribution ˜
f
is unambiguous. Some authors may not share these doubts (see Rudin [7]) and in
any case, the matter can be settled in the affirmative. However, we prefer to leave
the technical details to the problems and proceed by taking the affirmative answer
for granted. Readers who do not feel any need for the technical details of the matter
would do well to avoid engaging with Problems 1.1.P6, 1.1.P7, 1.1.P8 and 1.1.P9.
For the function f in part (a) of the exercise above, the set {x ∈ [0, 4]: f (x)t} has
been described at the beginning of the preceding paragraph and one can conclude
therefrom that the distribution ˜
f : [0, ∞) → R is given by

4 1 Preliminaries
˜
f (t)
⎧
⎪
⎨
⎪
⎩
4 if 0 ≤ t 2
2 if 2 ≤ t 6
0 if 6 ≤ t.
Since sup f 6, the last line in this description of ˜
f reflects the earlier observation
that, in general, ˜
f (t) 0 when t ≥sup f . We draw attention to the fact that
4
0
f 6 · 1 + 2 · 2 + 6 · 1 16 and
sup f
0
˜
f
6
0
˜
f 4 · 2 + 2 · 4 16,
so that
4
0 f and
sup f
0
˜
f are equal to each other. The figures below are intended
to illustrate how the two sums lead to the area of the same region broken up into
rectangles in two different ways. The figure on the left shows the “area under the
x x f (t)
f(x)
f(x) t
graph” of f in the usual manner as a union of rectangles. Adding up the areas of
the rectangles is the same computation as in the computation of
4
0 f above. The
middle figure shows the same area broken up into rectangles differently. The figure
on the right shows the graph of ˜
f with the axes interchanged and the area that would
have been under the graph if the axes had not been interchanged. The darker shaded
rectangle has been broken up into two by a vertical line so as to show the relation to
the rectangles in the middle figure. Again, adding up the areas of the rectangles is the
same computation as in the computation of
sup f
0
˜
f above. One can visually confirm
that the total shaded areas are the same in all three figures, which is a reflection of
the equality of integrals that was obtained above by computation. The forthcoming
proposition shows that the equality of the integrals of the nonnegative step function
and its distribution is not a coincidence.
1.1.3. Proposition. Any nonnegative-valued step function f : [a, b] → R has a dis-
tribution ˜
f and
b
a
f
sup f
0
˜
f .

Proof: The case when f is 0 everywhere is trivial because sup f 0 in this situation.
Therefore we consider only the case when f has at least one positive value.
The set {x ∈ [a, b]: f (x) t}, even if empty, is the union of a finite family of
disjoint bounded intervals. By Problem 1.1.P9, the total length of disjoint intervals
comprising the set is then uniquely determined, so that the distribution indeed exists.
Let P: a x0 x1 · · · xm b be a partition of [a, b] such that f has
constant value cj on the open interval (xj−1, xj). Then
b
a
f
m

j1
cj (xj − xj−1).
By grouping together the terms for which the values cj agree but are not 0, we rewrite
the above sum as
b
a
f
n

k1
Tkλk, (1.1)
where the Tk(1 ≤ k ≤ n) are the distinct positive values among c1, . . . , cm and λk is
the total length of the intervals on which f takes the value Tk. Any values of f that
are not among the Tk are taken only at one of the points xj of the partition P and
therefore the total length of the intervals where such values are taken is 0. Thus we
can include them in the sum with 0 as the corresponding λk. The numbers Tk then
represent all the distinct positive values of f . By renumbering if necessary, we may
assume that they are in increasing order. For reasons that will become clear in the
next paragraph, it will be convenient to set T0 0, although it does not appear in the
sum.
Since T1, . . . , Tn are the distinct positive values of f arranged in increasing order,
we have
Tn sup f and ˜
f (Tn) 0. (1.2)
Consider any integer k such that 1 ≤ k ≤ n. Surely, f (x) Tk_1 ⇔ f (x) ≥ Tk. In
other words,
{x ∈ [a, b]: f (x) Tk−1} {x ∈ [a, b]: f (x) Tk} ∪ {x ∈ [a, b]: f (x) Tk}.
(1.3)
Denote by F1 and F2 the families of disjoint intervals comprising the two sets on the
right side. Since the sets are disjoint, we deduce that the intervals in F1 are disjoint
from those in F2, so that F1 ∪ F2 is a family of disjoint intervals comprising the set
on the left side of (1.3). In particular, F1 and F2 can have no nonempty interval in

6 1 Preliminaries
common, so that the total length of the intervals in F1 ∪F2 is the sum of the separate
total lengths. This means ˜
f (Tk−1) ˜
f (Tk) + λk, and hence by (1.1),
b
a
f
n

k1
Tk( ˜
f (Tk−1 − ˜
f (Tk)). (1.4)
If n 1, then by (1.2), ˜
f (T1) 0; therefore, keeping in mind that we have set T0
0, we find that the sum in (1.4) becomes
T1( ˜
f (T0) − ˜
f (T1)) (T1 − T0) ˜
f (T0)
n

k1
˜
f (Tk−1)(Tk − Tk−1).
Hence,
b
a
f
n

k1
˜
f (Tk−1)(Tk − Tk−1) (1.5)
when n 1. Suppose n ≥2. By (1.2), ˜
f (Tn) 0, and therefore the equality (1.4)
can, by separating the last term in the sum, be recast as
b
a
f
n−1

k1
Tk( ˜
f (Tk−1) − ˜
f (Tk)) + Tn
˜
f (Tn−1).
Also, Tk (Tk − Tk−1) + · · · + (T1 − T0), remembering that T0 0. Using the Abel
summation formula of Problem 1.1.P5 (with ak ˜
f (Tk−1) and bk Tk −Tk−1), we
find that (1.5) holds in the present case too, namely, n ≥2.
It remains to show only that
sup f
0
˜
f
n
k1
˜
f (Tk−1)(Tk − Tk−1).
In view of the first equality in (1.2), we have
sup f
0
˜
f
Tn
0
˜
f . Since f does not
take any value between Tk−1 and Tk, it is true of any t ∈[Tk−1, Tk) that f (x)
t ⇔ f (x) Tk−1. It follows that ˜
f (t) ˜
f (Tk−1) for every t ∈[Tk−1, Tk), which
is to say, the restriction of ˜
f to [0, sup f ] is a step function having constant value
˜
f (Tk−1) on each interval [Tk−1, Tk). The equality that remained to be shown is now
an immediate consequence.
For a monotone function f , the set {x: f (x)t} is an interval and therefore the
function has a distribution ˜
f . The function of Exercise 1.1.2(d) is not even mono-
tone and yet the aforementioned set is an interval. It is natural to ask whether the
equality proved above for step functions is valid for other (nonnegative) functions
that have a distribution. We shall not prove this yet, because more technical hassles
are involved than proving that the total length of a disjoint union of intervals is deter-
mined uniquely, i.e. independently of the particular representation as such a union.
One cannot imitate the proof for the case of step functions because the integral is no

longer a finite sum. Although the proof is deferred until a proper formulation and
argument are possible (appearing as Theorem 4.3.4 later on), we shall illustrate some
cases now.
1.1.4. Example. (a) As noted in the paragraph following Exercise 1.1.2, for the
function f : [0, 2] → R such that f (x)x2
, the set S {x ∈[0, 2]: f (x)t}
is the interval (
√
t, 2] if 0≤t 4 and ∅ if 4≤t. Also sup f 4. Therefore
˜
f : [0, 4] → R is given by ˜
f (t) (2 −
√
t) for 0≤t ≤4, and thus
b
a
f
2
0
x2
dx and
sup f
0
˜
f
4
0
(2 −
√
t)dt.
Elementary computations show that both are 8
3
.
(b) As seen earlier in Exercise 1.1.2(d), for the function f : [0, π] → R, where
f (x)1+sin x, the set S {x ∈[0, π]: f (x)t} is the
interval [0, π] when 0 ≤ t 1,
interval (sin−1
(t − 1), π − sin−1
(t − 1)) when 1 ≤ t 2 and
∅ when 2 ≤ t.
Of course, sup f 2. Therefore ˜
f : [0, sup f ] → R is given on [0, sup f ) by
˜
f (t)
π if 0 ≤ t 2
π − 2 sin−1
(t − 1) if 1 ≤ t 2.
Thus
b
a
f
π
0
(1 + sin x)dx and
sup f
0
˜
f π(1 − 0) +
2
1
(π − 2 sin−1
(t − 1))dt.
Elementary computations show that both are π+2.
When f is bounded, the distribution ˜
f is 0 on [sup f , ∞) and the integral
M
0
˜
f is
the same for all M ≥sup f . Thus the restriction of ˜
f to [0, sup f ] can “do everything for
us” that the unrestricted function can and therefore will also be called the distribution
˜
f .
The Exercise about to be presented will be used to obtain a glimpse into the
technical hassles mentioned above. The concept of countability will play an essential
role in the exercise as well as further on. In the terminology of this book, “countable”
includes finite. Theelements of acountablyinfiniteset Acanbydefinitionbearranged
in a sequence with no repetitions and range equal to the entire set A. Any such
sequence will be called an enumeration of A.

8 1 Preliminaries
1.1.5. Exercise. Let {rk}k≥1 be an enumeration of the set of all rational numbers in
(0, 1). For each n ∈ N, let fn: [0, 1] → R be the function which vanishes at the first
n numbers of the sequence {rk}k≥1 and also at 0 and 1, while having the value 1 at
every other point of its domain. Also, let f : [0, 1] → R be the function for which
f (x) inf{ fn(x): n ∈ N} for each x ∈[0, 1]. Obviously, 0≤f (x)≤1. Show that
(i) for each x ∈[0, 1] and n ∈ N, the inequality f n(x)≥f n+1(x) holds;
(ii) for each x ∈[0, 1], f (x) limn→∞ fn(x);
(iii) each f n is a step function with ˜
fn(t) 1 when 0≤t 1;
(iv) S {x ∈[0, 1]: f (x)t} is not a countable union of intervals when 0≤t sup f .
Solution:Thefunctionsf n andf n+1 differonlyatrn+1 andf n(rn+1)10f n+1(rn+1).
This proves (i). It now follows from elementary facts about monotone sequences that
(ii) holds (see Theorem 3.3.3 of Shirali and Vasudeva [9] or 3.14 of Rudin [7]). The
first n numbers of the sequence {rk}k≥1 provide a partition of [0, 1] when arranged
in increasing order. By definition, f n is 1 on each open subinterval of the partition,
so that it is a step function. Besides, for any t such that 0≤t 1, the set Sn {x ∈[0,
1]: f n(x)t} is the union of the open subintervals of the partition, which means its
total length is 1. This proves (iii).
To prove (iv), we first note that f (x)0 if x is rational and 1 otherwise. This is
because, on the one hand, since the range of {rk}k≥1 is the set of all rational numbers
in (0, 1), any rational x equals rk for some k and hence we have f n(x)f n(rk)0 for
every n ≥k; on the other hand, since any irrational x differs from rk for every k, we
have f n(x)1 for every n ∈ N. From the foregoing description of f , we observe that
sup f 1. Moreover, when 0≤t 1, the set S {x ∈[0, 1]: f (x)t} consists of all
the irrational numbers in [0, 1]. Thus S ∩∞
n1Sn. If S were to be a countable union
of intervals, then each interval would have to consist of at most one point (because
any interval consisting of more than one point contains a rational number, which
cannot be in S), thereby making their union a countable set, which S is not.
The above exercise illustrates that it can happen for a sequence {f n}n≥1 of step
functions converging at each point to a limit function f that all functions of the
sequence have perfectly simple distributions and yet the limit function has none.
This jarring situation has arisen because although (in the notation of the solution
presented above) S ∩∞
n1Sn, where each Sn is a finite union of intervals, the set S
is not even a countable union of intervals. Of course, one has the option of switching
over to the complement of S in [0, 1], which consists of all the rational numbers in
[0, 1] and is thus a countable union of the single point intervals [0, 0], [1, 1] and
[rn, rn], whose total length can be taken as 0+0+…0. This makes it reasonable
to think of the “total length” of S as being 1, although it has not been arrived at as a
sum of lengths.
Whatever we understand by total length of a more general kind of a set has to go
beyond a mere totaling process.
The obstacle will eventually be surmounted in Chap. 4, but only after we have
looked at some other benefits that will accrue from considering distributions, some-
thing we begin doing in the next section.

1.1.6. Remark. In summary: In order to arrange for a more general class of functions
to have distributions that are sure to be decreasing functions, what we would like
is that the total length idea be extended—if necessary by going beyond a totaling
process—so as to ensure that:
(1) it is applicable to a class of sets that is broad enough to include (i) unions of
countable families of sets in the class and (ii) complements of sets in the class;
(intersections of countable families of sets in the class will then necessarily be
included);
(2) the total length (in the extended sense) of a set is never less than that of a subset,
so that a distribution is sure to be a decreasing function.
The complement of the set S occurring in Exercise 1.1.5 was seen to be the union
of a countable number of disjoint single point intervals, and the totaling process to
obtain its length led to a series with its nth partial sum representing the sum of the
lengths of the first n intervals. This would have been so even if the lengths of the
intervals were not 0. This suggests that the total length in the extended sense should
also have the property that
(3) the (extended) total length of a set S ∪∞
n1Sn, where any two among the sets
Sn are disjoint, agrees with the sum of the series formed by (extended) total
lengths of Sn.
The other concepts that we shall develop by using distributions have far- reaching
consequences. For one of them, the reader may go through the statement of Exercise
3.1.10 right away.
Problem Set 1.1
1.1.P1. For each t ≥0, find S {x ∈[0, 4]: f (x)t}, where f : [0, 4] → R is the
function given by:
f (x) 6 if 0 ≤ x 1, f (1) 5, f (x) 2 if 1 x 3, f (3) 1,
f (x) 6 if 3 x ≤ 4.
1.1.P2. For f : [1, 2] → Rgivenbyf (x)ex
,findthedistribution ˜
f : [0, sup f ] → R
and show that
sup f
0
˜
f agrees with
2
1 f .
1.1.P3. Let A 0. For the function f : [0, A] → R given by f (x) e−x2
, find the
distribution ˜
f : [0, sup f ] → R and show that
sup f
0
˜
f agrees with
A
1 f .
Hint: Use an appropriate substitution and integrate by parts.
1.1.P4. Abel summation formula. If a1, . . . , an and b1, . . . , bn are two finite
sequences of n terms each (n ≥2) and Sk b1 + · · · + bk, then bk Sk −Sk−1
for k 2, . . . , n and therefore
a1b1 + · · · + anbn a1S1 + a2(S2 − S1) + a3(S3 − S2) + · · · + an(Sn − Sn−1)
(a1 − a2)S1 + (a2 − a3)S2 + (a3 − a4)S3

10 1 Preliminaries
+ · · · + (an−1 − an)Sn−1 + an Sn.
In notation:
Let {ak}n
k=1 and {bk}n
k=1 be finite sequences of numbers, where n ≥2, and let
Sk
k
j1 bj for 1≤k ≤n. Then
n

k1
akbk
n−1

k1
(ak − ak+1)Sk + an Sn.
Prove this in notation without resorting to algebraic manipulations concealed
behind an ellipsis · · · How should Tk be defined in order to obtain the alternative
version
n
k1 akbk a1T1 +
n
k2 (ak − ak−1)Tk?
1.1.P5. (Relevant for Problem 1.1.P7 and to be used in Theorem 4.3.1) If one visu-
alizes two disjoint intervals on the real line, one of them lies entirely on the left of the
other. Prove the following precise formulation of this idea for bounded intervals: If
two bounded nonempty intervals are disjoint, then the right endpoint of one of them
is less than or equal to the left endpoint of the other.
Deduce that, if the union of two disjoint intervals is an interval, then the sum of
their lengths is equal to the length of the union.
1.1.P6. Suppose F is a finite family of disjoint bounded intervals. Show that there
exists a nonempty finite family G of bounded intervals such that
(a) a union of two distinct intervals in G is never an interval;
(b) ∪G ∪F (Here “∪H” means union of all the intervals in H);
(c) the total length of the intervals in G is the same as for F.
[Note that the intervals in G are necessarily disjoint in view of (a).]
1.1.P7. If an interval I has an endpoint that belongs to an interval J, show that the
union I ∪ J is an interval. (Only a left endpoint need be considered.)
1.1.P8. Let G be a finite nonempty family of bounded intervals such that a union of
two distinct intervals of G is never an interval. Suppose F is a finite family of disjoint
bounded intervals such that ∪F ∪G. Show that every interval of F is contained in
some interval of G. What more can be said if F too is nonempty and has the property
that a union of two distinct intervals of it is never an interval?
1.1.P9. Let F1 and F2 be finite families of bounded intervals, each having the prop-
erty that its intervals are disjoint. If ∪F1 ∪F2, show that the total length of the
intervals of each family is the same.
1.2 Improper Integrals
Towards theendof theprecedingsection, weencounteredsomedifficultyinassigning
a size, or total length, to certain subsets of R. What we did have was an instance

1.2 Improper Integrals 11
of a family F of subsets of an interval, with each set A ∈ F having an associated
nonnegative real number; in effect, a function μ: F → R such that μ(A)≥0 for every
A ∈ F. It is easy to conceive of other instances of a family F of subsets of some set
X with a nonnegative-valued function μ: F → R. In fact, here is a trivial example:
Let X consist of two distinct elements a and b, and let F consist of all subsets.
Define μ by setting μ(∅)0, μ({a})2, μ({b})3 and μ(X)5.
The fact that so trivial an example, lacking any ostensible purpose, is possible
testifies to the wide sweep of the idea. It turns out however that examples that do
have an ostensible purpose and are equally far removed from the original one in
which X is an interval are also possible. Readers who have worked with probability
will recognize that, in what is called a sample space, the family F of subsets called
“events” has a probability μ: F → R defined on it. In recent years, there has been
great interest in what are called belief measure, plausibility measure and possibility
measure, all of which are cases of a number μ(A) being assigned to every subset A
of some set. For details, the reader may consult the book “Fuzzy Measure Theory”
by Wang and Klir [11].
A function whose domain consists of some or all subsets of a set is sometimes
called a set function. All the functions in the above three paragraphs are set functions.
What is of immediate interest against the background of the preceding section
is that the concept of distribution makes sense in such a general framework. If a
nonnegative-valued function f : X → R has the property that
{x ∈ X : f (x) t} belongs to the family Ffor every t ≥ 0,
then μ({x ∈X : f (x)t}) is meaningful and we can take it as ˜
f (t) for t ≥0.
If f takes only finitely many values, then as observed before for step functions,
the set {x ∈X : f (x)t} remains unchanged as t varies in the open interval between
two consecutive values; thus ˜
f : [0, ∞) → R is a step function and therefore has a
Riemann integral on any bounded subinterval of its domain. Since it is 0 on [sup f ,
∞), the integral
M
0
˜
f is the same for all M ≥sup f . Therefore
lim
M→∞
M
0
˜
f
sup f
0
˜
f .
The limit here is of course the improper integral
M
0
˜
f .
To accommodate functions that may take infinitely many values, in which case
the distribution may fail to be a step function, we would like μ to be monotone in the
following sense: A ⊆B ⇒μ(A)≤μ(B); this ensures that ˜
f is decreasing, because
t1 t2 ⇒ {x ∈ X: f (x) t1} ⊆ {x ∈ X: f (x) t2}.
Since ˜
f is decreasing, we know that the integral
M
0
˜
f exists for all M (see Theo-
rem 9.3.3 of Shirali and Vasudeva [9] or 6.9 of Rudin [7]) and increases with M in

12 1 Preliminaries
view of the nonnegativity of ˜
f . Consequently, the limit
∞
0
˜
f lim
M→∞
M
0
˜
f exists
either as a real number or as ∞.
1.2.1 Remark. We have not ruled out the possibility that

˜
f ∞. We also do not
wish to rule out the possibility that the size of a subset is ∞, so that we can take X
to be R or [0, ∞); this means allowing ˜
f to take ∞ as a value. We shall therefore
need some conventions about ∞ when it appears in a computation, and we state them
below. If one or more of the terms in a statement or computation turn out to be ∞, then
the validity of the assertion will generally have to be checked separately by applying
the conventions we are about to articulate. Doing so is usually straightforward and
details will be left to the reader. In adopting these conventions, we are introducing
∞ as a mathematical object and no longer regarding it as just convenient shorthand.
(a) x ∞ for every x ∈ R; ∞ ≮ ∞;
(b) x + ∞ ∞ + x ∞ for every x ∈ R;
(c) ∞ + ∞ ∞;
(d) ∞ · ∞ ∞; ∞p
∞ for every p 0;
(e) x · ∞ ∞ · x ∞ for every positive x ∈ R;
(f) ∞ · 0 0 · ∞ 0.
Because of (a), the statement

˜
f ∞ [or ˜
f (t) ∞ ] is equivalent to

˜
f ∞
[or ˜
f (t) ∞], which is also expressed by saying that

˜
f [or ˜
f (t)] is finite.
Furthermore, 0∞ and therefore ∞ is regarded as positive but only “extended
real”. Thus, a function taking values in {x ∈ R: x ≥ 0} ∪ {∞}, a set to be denoted
henceforth by R∗+
, is said to be nonnegative extended real-valued. Note that such
a function need not actually take the value ∞. The symbol R+
will denote the set
{x ∈ R: x ≥ 0} of nonnegative real numbers; a function taking values in R+
will be
described as nonnegative real-valued.
We also adopt the convention that the supremum of a nonempty set that is not
bounded above is ∞. Moreover, for a sequence {sn}n≥1 in R∗+
, we take limn→∞ sn
∞ to mean that
for every A 0 there exists a natural number n0 such that n n0 ⇒ sn A.
Then it is easy to see that, as far as an increasing sequence {sn}n≥1 is concerned, the
statements
lim
n→∞
sn ∞ and sup{sn : n ≥ 1} ∞
both mean that the sequence is unbounded above. Therefore, for such a sequence,
we have
lim
n→∞
sn sup{sn : ≥ 1}.
It is elementary that this equality holds good for a bounded increasing sequence (see
Theorem 3.3.3 of Shirali and Vasudeva [9] or 3.14 of Rudin [7]). Thus it holds for
any increasing sequence, whether bounded or not. In particular, the sum of a series

of nonnegative terms is the supremum of its partial sums, and the sum is ∞ if and
only if the partial sums are unbounded. We shall later need the consequence of the
“rearrangement theorem” (see Theorem 4.2.4 of Shirali and Vasudeva [9] or 3.55 of
Rudin [7]) that the sum of a series of nonnegative terms is independent of the order
of terms even if the sum is ∞.
For increasing sequences, we have limn→∞(sn + tn) limn→∞ sn + limn→∞ tn
whether the limits are finite or not; similarly, for nonempty sets, we have sup {s+t: s ∈
S, t ∈ T } sup S + sup T , whether the suprema are finite or not (obvious when one
supremum is ∞; for finite suprema, see Problem 1.6.P6 of Shirali and Vasudeva [9]).
The first part can be summarized as “the limit of a finite sum of increasing sequences
is the sum of their separate limits”.
If one term in a series of nonnegative terms is ∞, then every partial sum up to
that term and beyond is ∞ and hence so is the sum of the series.
Suppose that for each n ∈ N, we have a series
∞
k1 an,k of nonnegative terms in
R∗+
.Thentheirsumsformasequenceofnonnegativeterms,usingwhichonecouldset
up a series
∞
n1
∞
k1 an,k. What this means is that we have a function a: N×N →
R∗+
, called a double sequence, and an associated repeated sum
∞
n1
∞
k1 an,k. By
reversing the roles of n and k, we get another repeated sum
∞
k1
∞
n1 an,k. It will
be useful to note that values of the two sums are the same. This is of course trivial
when one of the terms an,k is ∞. To see why it is true when every term is finite,
consider any N, K ∈ N. It is elementary that
N
n1
K
k1 an,k
K
k1
N
n1 an,k.
Therefore
∞

k1
N

n1
an,k lim
K→∞
K

k1
N

n1
an,k lim
K→∞
N

n1
K

k1
an,k
N

n1
∞

k1
an,k,
where the final equality is based on the fact that the limit as K→∞ of the sum of the N
sequences
K
k1 an,k is thesumof their separatelimits. Sinceeachan,k is nonnegative,
we can now argue that
∞
k1
∞
n1 an,k ≥
∞
k1
N
n1 an,k
N
n1
∞
k1 an,k.
Upon taking the limit as N→∞, we obtain
∞
k1
∞
n1 an,k ≥
∞
n1
∞
k1 an,k.
The reverse inequality can be established by an analogous argument and therefore
∞
n1
∞
k1 an,k
∞
k1
∞
n1 an,k (“interchanging the order of summation”). For
a detailed discussion of this topic, particularly when an,k is real or complex and not
restricted to be nonnegative when real, the reader may consult the article “Double
sequences and double series” by Habil [4].
The concepts of liminf and limsup carry over quite smoothly and we shall
encounter the liminf in Theorem 3.2.3.
We have not introduced −∞, although we could have done so by extending appro-
priately the list (a)–(e) above. Therefore, care must be taken to avoid such expressions
as −

˜
f or − ˜
f without ensuring first that the integral or function concerned is finite-
valued.
We have encountered
∞
0
˜
f before and now we need to lay down what such an
“improper” integral means when the integrand can have an infinite value and need
not vanish beyond some M 0. However, we need consider only functions that are

14 1 Preliminaries
decreasing and take nonnegative values, ∞ included; in other words, decreasing
nonnegative extended real-valued functions. They need not be distributions of any
functions, and accordingly, we shall dispense with the tilde in the rest of this section.
1.2.2. Definition. For a decreasing nonnegative extended real-valued function f on
[0, ∞), we define the integral from 0 to ∞ as
∞
0
f sup{
b
a
g: 0 a b ∞; g: [a, b] → R Riemann integrable
and 0 ≤ g ≤ f on [a, b]}. (1.6)
It is understood here that g is real-valued, because otherwise it would not be Riemann
integrable.
This is similar to the “extended integral” defined by Munkres [5, p. 121], except
that the domain of integration here is always [0, ∞) and the integrand must be
decreasing, but the integrand as well as the integral may take ∞ as a value.
Consider the case when f is real-valued except possibly at 0. Since it is decreasing,
its restriction to any [a, b], where 0a b ∞, is Riemann integrable. Moreover,
b
a f ≥
b
a g for any g of the kind mentioned in (1.6). It follows that
f real-valued on (0, ∞) ⇒
∞
0
f sup{
b
a
f : 0 a b ∞}. (1.7)
If f is not real-valued on (0, ∞), that is to say, f (α)∞ for some real α0, then by
considering a sequence of nonnegative functions gn ≤f such that gn(x)n on [0, α],
we find that
∞
0 f ∞. Therefore
∞
0
f ∞ ⇒ f real-valued on (0, ∞). (1.8)
Now suppose f is real-valued on (0, ∞). If f (0) is also real, then f is Riemann
integrable over [0, A] for every A 0, because it is decreasing. The usual meaning of
the improper integral
∞
0 f is then limA→∞
A
0 f which agrees with the supremum in
(1.6) because f is nonnegative-valued. If f (0)∞, then f is Riemann integrable over
[x, B] whenever 0x B ∞. Two possibilities arise. One is that f vanishes beyond
some B 0; the usual meaning of the improper integral
∞
0 f is then limx→0
B
x f ,
which again agrees with the supremum in (1.6). The other possibility is that there
is no such B. In this situation, the usual meaning of the improper integral
∞
0 f is
limx→0
B
x f +limA→∞
A
B f , which is independent of the chosen B 0. With a little
effort, one can see that this too agrees with the supremum in (1.6). Here, the result
that the sum of the integrals of a function on contiguous intervals [α, β] and [β, γ]
equals the integral of that function over [α, γ] is relevant.

We conclude from the above paragraph that when f is real-valued on (0, ∞), the
integral
∞
0 f in the sense of Definition 1.2.2 can be computed as usual.
1.2.3. Lemma. If {gn}n≥1 is a sequence of real-valued functions on an interval [a,
b] such that
(a) gn is a decreasing function on [a, b] for each n ∈ N,
(b) gn(x) ≤ gn+1(x) for each x ∈[a, b] and each n ∈ N,
(c) limn→∞ gn(x) g(x) for each x ∈[a, b] and g(a)∞,
then g is Riemann integrable and
b
a
g lim
n→∞
b
a
gn.
Proof: Since (a) and (c) together imply that g is a real-valued decreasing function on
[a, b], it follows that it is Riemann integrable. We need prove only the limit. In view
of (b) and (c), we already know that limn→∞
b
a gn exists and
b
a
g ≥ lim
n→∞
b
a
gn. (1.9)
The proof will be complete as soon as we prove the reverse inequality. For this
purpose, consider any ε0. There exists a partition P: a x0 x1 . . . xm b
of [a, b] such that the lower sum L(g, P) of g over P satisfies
b
a
g − L(g, P)
ε
2
. (1.10)
Since g is decreasing, its infimum on any interval [xj−1, xj] is g(xj) and therefore
L(g, P)
m

j1
g(xj )(xj − xj−1),
and correspondingly for each gn. Hence for each n ∈ N, the following equality holds:
L(g, P) − L(gn, P)
m

j1
(g(xj ) − gn(xj ))(xj − xj−1). (1.11)
In view of (c), there exists an N ∈ N such that N ≤ n ∈ N implies
g(xj ) − gn(xj )
1
2
·
ε
b − a
for 1 ≤ j ≤ m.

16 1 Preliminaries
Together with (1.11), this implies
L(g, P) − L(gn, P)
ε
2
,
which, together with (1.10), yields
b
a g − L(gn, P) ε for n ≥N. Consequently,
b
a
g − ε L(gn, P) ≤
b
a
gn for n ≥ N.
Since limn→∞
b
a gn exists, it follows that
b
a
g − ε ≤ lim
n→∞
b
a
gn.
This has been shown to be true for every ε0. Therefore the reverse of the inequality
(1.9) must hold. As noted earlier, this is all that needed to be proved.
1.2.4. Proposition. If {f n}n≥1 is a sequence of nonnegative extended real-valued
functions on the interval [0, ∞) such that
(a) f n is a decreasing function on [0, ∞) for each n ∈ N,
(b) 0≤f n(t)≤f n+1(t) for each t ∈[0, ∞) and each n ∈ N,
(c) limn→∞ fn(t) f (t) for each t ∈[0, ∞),
then
∞
a f limn→∞
∞
a fn.
Proof: First suppose
∞
a f is not ∞. Then by the consequence (1.8) of Definition
1.2.2, f is real-valued on (0, ∞) and hence by (1.7), for any ε0, there exist a and b
such that 0a ≤b ∞ and
0 ≤
∞
0
f −
b
a
f
ε
2
.
Since f is real-valued on (0, ∞), it follows by (b) and (c) of the hypothesis that each
f n is real-valued on (0, ∞). Therefore by Lemma 1.2.3 and by (a) of the hypothesis,
there exists an N ∈ N such that
n ≥ N ⇒ 0 ≤
b
a
f −
b
a
fn
ε
2
⇒ 0 ≤
∞
0
f −
b
a
fn ε.
But 0 ≤
∞
0 f −
∞
0 fn ≤
∞
0 f −
b
a fn. Therefore

n ≥ N ⇒ 0 ≤
∞
0
f −
∞
0
fn ε.
This proves the contention when
∞
a f is not ∞.
Now, suppose
∞
a f ∞ and that f is real-valued on (0, ∞). Then the same is
true of each f n in view of (b) and (c) of the hypothesis. By (1.7), for any K 0, there
exist a and b such that 0a ≤b ∞ and ∞
∞
a f K. The foregoing lemma
yields an integer N ∈ N such that
n ≥ N ⇒ 0 ≤
b
a
f −
b
a
fn
b
a
f − K ⇒
b
a
fn K.
Therefore
∞
a f limn→∞
∞
a fn also when
∞
a f ∞ and f is real-valued on (0,
∞).
Finally, suppose
∞
a f ∞ but f is not real-valued on (0, ∞). Then f (α)∞
for some real α0. From (b) and (c) of the hypothesis, we deduce that for any M
0, there exists an N ∈ N such that n ≥ N ⇒ fn(α) ≥ M. It follows from (a) that
n ≥ N ⇒ fn ≥ M on [α
2
, α], so that
∞
0 fn ≥
∞
α/2 M α
2
M. Since such an N exists
for any M 0, we have limn→∞
∞
a fn ∞
∞
a f .
Problem Set 1.2
1.2.P1. For the double sequence a: N × N → R given by
an,k 1 if n k, an,k −1 if k n + 1 and an,k 0 in all other cases,
show that the repeated sums
∞
n1
∞
k1 an,k and
∞
k1
∞
n1 an,k are unequal,
although
∞
n1 |
∞
k1 an,k| and
∞
k1 |
∞
n1 an,k| are convergent.
Hint: One can think of an,k as representing an infinite matrix with first three rows:
1 −1 0 0 . . .
0 1 −1 0 0 . . .
0 0 1 −1 0 0 . . .
1.2.P2. Let b 0 and g be a decreasing nonnegative function on [0, b] that is real-
valued except perhaps at 0. Denote by f the decreasing nonnegative function on [0,
∞) obtained by extending g to be equal to 0 to the right of b. Show that
∞
0
f lim
x→0
b
x
g.
Note: If g(0)∞, the function g is Riemann integrable on [0, b] and the limit here
is equal to
b
a g (see Problem 9.3.P2 of Shirali and Vasudeva [9]).

18 1 Preliminaries
1.2.P3. In Lemma 1.2.3, show that the condition g(a)∞ does not follow from the
remaining conditions in the hypothesis even if we impose the additional condition
that g is real-valued on (a, b]. If g(a)∞ but g (which must be decreasing) is real-
valued on (a, b], then the improper integral
b
a g in the usual sense is limx→0
b
x g,
though it may be ∞; show that
∞
0 g limn→∞
b
a gn. If g is not real-valued on (a,
b], which to say, g(α)∞ for some αa, show that limn→∞
b
a gn ∞.

Chapter 2
Measure Space and Integral
2.1 Measure and Measurability
The intent behind the definitions that are about to follow may be easier to appreciate
if Remark 1.1.6 is kept in view. Symbol-free descriptions of the properties in the
definitions are given alongside.
The complement of a set will be indicated by a superscript “c”; thus, the symbol
Ac
will mean the complement of the set A.
2.1.1. Definition. A family F of subsets of a nonempty set X is called a σ-algebra
(or a σ-field) if
(a) ∅ ∈ F (contains the empty set),
(b) for every sequence {Ak}k≥1 of sets, each belonging to F,
∪∞
k1 Ak ∈ F (closed under countable unions),
(c) Ac
∈ F whenever A ∈ F (closed under complements).
Since X ∅c
, conditions (a) and (c) in the definition together imply that X ∈ F.
Also, conditions (b) and (c) together imply closure under countable intersections,
considering that
∞

k1
Ak (
∞

k1
Ac
k)c
.
From conditions (a) and (b), we infer that a σ-algebra is closed under finite unions
as well:
if Ak ∈ F for 1 ≤ k ≤ n, then
n

k1
Ak ∈ F.
19

20 2 Measure Space and Integral
This is true because we may take Ak ∅ for k n. The same goes for finite inter-
sections as well. In particular, if A and B belong to F, then so does the set-theoretic
difference AB, which is defined to be A ∩ Bc
. These facts and their simple conse-
quences will be used without further ado.
We shall speak of three or more sets as being disjoint when no two of them have an
element in common. This is the same as what many authors call “mutually disjoint”
or “pairwise disjoint”.
2.1.2. Definition. A nonnegative extended real-valued set function μ: F → R∗+
,
where F is a σ-algebra of subsets of a set X, is called a measure if
(a) μ(A)≥0 for every A ∈ F (nonnegative),
(b) μ(∅)0 (vanishes at the empty set),
(c) for every sequence {An}n≥1 of disjoint sets, each belonging to F,
μ
∞

n1
An

∞

n1
μ(An)(countably additive).
The triplet (X, F, μ) is called a measure space.
The backdrop for countable additivity is requirement (3) of Remark 1.1.6.
Given any finite sequence A1, …, An of disjoint sets belonging to F, it is posible
to extend it to an infinite sequence of disjoint sets belonging to F by appending ∅,
∅, …, and hence properties (b) and (c) of Definition 2.1.2 justify the assertion that
μ
n

k1
Ak

n

k1
μ(An)(finitely additive).
This has the further consequence that A ⊆B ⇒B A ∪(BA)⇒μ(B)μ(A)+
μ(BA) by finite additivity, because A ∩(BA)∅; monotonicity (which means A
⊆B ⇒μ(A)≤μ(B)) now follows from property (a) of Definition 2.1.2.
In the terminology of these definitions, Remark 1.1.6 says that we would like
the total length idea to be extended so as to become a measure μ on a σ-algebra F
that includes all those subsets of [a, b] that are finite unions of disjoint intervals. By
“extend” we mean that for such finite unions, μ must agree with total length.
Elements of a σ-algebra F are called F-measurable, or just measurable espe-
cially when the σ-algebra is understood and has not been denoted by any symbol.
When this is so, it is easier to speak of a measure on a set X, with the tacit understand-
ing that some σ-algebra is intended but may or may not have been named. Similarly,
it is often convenient to speak of X as a measure space, with the tacit understanding
that some σ-algebra and measure are intended but may or may not have been named.
It will be convenient to abbreviate μ(A) as μA when there is no risk of confusion.

2.1 Measure and Measurability 21
2.1.3. Example. (a) Consider the example mentioned at the beginning of Sect.
1.2, in which X consists of two distinct elements a and b, and F consists of all
subsets. The set function μ is defined by setting μ(∅)0, μ({a})2, μ({b})
3 and μ(X)5. It is easy to check that μ is a measure. Let the function f on X
be defined as f (a)1, f (b)3. Then the set S {x ∈X: f (x)t} is seen to be
as described in the next three lines:
X if 0 ≤ t 1
{b} if 1 ≤ t 3 sup f
∅ if 3 ≤ t.
Since F consists of all subsets of X, there is such a thing as μS whatever non-
negative number t may be. Therefore there exists a distribution ˜
f , as explained
in Sect. 1.2.
(b) Let X consist of three distinct elements a, b, c and let F consist of all subsets.
Define μ by setting μ∅ 0, μ{a}2, μ{b}3, μ{c}4, μ{b, c}6, μ{c,
a}5, μ{a, b}4 and μX 8. Obviously, μ is nonnegative and vanishes at the
empty set; one can easily check case by case that it is also monotone. However,
it is not a measure, as it fails to be finitely additive: μ{a}2, μ{b}3 but μ{a,
b}42+3. Next, consider the function f on X defined as f (a)1, f (b)3,
f (c)4. For this function, the set S {x ∈X: f (x)t} is seen to be as described
in the next four lines:
X
{b, c}
{c}
∅
if 0 ≤ t 1
if 1 ≤ t 3
if 3 ≤ t 4 sup f
if 4 ≤ t.
Since F consists of all subsets of X, there is such a thing as μS whatever
nonnegative number t may be. Therefore there is a distribution ˜
f . If F were
to consist of selected subsets, then f would have to have the property that S is
always one of the selected subsets before we could speak of its distribution.
2.1.4. Definition. Let F be a σ-algebra of subsets of a set X. A function f : X →
R ∪ {∞} is said to be F-measurable (or just measurable) if ,
for every t ∈ R, the set S {x ∈ X: f (x) t} is F−measurable,
i.e. belongs to the σ-algebra F.
If there is also an extended real-valued function μ: F → R∗+
, then the distri-
bution of a measurable function f on X is the function f̃ : [0, ∞) → R∗+
such
that
˜
f (t) μ({x ∈ X: f (x) t}).

The distribution of a sum f+g will be denoted by ( f + g
) or (
f + g), depending on
convenience. When we speak of a distribution, μ will usually be understood to be a
measure but will occasionally be just nonnegative, monotone and vanishing at ∅, as
was the case in Example 2.1.3(b). As was indicated just before Remark 1.2.1, this is
enough to ensure that the distribution is nonnegative and decreasing.
If f takes only negative values, then {x ∈X: f (x)t} is empty for any t ≥0 and
hence ˜
f is 0 everywhere; however, we shall have no occasion to consider such a situ-
ation. In fact, we shall be dealing with distributions only of measurable nonnegative
functions although measurability has been defined to be applicable to functions that
may take negative values.
For reasons set forth in Sect. 1.1, when f is nonnegative and bounded, the restric-
tion of ˜
f to [0, supf ] will also be called the distribution.
The set {x ∈X: f (x)t}, sometimes called the “t-cut” of f , will be denoted by X(f
t) when convenient. Without explicit mention of F, one can state that an R ∪ {∞}-
valued function f on a set X is measurable if its t-cut is measurable for every real
number t. Thus, when we speak of a measurable function on a set X, it is understood
that some σ-algebra of subsets of X is intended.
2.1.5. Exercise. Let f , g be measurable functions on a space X with measure μ, and
f (x)g(x) for all x ∈A, where A is a measurable subset of X satisfying μ(Ac
)0.
Show that ˜
f g̃.
Solution: Consider any real t and the t-cuts B and C of f and g respectively. Since
a measure is monotone and μ(Ac
)0, it follows that μ(B ∩Ac
)0. Now B (B
∩Ac
)∪(B ∩A) and additivity of measure implies μ(B)μ(B ∩Ac
)+μ(B ∩A)
μ(B ∩A). This means ˜
f (t) μ(B ∩ A). Similarly, g̃(t) μ(C ∩ A). By hypothesis,
f and g agree on the set A, which implies that the t-cuts B and C satisfy B ∩A C
∩A.
2.1.6. Definition. For any subset A ⊆X, the characteristic function χA is the func-
tion with domain X, for which
χA(x)

1 if x ∈ A
0 if x /
∈ A.
For the characteristic function χA of any subset A ⊆X, the supremum is 1, unless
A is empty, in which case the supremum is 0. Thus the interval 0, sup χA is [0, 1]
if A is nonempty and is the single point interval [0, 0] when A is empty.
It is easy to verify that (i) if A ⊆B, then χA ≤χB (ii) if A ∩B ∅, then χAχB
0 everywhere and χA+χB χA∪B.
In Exercise 1.1.5, the relation between the function f n and the complement Sn
of the set formed by the first n terms of the sequence {rk}k≥1 is described. In the
language of the above definition, the relation is that f n χSn
. The relation described

there between the function f limn→∞ fn and the set S ∩∞
n1Sn is that f χS.
On the basis of the inclusion Sn ⊇Sn+1, we obtain χSn
≥χSn+1
, which is simply a
restatement of the inequality f n ≥f n+1 of that exercise (cf. Problem 2.1.P1).
2.1.7. Proposition. The characteristic function of a subset A ⊆X is measurable
if and only if the subset A is measurable. When there is a measure μ on X, the
characteristic function has a distribution χA, whose restriction to [0, 1] is a step
function having value A on [0, 1) and 0 at 1. When μA 0, this means χA is 0
everywhere. Moreover,
∞
0
χA
sup χA
0
χA μA.
Proof: For t 0, the t-cut X(χA t) is X, which is always measurable (because any
σ-algebra has X as an element). For 0≤t 1, the set X(χA t) is A and for t ≥
sup χA, the set X(χA t) is ∅, which is always measurable (because any σ-algebra
has ∅ as an element). One consequence of this observation is that χA is measurable
if and only if A is. Another consequence is that χA(t) is μA on [0, 1) and 0 at 1. Thus
χA is a step function on [0, 1].
If A ∅, then sup χA 1 and hence
sup χA
0 χA
1
0 χA μA; if A ∅, then
sup χA 0 and hence
sup χA
0 χA
0
0 χA 0 μA.
It is worth noting that the above proposition is valid for any set function μ:
F → R∗+
that vanishes at the empty set.
2.1.8. Example. (a) Let X consist of two elements a and b and F consist of only
∅ and X. It is trivial that F is a σ-algebra. The function f for which f (a)1,
f (b)3, is not measurable because X( f 1) {b} /
∈ F. Therefore it has no
distribution.
(b) Let X, F and μ be as in Example 2.1.3(a). Define the nonnegative functions f
and g as below:
f (a) 1, f (b) 3 and g(a) 5, g(b) 1.
Then (f+g)(a)6, (f+g)(b)4. Also,
˜
f (t)
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
5 0 ≤ t 1
3 1 ≤ t 3
0 3 ≤ t ∞
, g̃(t)
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
5 0 ≤ t 1
2 1 ≤ t 5
0 5 ≤ t ∞
and
( f + g)(t)
⎧
⎪
⎨
⎪
⎩
5 0 ≤ t 4
2 4 ≤ t 6
0 6 ≤ t ∞.

Hence
sup f
0
˜
f 5 + 6 11,
sup g
0
g̃ 5 + 8 13,
and
sup( f +g)
0

( f + g) 20 + 4 24.
Thus
sup f
0
˜
f +
sup g
0
g̃
sup( f +g)
0

( f + g).
(c) Let X, F and μ be as in Example 2.1.3(b). As noted there, μ is not a measure
but is monotone. It also provides an example of what we shall call an outer
measure later on. Define the nonnegative functions f and g as below:
f (a) 1, f (b) 3, f (c) 4 and g(a) 2, g(b) 1, g(c) 4.
Then (f+g)(a)3, (f+g)(b)4, (f+g)(c)8. Also,
˜
f (t)
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
8 0 ≤ t 1
6 1 ≤ t 3
4 3 ≤ t 4
0 4 ≤ t ∞
, g̃(t)
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
8 0 ≤ t 1
5 1 ≤ t 2
4 2 ≤ t 4
0 4 ≤ t ∞
and

( f + g)(t)
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
8 0 ≤ t 3
6 3 ≤ t 4
4 4 ≤ t 8
0 8 ≤ t ∞.
Hence

sup f
0
˜
f 8 + 12 + 4 24,
sup g
0
g̃ 8 + 5 + 8 21,
and
sup( f +g)
0

( f + g) 24 + 6 + 16 46.
Thus
sup f
0
˜
f +
sup g
0
g̃
sup( f +g)
0

( f + g).
The fact that we do not have equality here has to do with μ not being a measure.
One might have liked the integral of the sum to appear on the smaller side of
the inequality. However, the inequality

sup( f +g)
0

( f + g) ≤

sup f
0
˜
f +

sup g
0
g̃,
in which the integral of the sum does appear on the smaller side, is found to
hold in the present instance. Its validity for general nonnegative f and g can be
traced to a property that μ has, called subadditivity (see Sect. 3.3).
Problem Set 2.1
2.1.P1. Let {Sn}n≥1 be a sequence of subsets of a set X such that Sn ⊆Sn+1 for
every n ∈ N (no σ-algebra intended) and let S ∪∞
n1Sn. Denote the characteristic
functions of Sn and of S respectively by f n and f . For each x ∈X, show that
(i) fn(x) ≤ fn+1(x) and (ii) f (x) lim
n→∞
fn(x).
2.1.P2. If f and g are functions such that f ≤g everywhere, show for any real t that
the t-cut of f is a subset of the t-cut of g. If f and g are also measurable and μ is a
measure on their domain, show that ˜
f (t) ≤ g̃(t) for every t.
2.1.P3. Let {f n}n≥1 be a sequence of functions on X such that, for every x ∈X and
n ∈ N, the inequality f n(x)≤f n+1(x) holds. For any t ∈ R , denote the t-cut of f n by
Sn(t). Show that, Sn(t)⊆Sn+1(t) for every n ∈ N. If furthermore, f is a function on X
such that f (x) limn→∞ fn(x) for every x ∈X, show that X( f t) ∪∞
n1Sn(t).
Hence conclude that if each f n is measurable, then f is measurable. (The latter part is
true without the hypothesis that f n(x)≤f n+1(x), as is to be proved in Problem 2.2.P4.)

2.1.P4. (Note: This problem is of interest in Combinatorics rather than Measure
Theory, because it relates to the Inclusion-Exclusion Principle there; see Problem
3.1.P1.) Let A1, …, An be subsets of X. Using the symbol χ[B] to denote the char-
acteristic function of a set B, show that
χ[A1 ∪ · · · ∪ An]
n

r1
⎛
⎝(−1)r−1

1≤ j1··· jr ≤n
χ[Aj1
∩ · · · ∩ Ajr
]
⎞
⎠.
2.1.P5. If μ: F → R is monotone as well as finitely additive, then show that it is
also finitely subadditive in the sense that μ(A ∪B)≤μ(A)+μ(B) for measurable sets
A and B, disjoint or not.
2.1.P6. Let μ be as in Example 2.1.3(b) and the functions f , g be defined on X
{a, b, c} as
f (a) 1, f (b)
√
3, f (c) 2, g(a)
√
2, g(b) 1, g(c) 2.
Show that
3

sup( f +g)2
0

( f + g)
2

3

sup f 2
0
( f 2) +
3

sup g2
0
(
g2)
but

sup( f +g)2
0

( f + g)
2

sup f 2
0
( f 2) +

sup g2
0
(
g2).
2.1.P7. Show that a measure is countably subadditive in the following sense: If
(X, F, μ) is a measure space and

Aj

j∈N
is a sequence of measurable sets, then
μ(∪∞
j1 Aj ) ≤
∞
i1 μ

Aj

. Is the corresponding statement true regarding a finite
sequence of measurable sets?
2.1.P8. Let F be a σ-algebra of subsets of a set X. Suppose μ: F → R∗+
is finitely
additive and vanishes at ∅. If it is also countably subadditive (as defined in Problem
2.1.P7), show that it is a measure.
2.1.P9. Suppose that, for each j ∈ N, Fj is a σ-algebra of subsets of Xj. Define the
family F of sets to consist of those subsets A of the union X ∪∞
j1 X j for which A
∩Xj ∈ Fj for each j ∈ N. Show that F is a σ-algebra.
2.1.P10. For each j ∈ N, let (Xj, Fj ,μj) be a measure space and denote the union
∪∞
j1 X j by X. Set F {A ⊆X: A ∩Xj ∈ Fj for each j ∈ N } and for each A ∈ F,
set μ(A)
∞
i1 μj

A ∩ X j

. Show that μ is a measure.

2.1.P11. Let F be a σ-algebra of subsets of a nonempty set X and Y any set with
a map F: X→Y. Take G to be the family of subsets {B ⊆Y: F−1
(B)∈ F } of Y.
Here, F−1
(B) means {x ∈X: F(x)∈B}, as usual. Show that G is a σ-algebra. If μ is
a measure on F, show that the map ν: G → R∗+
defined by ν(B)μ(F−1
(B)) is a
measure on G.
2.1.P12. Let F be a σ-algebra of subsets of a set X, and E ∈ F have the property
that every subset of it belongs to F.
(a) If B ⊆A ∪E, show that Ac
∩B ∈ F.
(b) If B ⊆A ∪E, B ∈ F, show that A ∩B ∈ F.
(c) If B ⊆A ∪E, A ⊆B ∪E, B ∈ F, show that A ∈ F.
2.1.P13. Prove that the result of Exercise 2.1.5 holds even if the set function μ: F →
R∗+
is assumed to be only finitely additive provided it is monotone and vanishes at
∅.
2.2 Measurability and the Integral
As noted just after Definition 2.1.4, the monotonicity of measure guarantees that
a distribution of a measurable nonnegative function f is a decreasing nonnegative
function and therefore
∞
0
˜
f exists either as a real number (we then call it finite) or
as ∞. Therefore the following definition makes sense.
2.2.1. Definition. Suppose μ is a measure on a set X. For a measurable nonnegative
function f : X → R∗+
, the measure space integral is defined to be
f
∞
0
˜
f .
Oftenitmakesforeasierreadingifwewrite X f, X f dμor X f (x)dμ(x)inplace
of f , especially if two or more measure spaces are involved in the discussion.
For a discussion of the equivalence of this definition with the more traditional
one, see 4.7.1 of Craven [1, p. 128]. The Abel summation formula is used.
We shall abbreviate the name to simply integral. When it is necessary to specify
the measure μ being used, as may happen if there is another measure relevant to the
context, we write the integral as f dμ. Sometimes, the function is known by its
expression f (x) without any symbol such as f having been introduced for it; when this
is so, the integral may be more conveniently denoted by f (x)dx or f (x)dμ(x).
In terms of the integral, the conclusion of Example 2.1.8(b) can be rephrased as
f 11, g 13 and ( f + g) 24 f + g. Also, Proposition 2.1.7 can be
phrased as: χA μA for any measurable set A ⊆X.

If the desired extension indicated in Remark 1.1.6 works out to be a measure on
a σ-algebra, then Proposition 1.1.3 says that the Riemann integral of a step function
agrees with its measure space integral. Moreover, Examples 1.1.4 would show that
the same is true of the functions discussed there.
Henceforth a measurable nonnegative-valued function will be understood to be
extended real-valued unless the context indicates that the function is not permitted
to take the value ∞.
2.2.2. Proposition. (a) Measurability of f : X → R ∪ {∞} is equivalent to each
of the following:
(i) X(f ≥ t) is measurable for every t ∈ R;
(ii) X(f t) is measurable for every t ∈ R;
(iii) X(f ≤ t) is measurable for every t ∈ R.
(b) Measurability of f : X → R implies that of − f ; measurability of f : X →
R ∪ {∞} implies that of |f |.
Proof: (a) Assume f measurable and let t ∈ R. Then for each n ∈ N, the set
X( f t − 1
n
) is measurable. Now, f (x) ≥ t ⇔ f (x) t − 1
n
for every n ∈ N.
Therefore X( f ≥ t) ∩∞
n1 X( f t − 1
n
). Since a σ-algebra is closed under
countable intersections, the set X(f ≥t) is measurable. Thus measurability of f
implies the statement (i).
(i)⇒(ii) because a σ-algebra is closed under complements.
By an argument analogous to the one above, we can prove (ii)⇒(iii).
The reason why (iii) implies measurability of f is the same as why (i)⇒(ii),
namely, that a σ-algebra is closed under complements.
(b) Left as Problem 2.2.P1.
We shall next show that certain simple algebraic combinations of measurable
function are measurable (“algebra of measurable functions”). Their limits are
deferred to a later section.
2.2.3. Theorem. If f : X → R ∪ {∞} and g: X → R ∪ {∞} are measurable
functions and c ∈ R, then the following functions are also measurable:
(a) cf provided that either f is real-valued or c ≥0;
(b) f+g;
(c) fg provided that either both are real-valued or both are nonnegative;
(d) min {f , g} and max {f , g}.
Proof: (a) If c 0, the statement is trivial. If c 0, then
X(cf t) X( f
t
c
)
and the set on the right side is measurable. For c 0, apply Proposition 2.2.2(b).

2.2 Measurability and the Integral 29
(b) Let X1 X(g ∞) and X2 be the complement. Consider any x ∈ X1. If there
exists a rational number r such that
f (x) r and g(x) t − r, (2.1)
then the inequality
( f + g)(x) t (2.2)
also holds. Conversely, if (2.2) holds, then f (x)t − g(x) and so, there exists a
rational number r such that f (x)r t −g(x), i.e. (2.1) holds. Thus the existence
of a rational number r satisfying (2.1) is equivalent to (2.2). This means
X1 ∩ X( f + g t) X1 ∩

r∈Q
(X( f r) ∩ X(g t − r))

.
On the other hand, the same equality holds with X1 replaced by its complement
X2 because both sides are then equal to X2. Upon equating the unions of the left
sides and the right sides of the two equalities, we obtain
X( f + g t)

r∈Q
(X( f r) ∩ X(g t − r)).
The sets X(f r)∩X(g t −r) on the right side of this equality are measurable
and are countable in number (i.e. can be arranged in a sequence). Therefore
their union, which has been shown to be X(f +g t), must be measurable.
(c) Here we shall prove only the case when both functions are real-valued. The case
when both are nonnegative extended real-valued is left as Problem 2.2.P11. First
we shall prove that measurability of f implies that of f 2
. If t 0, then X(f 2
t)
X, which is measurable. If t ≥0, then f (x)2
t is equivalent to the assertion that
either f (x)
√
t or f (x) −
√
t and hence we have X( f 2
t) X( f
√
t) ∪ X( f −
√
t), which is measurable by Proposition 2.2.2(a). To prove
measurability of the product fg, we note that
f g
1
2

( f + g)2
− f 2
− g2

when f and g are both real-valued, and apply parts (a) and (b).
(d) For an arbitrary real number t, we have
max{ f, g}(x) t if and only if f (x) t or g(x) t.
Therefore
X(max{ f, g} t) X( f t) ∪ X(g t).

Since both sets on the right are measurable, the one on the left is measurable. It
follows that max{f , g} is measurable. The argument for min{f , g} is similar.
We need the following simple fact to proceed further.
2.2.4. Proposition. For any x ∈ R ∪ {∞}, let
x+
max{x, 0}, x−
max{−x, 0} i f x ∈ R,
and
∞+
∞, ∞−
0.
(a) Then
x+
≥ 0, x−
≥ 0,
x x+
− x−
and |x| x+
+ x−
,
where |∞| means ∞. In particular,
x+
≤ |x| and x−
≤ |x|.
(b) If x ∈ R,
x+

1
2
(|x| + x) and x−

1
2
(|x| − x).
(c) Moreover, if x y −z, where y ≥0 and z ≥0, then y ≥x+
and z ≥x−
. (Obviously,
z cannot be ∞ here.)
Proof: (a) Since this is trivial when x ∞, we need only consider x ∈ R. If x ≥0,
then x+
max{x, 0}x ≥0, x−
max{−x, 0}0≥0 and |x|x. Therefore
x+
− x−
x −0x and x+
+ x−
x + 0x |x|. In the contrary case when x
0, we have x+
max{x, 0}0≥0, x−
max{−x, 0}−x ≥0 and |x|−x.
Therefore x+
− x−
0−(−x)x and x+
+ x−
0+(−x)−x |x|. Thus the
two equalities hold in both cases.
(b) This follows from (a) by an elementary computation.
(c) Since z cannot be ∞ here (otherwise “y − z” would make no sense), x and y
are both ∞ or both finite. If both are ∞, the truth of the assertion is trivial. In
the contrary case when x, y, z are all finite, y x+z ≥x and y ≥0, whence y
≥max{x, 0}x+
. Also, z y −x ≥−x and z ≥0, whence z≥max{−x, 0}
x−
.

Using Proposition 2.2.4, for any real-valued function f on any set X, one can form
the associated functions f +
max{f ,0} and f _
max{_
f ,0}, which then have the
property that
f f +
− f −
and | f | f +
+ f −
.
When f is nonnegative, we have f +
f everywhere and f −
0 everywhere. It
follows that if X is a measure space and f ≥0 is measurable, then f +
f and
f −
0, so that f f +
− f −
. For a general measurable f that need not be
nonnegative, it is still true in view of Theorem 2.2.3 that the nonnegative functions
f +
and f −
are measurable and f +
− f −
makes sense at least when both integrals
are real numbers.
2.2.5. Definition. Suppose μ is a measure on a set X. For any measurable real-
valued function f : X → R, the measure space integral is defined to be
f f +
− f −
,
provided that f −
is finite (i.e. is not ∞). The function f is said to be summable or
integrable if both f +
and f −
are finite, or equivalently, if f is finite.
Often it makes for easier reading if we write X f, X f dμ, or X f dμ(x) in place
of f , especially if two or more measure spaces are involved in the discussion.
Until we reach Sect. 3.2, we shall be dealing mainly with integrals of nonnegative
(extended real-valued) measurable functions. Definition 2.2.5 has been stated here
only for completeness.
An R∗+
-valued measurable function with a finite integral will not be called inte-
grable unless it is known to take only finite values. The restriction that a function be
finite-valued in order to be regarded as integrable can be removed but will be retained
here so that we can consider −f without having to introduce −∞ and handle the
attendant complications with “∞−∞”.
At this juncture, it would be well to recall that for a bounded function f : [a, b] →
R, the supremum (resp. infimum) of lower (resp. upper) sums, taken over all partitions
of [a, b], is called the lower (resp. upper) integral of the function over the interval
and is denoted by
b
_ a f

resp. ¯ b
a f

. In general,
b
_ a f ≤ ¯ b
a f . If equality holds, the
function is said to be Riemann integrable and the common value of the lower and
upper integrals is the Riemann integral
b
a f .
2.2.6. Proposition. Let f : [a, b] → R be Riemann integrable and c 0. Then the
function g: [ca, cb] → R given by g(u) f (u
c
) is Riemann integrable and
cb
ca
g c
b
a
f,
which is to say,

cb
ca
f
u
c

du c
b
a
f (u)du.
Proof: Let P: a x0 x1 ··· xn b be any partition of [a, b]. Since c 0,
Q: ca cx0 cx1 · · · cxn cb
is a partition of [ca, cb]. For any x ∈ [xj−1, xj], we have u cx ∈ [cxj−1, cxj] and
f (x) f (cx
c
) g(cx) g(u). Conversely, for any u ∈ [cxj−1, cxj], we have
x u
c
∈ xj−1, xj and g(u) f (u
c
) f (x). Therefore
{ f (x): x ∈ [xj−1, xj ]} {g(u): u ∈ [cxj−1, cxj ]},
so that
inf{ f (x): x ∈ [xj−1, xj ]} inf{g(u): u ∈ [cxj−1, cxj ]}
and similarly for the suprema. Upon multiplying by cxj −cxj-1 c(xj −xj−1), and
taking the summation over all j from 1 to n, we get
cL( f, P) L(g, Q) ≤
cb
_ ca
g and cU( f, P) U(g, Q) ≥
¯ cb
ca
g.
Upon taking the supremum and infimum respectively over all P, it follows from here
that
c
b
_ a
f ≤
cb
_ ca
g and c
¯ b
a
f ≥
¯ cb
ca
g.
Since
cb
_ ca g ≤ ¯ cb
ca g and
b
_ a f ¯ b
a f
b
a f by hypothesis, we conclude that
c
b
a
f
cb
_ ca
g
¯ cb
ca
g.
2.2.7. Proposition. Suppose μ is a measure on a set X. For a measurable f : X →
R∗+
and for 0≤c ∞, we have
(cf ) c f.
Proof: For c 0, the function cf is 0 everywhere and equality is seen to hold trivially,
keeping in mind that ∞·00·∞0 (Remark 1.2.1(f)). So, suppose c 0.

We have (cf )(x)t ⇔ f (x) t
c
, so that X(cf t)X(f t
c
), which implies that cf
is measurable and that
(cf )(t) ˜
f ( t
c
). In particular, ˜
f is real-valued wherever
(cf )
is. If ˜
f is not real-valued on (0, ∞), that is to say, ˜
f (α) ∞ for some real α0,
then the same is true of
(cf ), in which case the required equality holds because both
sides are equal to ∞. Consider the case when ˜
f is real-valued except possibly at 0.
The same must be true of
(cf ), and it follows by equality (1.7) of Sect. 1.2 and from
Proposition 2.2.6 that the required equality holds.
It is an immediate consequence of Proposition 2.1.7 and Definition 2.2.1 that
for any measurable set, χA μA; it now follows from Proposition 2.2.7 that
(cχA) c(μA) when 0≤c ∞.
2.2.8. Proposition. Suppose μ is a measure on a set X and the nonnegative measur-
able functions f and g on X satisfy f (x)≤g(x) for every x ∈X. Then f ≤ g
Proof: From the hypothesis that f (x)≤g(x) for every x ∈ X, it follows that ˜
f (t) ≤ g̃(t)
for every t. The required inequality follows directly from this.
2.2.9. Proposition. Suppose μ is a measure on a set X and f : X → R∗+
is mea-
surable. If f ∞, then there exists a (measurable) set A such that μA 0 and
f (x)∞ for x /
∈ A.
Proof: Let A {x ∈ X: f (x)∞}, so that f (x)∞ for x /
∈ A. The set A is mea-
surable because it equals ∩∞
n1 X( f n). We need only prove that μA 0. Now,
nχA(x)≤f (x) for every x ∈ X and every positive integer n. It follows in view of
Proposition 2.2.8 that nχA ≤ f . But nχA n(μA) and therefore n(μA)≤ f .
Since this must hold for every positive integer n and f ∞, we get a contradiction
unless μA 0.
2.2.10. Proposition. Suppose μ is a measure on a set X and f : X → R∗+
is mea-
surable. If f ∞, then there exists a sequence {An}n≥1 of measurable sets such
that
(a) μAn ∞ for every n,
(b) f (x)0 for x /
∈ ∪∞
n1 An,
(c) f f χA, where A ∪∞
n1 An.
Proof: Put An X( f 1
n
), so that 1
n
χAn (x) ≤ f (x) for every x ∈ X. By Proposition
2.2.8, we obtain (1
n
χAn
) ≤ f . But (1
n
χAn
) 1
n
(μAn) and therefore 1
n
(μAn) ≤
f . Since f ∞, it follows that 1
n
(μAn) ∞, and hence (μAn)∞. Thus (a)
must hold.
If x /
∈ ∪∞
n1 An, then x /
∈ An for every n, which means f (x) ≤ 1
n
for every n. Since
f (x)≥0, it follows that f (x)0. So, (b) must hold.
Consider any x ∈ X. If x ∈ A ∪∞
n1 An, we have χA(x)1 and hence f (x)
f (x)χA(x)(f χA)(x). In the contrary case, χA(x)0 by definition of a characteristic
function and f (x)0 by (b), which leads to f (x)0f (x)χA(x)(f χA)(x). Thus
(c) must also hold.

Problem Set 2.2
2.2.P1. Prove Proposition 2.2.2(b).
2.2.P2. If f : X → R∗+
has the property that X(f ≥t) is measurable for every t ∈ Q,
show that f is measurable.
2.2.P3. Let {f n}n≥1 be a sequence of extended real-valued measurable functions on
X. Then the function f defined on X as f (x)inff n(x), i.e. inf{ fn(x): n ∈ N}, is
measurable.
2.2.P4. In Problem 2.1.P3, show that the measurability of the limit function f holds
without the hypothesis that f n(x)≤f n+1(x).
2.2.P5. If f : X → R is measurable and φ: R → R is increasing, show that the
composition φ ◦ f : X → R is measurable.
2.2.P6. Suppose μ is a measure on a set X. Prove that
(a) For a measurable function f : X → R+∗
and any real p 0, the nonnegative
function f p
: X → R∗+
is measurable; moreover,
f 0 if and only if ˜
f (t) 0 for every t 0,
which, in turn, is equivalent to f p
0 for any p 0.
(b) If f : X → R∗+
and g: X → R∗+
are measurable functions satisfying f
0 g, then ( f + g) 0.
2.2.P7. (a) Let f , g be nonnegative measurable functions on a space X with measure
μ, and f (x)g(x) for all x ∈ A, where A is a measurable subset of X satisfying
μ(Ac
)0. Show that f g.
(b) Let f , g be measurable functions on a space X with measure μ, and f (x)g(x)
for all x ∈ A, where A is a measurable subset of X satisfying μ(Ac
)0. If g is
integrable, show that f is also integrable and f g.
2.2.P8. Let f , g be nonnegative measurable functions on a space X with measure μ,
and f (x)≤g(x) for all x ∈ A, where A is a measurable subset of X satisfying μ(Ac
)
0. Show that f ≤ g.
2.2.P9. For a function f : X → R ∪ ∞, let A be the set X(f ∞). Show that
(a) If t 0, then X(f χA t)Ac
∪(A∩X(f t)).
(b) If t ≥0, then X(f χA t)A∩X(f t).
(c) The product f χA is real-valued.
(d) If f is measurable, then A is measurable and f χA is measurable.
2.2.P10. Let the function f : X → R ∪ ∞ have the property that X(t f ∞) is
measurable for every t ∈ R. Show that f is measurable. (The converse is trivial.)

2.2.P11. If f : X → R+∗
and g: X → R+∗
are both measurable, show that their
product f g: X → R+∗
is measurable.
2.2.P12. For an integrable function f : X → R∗
, show that
α · μ(X(| f | α)) ≤ | f | for all α 0.
2.2.P13. Suppose f : X → R∗+
is a measurable function such that, for some α0,
the function min{f , α} has a finite integral. Show that f has a finite integral if and
only if the series
∞
k1
˜
f (k) converges.
2.3 The Monotone Convergence Theorem
It follows from the result of Problem 2.1.P2 that, for any sequence {f n}n≥1 of mea-
surable functions such that f n ≤f n+1 everywhere, we have ˜
fn ≤ ˜
fn+1 everywhere. If
limn→∞ fn(x) is a real-valued function f , then it follows from Problem 2.1.P3 that
it is measurable, so that it has a distribution ˜
f . Since f n ≤f everywhere, it further
follows from the result of Problem 2.1.P2 that the distribution satisfies ˜
f ≥ ˜
fn. The
fact that a measure has a property called inner continuity makes it possible to obtain
something more satisfying; see Theorem 2.3.2 further below.
2.3.1. Proposition. Let μ be a measure. Then for a sequence of measurable sets
{Sn}n≥1, we have
μ(∪∞
n1Sn) limn→∞ μ(Sn) provided that Sn ⊆Sn+1 for every n. (inner continu-
ous)
Proof: As observed immediately after the definition of measure (Definition 2.1.2), μ
is finitely additive. Now consider a sequence {Sn}n≥1 of measurable sets such that
Sn ⊆Sn+1 for every n ∈ N. Let A1 S1 and An SnSn−1 for n 1. Then each An
is measurable and the condition Sn ⊆Sn+1 has the consequence that the sets An are
disjoint and that
Sn
n

j1
Aj . (2.3)
Therefore by finite additivity, we have
μ(Sn)
n

j1
μ(Aj ). (2.4)
Using countable additivity and then using (2.4), we get

μ
⎛
⎝
∞

j1
Aj
⎞
⎠
∞

j1
μ(Aj )
lim
n→∞
μ(Sn). (2.5)
But
∞

n1
Sn
∞

j1
Aj
in view of (2.3). It follows from this equality and from (2.5) that
μ
∞

n1
Sn

lim
n→∞
μ(Sn).

The functions μ in all our examples so far are inner continuous, including the
one in Example 2.1.3(b), although it is not even a measure (it fails to be countably
additive). The proof below does not use additivity directly but depends crucially on
inner continuity.
2.3.2. Monotone Convergence Theorem. Let μ be a measure on a set X. Suppose
f is a nonnegative extended real-valued function on X and {f n}n≥1 a sequence of
measurable nonnegative extended real-valued functions such that
(i) fn ≤ fn+1(x) and (ii) f x lim
n→∞
fn(x) f or every x ∈ X.
Then f is measurable, ˜
f (t) limn→∞
˜
fn(t) for every t ≥0, and also f (t)
limn→∞ fn(t).
Proof: Measurability of f is a simple consequence of Problem 2.2.P4. In the sequence
{ ˜
fn}n≥1 of nonnegative extended real-valued functions, each term ˜
fn is a decreasing
function on [0, ∞). By Problem 2.1.P3, we know that 0 ≤ ˜
fn(t) ≤ ˜
fn+1(t) for each
t ∈ [0, ∞) and each n ∈ N. By inner continuity (Proposition 2.3.1), in conjunction
with Problem 2.1.P3, we also know that limn→∞
˜
fn(t) ˜
f (t) for each t ∈ [0, ∞).
Therefore Proposition 1.2.4 yields
∞
0
˜
f limn→∞
∞
0
˜
fn. But by Definition 2.2.1,
this means f limn→∞ fn.
2.3.3. Exercise. Show that the “monotonicity” hypothesis (i) of the Monotone Con-
vergence Theorem 2.3.2 cannot be dropped.
Solution: Let X N and μ(A)

j∈A
1
2j

. Then μ is a measure. One way to see
this is to note that it is obtained by applying Problem 2.1.P10 with X j { j} ⊂ N
and μj defined by μj

X j

1
2j . Define f n to be 2n
times the characteristic function
of {n}. Then fn 1 for each n. Also, for each j ∈ X N, n ≥ j + 1 ⇒
fn( j) 0; so limn→∞ fn( j) 0 for each j ∈ X. Thus limn→∞ fn 0 although
limn→∞ fn 0.

2.3 The Monotone Convergence Theorem 37
Problem Set 2.3
2.3.P1. (Needed in Problem 3.2.P1 and Theorem 4.3.4) Let μ be a measure on X
and f a nonnegative measurable function such that f 0. Prove μ(X(f 0))0.
2.3.P2. Let {f n}n≥1 be a sequence of real-valued functions on an arbitrary nonempty
set X. Suppose f is a real-valued function on X such that, for every x ∈ X, some
subsequence of the real sequence {f n(x)}n≥1 converges to f (x). Show for any real t
that X(f t)⊆∪∞
n=1X(f n t).
2.3.P3. In a measure space, suppose that every set of infinite measure contains a
subset of finite positive measure. Show that every set of infinite measure contains a
subset of arbitrarily large finite measure.
Hint: For a given set of infinite measure, show that there exists a subset whose
measure is the supremum of the measures of all subsets of finite measure. If its
measure is finite, then the measure of its complement in the given set must be infinite.
2.3.P4. Let μ be a measure. For a sequence of measurable sets {Sn}n≥1 with
μ(S1)∞, show that
μ(∩∞
n1Sn) limn→∞ μ(Sn) provided that Sn ⊇Sn+1 for every n. (outer continu-
ous).

Chapter 3
Properties of the Integral
3.1 Simple Functions
So far, we have proved the Monotone Convergence Theorem but not the seemingly
simpler property

( f + g)

f +

g. In order to prove this, we need to make full
use of the finite additivity of measure, which we did not have to do for the Monotone
Convergence Theorem, because inner continuity of measure, a consequence of its
countable additivity, was sufficient.
In this section, we shall establish the property in question for a restricted class of
functions and extend it to a broader class only later.
3.1.1. Definition A measurable function f : X → R is said to be simple if its range
is finite.
Simple functions are a generalization of step functions, which also take finitely
many values, each value being taken on a union of finitely many intervals; thus,
step functions are linear combinations of characteristic functions of intervals. For
example, if s : [1, 3] → R is the step function that equals 3 on [1, 2), 4 on [2, 3)
and s(3)3, then s 3χ[1,2)∪[3,3] + 4χ[2,3). The generalization consists in replacing
unions of finitely many intervals by measurable sets.
3.1.2. Examples. (a) The characteristic function χA of a nonempty measurable set
A ⊂ X is a simple function with range {0, 1}. If the measurable set A is empty
or equals X, the range of χA consists of only 0 or only 1.
(b) Suppose A and B are nonempty measurable sets, both distinct from X. As
observed in (a), each of χA and χB has range {0, 1}; but the range of the sum
χA + χB may consist of 1, 2 or 3 numbers. Indeed, the range of χA + χB is
{1} if A ∩ B ∅ and A ∪ B X
{0, 1} if A ∩ B ∅ and A ∪ B X
{0, 2} if A B (in this event, A ∩ B A X)
39

40 3 Properties of the Integral
{1, 2} if A ∪ B X and A ∩ B ∅
{0, 1, 2} if A ∪ B X, A B and A ∩ B ∅.
(c) The sum of two functions with finite range has a finite range. Since the sum
of two measurable functions is measurable (Theorem 2.2.3(b)), the sum of two
simple functions is simple. Therefore a finite sum
n
j1 αj χAj
, where the sets
A1, . . . , An are measurable and α1, . . . , αn are real numbers, is always a simple
function. It can have a value which is none of the numbers αj , as illustrated in
part (b), where χA+χB 1· χA+1·χB α1·χA+α2·χB with α1 1 α2 2,
but χA + χB can have 2 as a value.
3.1.3. Remarks. (a) Let α1, . . . , αn be the distinct values assumed by a simple
function s. Then the sets Aj X(s αj ) are nonempty and disjoint, and
have union
n
j1 Aj X; moreover, each of them is measurable by virtue of
Proposition 2.2.2(a), and
s(x)
n

j1
αj χAj
(x).
Conversely, suppose A1, . . . , An are any n measurable sets that are nonempty
and disjoint, having union
∞
j1 Aj X, and let α1, . . . , αn be any n distinct
numbers.Thenthesimplefunctions
n
j1 αj χAj hasthenumbersα1, . . . , αn
as its distinct values and satisfies X(s αj ) Aj . In particular, the range
s(X) of the function is {αj : 1 ≤ j ≤ n} . If the n distinct values α1, . . . , αn
are arranged in increasing order, we shall refer to the sum as the canonical
representation of the simple function s.
(b) The canonical representations of χA + χB in the five illustrations in Example
3.1.2(b) are respectively
1 · χX , 0 · χ(A∪B)c + 1 · χA∪B, 0 · χAc + 2 · χA, 1 · χ(A∩B)c + 2 · χA∩B,
0 · χ(A∪B)c + 1 · χAB + 2 · χA∩B.
Here A B means the “symmetric difference” (A ∩ Bc
) ∪ (Ac
∩ B).
3.1.4. Exercise. When X {a, b, c, d} and A X, B {c, d}, C {d}, find the
canonical form of the simple function s χA + χB + χC .
Solution: Since a and b belong to precisely one among the three sets A, B, C (namely,
A), we have s(a)s(b)1. Since c belongs to precisely two of the sets (namely, A
and B), s(c)2. Lastly, since d belongs to all three sets, s(d)3. To summarize,
s(a) s(b) 1, s(c) 2 and s(d) 3.
Thus s takes the three values 1, 2, 3. Now,

3.1 Simple Functions 41
A1 X(s 1) {a, b}, A2 X(s 2) {c} and A3 X(s 3) {d}.
So, the canonical form is
s 1 · χA1
+ 2 · χA2
+ 3 · χA3
1 · χ{a,b} + 2 · χ{c} + 3 · χ{d}.
The next proposition involving a canonical representation will be used to prove
the stronger result that the same conclusion is true with a representation that is not
canonical. The latter will make it transparent why integrals of simple nonnegative
functions add up as expected.
3.1.5. Proposition. Suppose (X, F, μ) is a measure space and s : X → R a non-
negative simple function with canonical representation
s(x)
n

j1
αj χAj
(x).
Then

s
n

j1
αj μ(Aj ).
Proof: The case when s takes only one value is trivial. Therefore we consider only
the case when s has at least two values, i.e. n ≥ 2.
Since the numbers α1, . . . , αn are in increasing order (by definition of canonical),
sup s αn. Therefore

s
sup s

0
s̃
αn

0
s̃. (3.1)
Another consequence of the numbers α1, α2 . . . , αn being in increasing order is
that, for j 1 and αj−1 ≤ t αj , the inequality s(x)t holds if and only if s(x) αk
for some k ≥j, i.e. x ∈ Aj ∪ · · · ∪ An, which is to say, {x ∈ X : s(x) t}
Aj ∪ · · · ∪ An. Therefore
s(t) μ({x ∈ X : s(x) t}) μ(Aj ∪ · · · ∪ An) when
αj−1 ≤ t αj . Since the sets Aj are disjoint, the additivity of μ yields
μ(Aj ∪ · · · ∪ An) μ(Aj ) + · · · + μ(An).
Thus the distribution
s satisfies

s(t) μ(Aj ) + · · · + μ(An) for t ∈ [αj−1, αj ) and 1 j ≤ n, (3.2)
and if α1 0, also satisfies

42 3 Properties of the Integral

s(t) μ(A1) + · · · + μ(An) for t ∈ [0, α1). (3.3)
If each μ(Aj) is finite,
s is a step function and it follows from (3.2) and (3.3) that,
whether or not α1 0, we have
αn

0

s α1(μ(A1) + · · · + μ(An)) +
n

j2
(αj − αj−1)

μ(Aj ) + · · · + μ(An) . (3.4)
One can transform the right side here by using the Abel summation formula of
Problem 1.1.P5 with
b1 α1, bj αj − αj−1, ( j ≥ 2) and aj μ(Aj ) + · · · + μ(An),
thereby obtaining
αn

0

s
n

j1
αj μ(Aj ). (3.5)
Now (3.1) and (3.5) together imply the required equality for the case when each
μ

Aj is finite.
Next, suppose μ

Aj ∞ for some j 1. In this event,
s(α) ∞ for α ∈
[αj−1, αj ) and hence both sides of the required equality are ∞.
Finally, suppose μ

Aj is finite for all j 1 but μ(A1) ∞. If α1 0, then

s(t) ∞ for t ∈ [0, α1) and again both sides of the equality in question are ∞. If
α1 0, then (3.2) shows that
s is a step function and (3.4) holds with μ(A1) replaced
by any real number. As before, we arrive at (3.5) but with μ(A1) replaced by a real
number. However, (3.5) continues to hold if μ(A1) is changed back to ∞ because
now α1 0. Once again, (3.1) and (3.5) together imply the required equality.
When μ(A) is the number of elements in A, the function μ is a measure and is
called the counting measure; it is understood that μ(A) ∞ when A is an infinite
set. In Example 3.1.4, if we use the counting measure,

s can be computed on the
strength of the preceding proposition as

s 1 · 2 + 2 · 1 + 3 · 1 2 + 2 + 3 7.
We note that s χA + χB + χC and

χA +

χB +

χC μ(A) + μ(B) + μ(C)
4 + 2 + 1 7, which agrees with

s.
3.1.6. Exercise. Let X {a, b, c, d} and A, B, C be subsets of X such that s

χA +

χB +

χC works out to be the same as in Exercise 3.1.4. If A {b, c, d}
and B {a, d}, find C and compare

χA +

χB +

χC with

s when μ is the
counting measure.

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On the other hand, how sad is the condition of vocal music in our
time! How few artistically 19 cultivated voices are there! And the few
that there are, how soon are they used up and lost! Artists like Lind,
and more recently Trebelli, are exceptions to be made.
Mediocre talent is now often sought, and rewarded far beyond its
desert. One is often tempted to think that the public at large has
wellnigh lost all capacity of judgment, when he witnesses the
representation of one of our operas. Let a singer, male or female,
only drawl the notes sentimentally one into another, execute a
tremulo upon prolonged notes, introduce very often the softest
piano and just where it is entirely out of place, growl out the lowest
notes in the roughest timbre, and scream out the high notes lustily,
and he or she may reckon with certainty upon the greatest applause.
In fact, we have become so easily pleased that even an impure
execution is suffered to pass without comment. Let the personal
appearance of the singer only be handsome and prepossessing, he
need trouble himself little about his art in order to win the favor of
the public. This decline of the art of singing is usually ascribed to the
want of good voices, and this poverty of voices to our altered modes
of living. To me it appears as 20 the natural consequence of the
whole manner and way in which the art of singing has been
historically developed since its earlier high state of perfection.
The human voice is, of all instruments, the most natural, the most
perfect, the most intimate in its relation to us, as, for the use of it,
we have a talent or faculty innate, which, in the case of other
instruments, can only be laboriously acquired, to say nothing of the
fact that these instruments are first to be invented and put together.
Hence vocal music appears to have been almost the only music

among the Greeks, and the rude instruments then in use served
merely for an accompaniment. The history of our so-called Western
music, which dates no farther back than the fourth century after
Christ, tells us hardly anything else than of vocal musicians and of
their compositions for concerted and chorus singing.
Our art, only slowly developing itself from those earliest times,
was cherished, mainly in Italy, for the sole purpose of exalting divine
worship. We have, at least, no account of any secular art of music in
those days. As yet unacquainted with harmony, the only singing was
21 in unison, as was the custom, at an earlier period, among the
Greeks; for not until the tenth century of the Christian era was it
attempted, and then by a Flemish monk (Hukbaldus), to harmonize
several and different notes; thus was invented and founded our
harmony, whose exponent was the organ. 1
From that time forward, history makes mention of many persons
who labored worthily, now more and now less, to create a theory of
music, seeking to found a system of harmony upon that rude
beginning, and by degrees to improve it. In the fourteenth and
fifteenth centuries music burst forth into blossom in the Netherlands,
and thenceforth rose steadily in excellence, when also it began to
branch out into the excesses of counterpoint. The fame of the
Netherlands soon spread over all the civilized countries of Europe.
The artists of the Netherlands were invited upon the most favorable
terms to Italy, France, Spain, and Germany, and thus the progress of
music spread over all these countries almost pari passu. For two
hundred years the 22 Netherlands maintained the reputation of the
best and highest culture in vocal music, and not until the middle of
the sixteenth century did there appear in Italy and Germany artists

who attained to a like renown. Up to that time prejudice denied to
the Italians all talent for music, as it has ever since exaggerated
their claims in this respect. Kiesewetter remarks, in his History of
Music, that, although the Netherlands in Italy no longer had the
monopoly, they nevertheless always maintained the supremacy in
music. Climate and language were, however, so favorable to vocal
music in Italy that it soon found there its peculiar home, and though
theoretical knowledge of music was advanced by the earlier singers,
now richness and power of voice were also attained. As it had
previously been with the Netherlanders, so it now became with the
Italians. They were drawn to all countries in which there was any
love of art; and they soon won that supremacy in music which they
maintained until the last century. Until the latter part of the sixteenth
century, good musicians were devoted almost exclusively to church
music, and held it beneath their dignity to take part in music of any
other kind. All but church music 23 they left to the minstrels and
strolling singers, who traveled over the country from place to place,
and in different lands were styled minstrels, minnesänger and
trovatori. They mostly sung love-songs, which they often
extemporized in word and tune, finding place and popularity on all
festive occasions. But under the impulse which music began to feel,
the desire among the educated class to revive the old Greek drama,
which just at that period had come to be well known, became more
and more urgent. Imbued with the spirit of that age, the whole
tendency of which was to exalt the ancient classic poets, a circle of
men of science and culture from the higher classes gave themselves
to the task of producing a style of music such as the Greeks must
have had in the representation of their dramas. In the mansion of

Count Bardi, in Florence, then the centre of union for all who had
any claims to cultivation, music was first arranged for a single voice
by a dilettante, the father of the renowned Galileo.
This attempt met with applause and imitation among the most
distinguished singers of the time, who thenceforth turned their
attention also to secular music. It thus came about that, 24 towards
the end of the sixteenth century, on festal occasions in Italy, and
even earlier in France, theatrical representations were given with
vocal music. This music was, however, always composed in the form
of the chorus, and the leading voice alone was represented by a
singer; the other voices were represented by instruments.
Such was the beginning of solo singing, which, growing ever more
in public favor, soon came to be introduced into the most solemn
church music; dramatic representations, religious and secular, grew
very popular, and were the forerunners of the opera and oratorio,
the richest inventions of the sixteenth century.
Up to this time, a singer of sound musical culture sufficed for
chorus singing, but by the introduction of solo singing a more
complete education of the organ of singing became a necessity.
Indeed, as early as the middle of the fifteenth century there existed
in Rome and Milan schools of music and professorships for the
education of singers; but with the introduction and diffusion of solo
singing similar conservatories were established in nearly all the more
considerable cities of Italy, and all the 25 energies of the musician
were devoted to the highest possible culture of the voice.
But, with solo singing, greater attention was paid to instruments,
which were already in those days constructed with the greatest care
and skill. With the higher cultivation of single voices, chorus singing

also became richer in harmony and embellishment, but as, in vocal
music, words accompany the music, the expression of the music
becomes more definite and intelligible for the hearer, and thus with
the higher cultivation of vocal music, and by means of it, even our
whole modern system of harmony has been developed.
Women were, by ecclesiastical law, excluded from participation in
church music, and as the voices of boys could be used only for a few
years, they did not suffice to meet the ever-increasing demands of
church music. At first it was attempted to supply the place of the
sopranists and contraltists with so-called falsettists. As, however,
these substitutes proved insufficient, the soprano and contralto of
boys were sought to be preserved in men. And so, in 1625,
appeared the first male sopranist in the Papal chapel in Rome. Such
sopranists and contraltists soon appeared in great numbers, and as
their 26 organs of singing continued soft and tender as those of
women, and their compass was the same, to them the education of
female voices was given over exclusively. Thenceforth women
became the richest ornament of the opera, then blooming into
beauty. But only when the ecclesiastical law forbidding women to
take part in church music was annulled, did women begin, in the
middle of the last century, to take the place of those male sopranists
and contraltists.
It thus became unnecessary to secure longer duration to the
voices of boys, especially as these were never able to attain to the
peculiar grace of the female voice, and so this class of singers
gradually died out. But still in the first half of the present century
there were many of them living and sought for as teachers of

singing. To the disappearance of this kind of singers, Rossini thinks
the decline of vocal art is to be mainly ascribed.
The art of singing rose in the course of the seventeenth century to
an extraordinary height of cultivation, and was diffused more and
more by means of the opera, then blooming, as we have said, into
beauty. But in that brilliant springtime of vocal art, it was not mere
externals, 27 such as beauty of tone, flexibility, etc., that were striven
for, but, above all, the correct expression of the feeling intended in
the composition. This rendered necessary to the singer the most
thorough æsthetic culture, going hand in hand with the culture of
the vocal organ. For only thus could he succeed in acting upon the
souls of his hearers, in moving them and carrying them along with
him in the emotions which the music awakened in his own mind.
The dramatic singer was now strongly tempted to neglect the
externals of his art for the æsthetic, purely inward conception of the
music. Certain, at least, it is that to the neglect of the training of the
voice (Tonbildung), and to the style of writing of our modern
composers—a style unsuited to the art of singing, and looking only
to its spiritual element—the decline of this art is in part to be traced.
Mannstein says that, with the disappearance of those great masters,
power and beauty of tone have fallen more and more into contempt,
and at the present day it is scarcely known what is meant by them.
True it is, that a beautiful tone of voice (Gesangston), which must be
considered the foundation and first requisition of fine singing, is
28 more and more rare among our singers, male and female, and yet
it is just as important in music as perfect form in the creations of the
sculptor.

But the complete technical education of the earlier singers misled
many of them into various unnatural artifices, in order to obtain
notice and distinction. The applause of the public caused such
trickeries to become the fashion among artists. The multitude,
accustomed to such effects, began to mistake them for art. By the
gradual disappearance of the male sopranists, instruction in singing
fell into the hands of tenor singers, who usually cultivated the
female voice in accordance with their own voices, which could not be
otherwise than injurious in the uncertainty existing as to the limits in
compass and the difference between the male and female organs of
voice.
Thus it has come to pass that people are now apt to imagine that
they know all that is to be known; and as teaching in singing is
generally best paid, the office has been undertaken, without the
slightest apparent self-distrust, by many persons who have not the
slightest idea what thorough acquaintance with the organs of
singing, what comprehensive knowledge of all the departments of
29 music and what æsthetic and general culture, the teacher of
singing requires. Very few persons indeed clearly understand what is
meant by the education of a voice, and what high qualifications both
teacher and pupil always require. The idea, for instance, is very
prevalent that every musician, whatever may be the branch of music
to which he is devoted, and especially every singer, is qualified to
give instruction in singing. And therefore a dilettanteism without
precedent has taken the place of all real artistic endeavors. Be this,
however, as it may, such is the wide diffusion and popularity of
music beyond all the other arts, that the want of singers artistically
educated, and consequently also of a recognized sound method of

instruction, becomes more and more urgent; and although we have
in these times distinguished singers, male and female, as well as
skilful teachers, yet the number is very small and by no means equal
to the demand.
But now, as every evil, as soon as it is felt to be such, calls forth
the means of its removal, already in various ways attempts are
making in the department of the Art of Singing to restore it as
perfectly as possible to its former high 30 position, and if possible to
elevate it to a yet higher state. It was natural that the attempt
should, first of all, be made to revive the old Italian method of
instruction, and that, by strict adherence in everything to what has
come down to us by tradition, we should hope for deliverance and
salvation; for to the Italians mainly vocal music was indebted for its
chief glory. Without considering in what a sadly superficial way music
—and vocal music especially—is now treated in Italy, many have
given in to the erroneous idea that any Italian who can sing anything
must know how to educate a voice. Thus many incompetent Italians
have become popular teachers in other countries.
The old Italian method of instruction, to which vocal music owed
its high condition, was purely empirical, i. e., the old singing masters
taught only according to a sound and just feeling for the beautiful,
guided by that faculty of acute observation, which enabled them to
distinguish what belongs to nature. Their pupils learned by imitation,
as children learn their mother tongue, without troubling themselves
about rules. But after the true and natural way has once been
forsaken, and for so long a 31 period only the false and the unnatural
has been heard and taught, it seems almost impossible by
empiricism alone to restore the old and proper method of teaching.

With our higher degree of culture, men and things have greatly
changed. Our feeling is no longer sufficiently simple and natural to
distinguish the true without the help of scientific principles.
But science has already done much to assist the formation of
musical forms of art. Mathematics and physics have established the
principal laws of sound and the processes of sound, in accordance
with which our musical instruments are now constructed.
Philosophical inquirers have succeeded also in discovering the
eternal and impregnable laws of Nature upon which the mutual
influences of melody, harmony and rhythm depend, and in thus
giving to composition fixed forms and laws which no one ventures to
question. And more recently Professor Helmholtz, in his great work,
“Die Lehre von den Tonempfindungen,” has given to music of all
kinds a scientific ground and basis. But for the culture of the human
voice in singing science has as yet furnished only a few lights. The
well-known experiments of Johannes Müller 32 upon the larynx gave
us all that was known, until very recently, respecting the functions of
the organ of singing. Many singing masters have sought to found
their methods of instruction upon these observations on the larynx,
at the same time putting forth the boldest conjectures in regard to
the functions of the organ of singing in the living subject. But they
have thus ruined more fine voices than those teachers who, without
reference to the formation of the voice, only correct the musical
faults of their pupils, and for the rest let them sing as they please.
This superficial treatment of science, and the unfortunate results
of its application, have injured the art of singing more than benefited
it, and created a prejudice against all scientific investigations in this
direction among the most distinguished artists and teachers, as well

as among those who take an intelligent interest in this department
of music. It is a pretty common opinion that science can do little for
the improvement of music, and nothing for the culture and
preservation of the voice in singing. And the habit of regarding
science and art as opposed to each other renders it extremely
33 difficult to secure a hearing for the results of thorough scientific
inquiry in this direction.
Science itself admits that it can neither create artistic talent, nor
supply the place of it, but only furnish it with aids. Besides, with the
whole inner nature of music, no forms of thought (reflection) have
anything to do. It has “a reason above reason.” This art transmits to
us in sound the expression of emotions as they rise in the human
soul and connect themselves one with another. It is the revelation of
our inmost life in its tenderest and finest processes, and is therefore
the most ideal of the arts. It appeals directly to our consciousness.
As a sense of the divine dwells in every nation, in every human
being, and is impelled to form for itself a religious cultus, so we find
among all nations the need of music dwelling as deeply in human
nature. The most uncivilized tribes celebrate their festivals with
songs as the expression of their devotion or joy, and the cultivated
nations of ancient times, like the Greeks, cherished music as the
ethereal vehicle of their poetry, and regarded it as the chief aid in
the culture of the soul.
But together with its purely internal character, 34 music has yet
another and formal side, for if our art consisted only in the æsthetic
feeling, and in representing this feeling, every person of culture,
possessing the right feeling, would be able to sing, just as he
understands how to read intelligibly.

Everything spiritual, everything ideal, as soon as it is to be made
present to the perceptions of others, requires a form which, in its
material as well as in its structure, may be more or less perfect, but
it can never otherwise than submit to those eternal laws to which all
that lives, all that comes within the sphere of our perceptions, is
subject. To discover and establish the natural laws which lie at the
basis of all our forms of art is the office of science; to fashion and
control these forms and animate them with a soul is the task of art.
In singing, the art consists in tones beautiful and sonorous, and
fitted for the expression of every variation of feeling. To set forth the
natural laws by which these tones are produced is the business of
physiology and physics.
Thus is there not only an æsthetical side to the art of singing, but
a physiological and a physical side also, without an exact knowledge,
35 appreciation, observance, and study of which, what is hurtful
cannot be discerned and avoided; and no true culture of art, and
consequently no progress in singing, is possible.
In the physiological view of vocal art, we have to do with the
quality and strength of the organ of singing in the act of uttering
sound, and under the variations of sound that take place in certain
tones (the register being transcended).
By the physical side is to be understood the correct use and skilful
management of the air flowing from the lungs through the windpipe,
and brought into vibration by the vocal chords in the larynx.
But the æsthetics of vocal art, and the spiritual inspiration of the
form (of the sound), comprise the whole domain of music and poetic
beauty.

1 Those who are interested in the history of music are referred to the
historical works already mentioned for a fuller account of what is only
alluded to above.
Return to text

36 II
PHYSIOLOGICAL VIEW
FORMATION OF SOUND BY THE ORGAN OF THE HUMAN VOICE
THE great physiologist, Johannes Müller, fastened a larynx, which
he had cut out with the whole trachea belonging to it, to a board,
and, stretching the vocal chords by a weight that could be increased
or diminished at pleasure, caused vibrations in it by blowing through
the trachea with a pair of bellows, or through a tube with his own
breath. In this way he succeeded in producing almost all the tones
of the human voice, and even some which are beyond the compass
of this organ.
He distinguished two different kinds of tones, to which he gave
the names of the chest register and the falsetto register. The chest
tones were produced when the vocal ligaments, slackly stretched,
were made to vibrate easily in their whole breadth; the falsetto
tones came merely 37 through the vibration of the fine inner edges of
the vocal chords when they were more tightly stretched. At a
moderate stretching of the vocal chords, it depended upon the
manner of blowing whether a sound corresponding to the chest
voice or to the falsetto were produced, or whether it were higher or
lower for several tones, often for a whole octave. A series of tones of

more than two octaves could thus be produced in the same larynx,
with, however, gaps and places at which the vocal chords, instead of
being stretched gradually, have to be stretched at once very
strongly, in order that the succeeding higher half tone may be
reached. Such a place Müller indicates from c2
to c 2
, or d2
to d 2
, with the remark that it differs in different larynxes,
being in some higher and in some lower. But in order to render
practicable the proper stretching of the exsected larynx, muscles and
membranes have to be cut, which sufficiently proves that the
functions of the organ of singing in the living must be differently
carried on.
Dr. Merkel, in Leipzig, has continued these experiments, and by
means of a peculiar contrivance 38 has succeeded in producing all
the tones in the exsected larynx, without mutilating it. But these
investigations, interesting as they are, throw no certain light upon
the formation of sound by the vocal organ in the living.
The celebrated singing master, Manuel Garcia, now living in
London, was the first to adopt the right mode of scientific inquiry in
this department, with favorable results. He undertook to apply the
laryngoscope (previously invented by the Englishman, Liston) to the
larynx in the act of singing. The interesting results of these
observations were published by him in the Philosophical Mag. and
Journal of Science, vol. x. p. 218. While men of science immediately
repeated Garcia’s experiments and applied them with the greatest
advantage to pathological purposes, they were received with
distrust, scarcely noticed, and in many instances entirely rejected, by
teachers of vocal music. The few who attempted to follow the path

thus opened soon gave it up, because they lacked either patience or
the anatomical knowledge necessary to such investigations.
39 The laryngoscope is well known among medical men. It is a
small plane mirror of glass or metal, having a long handle. Before it
is introduced into the throat, it is first warmed, to prevent its
becoming dimmed. The reflecting surface of this instrument is
directed downwards and forwards, so that it receives the reflection
caught from a concave mirror, and presents to the eye of the
observer a picture of the illuminated larynx. In using it upon oneself,
there is need of a second mirror, which must be so held that the
image may be seen in the laryngoscope.
The use of the laryngoscope requires in the observer a certain
adroitness and long-continued practice—almost more in the observer
than in the subject of observation. In self-observation one must first
learn to overcome the irritation always caused at the first by the
contact of the mirror with the back of the throat. Once accustomed
to the contact, one soon succeeds in obtaining a sight of the larynx,
sufficient for the most part for pathological purposes. But it requires
long practice before one can control those organs, usually not
immediately submissive to the will, and raise the epiglottis, so as to
be 40 able to see into the whole larynx. But this is absolutely
indispensable, in the observation of the formation of sound, to the
attainment of any substantial results. Garcia says himself that one-
third of the glottis was always hidden from him by the epiglottis, and
to this circumstance is the unsatisfactory character of his
observations to be ascribed. But even when, after long practice, one
is able at last to bring the whole glottis into view, this is not by any
means enough. Not until observation has been so long continued

that all the movements of the vocal organ are normal,
notwithstanding the unnatural drawing back of the epiglottis, and
not until the process that goes on is found again and again to be
always the same, can it be recognized as fact.
As Garcia is the most eminent of singing masters now living, and
as he has sought, solely in the interest of vocal music, to ascertain
the mechanism by which sound is formed, and as his observations
have been confirmed by men of science, I give them here in his own
words.
In order that what follows may be better 41 understood by those
unacquainted with anatomy, a brief anatomical description of the
vocal organ will be found in an Appendix to the present work.
OBSERVATIONS WITH THE LARYNGOSCOPE
BY MANUEL GARCIA
“At the moment when the person draws a deep breath, the
epiglottis being raised, we are able to see the following series of
movements: the arytenoid cartilages become separated by a very
free lateral movement; the superior ligaments are placed against the
ventricles; the inferior ligaments are also drawn back, though in a
less degree, into the same cavities; and the glottis, large and wide
open, is exhibited, so as to show in part the rings of the trachea.
But, unfortunately, however dexterous we may be in disposing these
organs, and even when we are most successful, at least the third
part of the anterior of the glottis remains concealed by the epiglottis.

“As soon as we prepare to produce a sound, 42 the arytenoid
cartilages approach each other, and press together by their interior
surfaces, and by their anterior apophyses, without leaving any
space, or inter-cartilaginous glottis; sometimes, even, they come in
contact so closely as to cross each other by the tubercles of
Santorini. To this movement of the anterior apophyses that of the
ligaments of the glottis corresponds, which detach themselves from
the ventricles, come in contact with different degrees of energy, and
show themselves at the bottom of the larynx, under the form of an
ellipse of a yellowish color. The superior ligaments, together with the
aryteno-epiglottidean folds, assist to form the tube which surmounts
the glottis; and being the lower and free extremity of that tube,
enframe the ellipse, the surface of which they enlarge or diminish
according as they enter more or less into the ventricles. These last
scarcely retain a trace of their opening. By anticipation, we might
say of these cavities that, as will afterwards appear clearly enough in
these pages, they only afford to the two pair of ligaments a space in
which they may easily range themselves. When the aryteno-
epiglottidean folds contract, they 43 lower the epiglottis and make
the superior orifice of the larynx considerably narrower.
“The meeting of the lips of the glottis, naturally proceeding from
the front towards the back, if this movement is well managed, will
allow, between the apophyses, of the formation of a triangular space
or inter-cartilaginous glottis, but one which, however, is closed as
soon as the sounds are produced.
“After some essays we perceive that this internal disposition of the
larynx is only visible when the epiglottis remains raised. But neither
all the registers of the voice, nor all the degrees of intensity, are

equally fitted for its taking this position. We soon discover that the
brilliant and powerful sounds of the chest register contract the cavity
of the larynx, and close still more its orifice; and, on the contrary,
that veiled notes, and notes of moderate power, open both, so as to
render any observation easy. The falsetto register especially
possesses this prerogative, as well as the first notes of the head
voice. So as to render these facts more precise, we will study in the
voice of the tenor the ascending progression of the chest register,
and in the soprano that of the falsetto and head registers.
44 EMISSION OF THE CHEST VOICE
“If we emit veiled and feeble sounds, the larynx opens at the
notes , and we see the glottis agitated by large and
loose vibrations throughout its entire extent. Its lips comprehend in
their length the anterior apophyses of the arytenoid cartilages and
the vocal chords; but, I repeat it, there remains no triangular space.
“As the sounds ascend, the apophyses, which are slightly rounded
on their internal side, by a gradual apposition commencing at the
back encroach on the length of the glottis, and as soon as we reach
the sounds they finish by touching each other throughout
their whole extent; but their summits are only solidly fixed one
against the other at the notes . In some organs these
summits are a little vacillating when they form the posterior end of

the glottis, and two or three half-tones which are formed show a
certain want of purity and strength, which is very well known to
singers. 45 From the vibrations, having become rounder
and purer, are accomplished by the vocal ligaments alone, up to the
end of the register.
“The glottis at this moment presents the aspect of a line slightly
swelled towards its middle, the length of which diminishes still more
as the voice ascends. We also see that the cavity of the larynx has
become very small, and that the superior ligaments have contracted
the extent of the ellipse to less than one-half.
“Thus the organs act with a double difference: 1. The cavity of the
larynx contracts itself more when the voice is intense than when it is
feeble. 2. The superior ligaments are contracted, so as to reduce the
small diameter of the ellipse to a width of two or three lines. But
however powerful these contractions may be, neither the cartilages
of Wrisberg, nor the superior ligaments themselves, ever close
sufficiently to prevent the passage of the air, or even to render it
difficult. This fact, which is verified also with regard to the falsetto
and head registers, suffices to prove that the superior ligaments do
not fill a generative part in the formation of the voice. We may draw
the 46 same conclusion by considering the position occupied by the
somewhat feeble muscles which correspond to these ligaments; they
cover externally the extremity of the diverging fibres of the thyro-
arytenoid muscles, and take part especially in the contractions of the
cavity of the larynx during the formation of the high notes of the
chest and head registers.

PRODUCTION OF THE FALSETTO
“The low notes of the falsetto show the glottis infinitely better
than the unisons of the chest voice, and produce vibrations more
extended and more distinct. Its vibrating sides, formed by the
anterior apophyses of the arytenoid cartilages and by the ligaments,
become gradually shorter as the voice ascends; at the notes
the apophyses take part only at their summits; and in
these notes there results a weakness similar to that which we have
remarked in the chest notes an octave below. At the notes
, the ligaments alone continue to act; then begins the
series of notes called the head voice. The moment in which the
action 47 of the apophyses ceases exhibits in the female voice a very
sensible difference to the ear and in the organ itself. Lastly, we verify
that up to the highest sounds of the register the glottis continues to
diminish in length and in width.
“If we compare the two registers in these movements, we shall
find some analogies in them; the sides of the glottis formed at first
by the apophyses and the ligaments become shorter by degrees,
and end by consisting only of the ligaments. The chest register is
divided into two parts, corresponding to these two states of the
glottis. The register of falsetto-head presents a complete similarity,
and in a still more striking manner.
“On other points, on the contrary, these same registers are very
unlike. The length of the glottis necessary to form a falsetto note
always exceeds that which produces the unison of the chest. The

movements which agitate the sides of the glottis are also
augmented, and keep the vibrating orifice continually half opened,
which naturally produces a great waste of air. A last trait of
difference is in the increased extent of that elliptic surface.
“All these circumstances show in the mechanism 48 of the falsetto
a state of relaxation which we do not find in the same degree in the
chest register.
MANNER IN WHICH THE SOUNDS ARE
FORMED
“As we have just seen—and what we have seen proves it—the
inferior ligaments at the bottom of the larynx form exclusively the
voice, whatever may be its register or its intensity; for they alone
vibrate at the bottom of the larynx.… By the compressions and
expansions of the air, or the successive and regular explosions which
it produces in passing through the glottis, sound is produced.” (The
London, Edinburgh and Dublin Phil. Mag. and Journal of Science, vol.
x. 4th Series, pp. 218–221, 1855.)
Garcia proceeds, in the same paper, to give an elaborate account
of his theory of the compression, expansion and explosion of the air
in expiration, together with his conjectures as to the action of the
muscles of the larynx in relation to the different registers. I omit
both here, for, since this publication of Garcia’s, the movements of
the breath generating sound in expiration have been thoroughly
investigated 49 and determined by Prof. Helmholtz; and in the

physical section of the present work all may be found that is of value
in the culture of the singing voice. Whatever can be definitely
communicated in regard to the working of the muscles of the larynx
may likewise be found in any anatomical work. An acquaintance,
however, with the action of these muscles is not directly necessary
to our purpose, and is of interest only to the physiologist.
It is not to be denied that Garcia’s observations do not, by any
means, lead to satisfactory conclusions as to the functions of the
vocal organ. He has, as we shall see in the sequel, attached special
importance to much that is unessential and abnormal, and the main
facts, the elucidation of which is particularly needed, he has scarcely
mentioned. Thus he tells us nothing of that series of tones which he
calls the head register. The transition also of the registers he has not
carefully examined and observed in different voices: the chest
register in the male and the falsetto of the female voice.
Nevertheless, these investigations possess much that is valuable,
and are of special value to the art of singing, because they teach a
method 50 hitherto unknown of observing the larynx, by which sure
and satisfactory results are reached. And when an acquaintance with
these results comes to be universally diffused, and the art of singing
is thereby led into the right direction, we shall owe it most especially
to the excellent experimental observations of Garcia.
Garcia has accepted the division made by Müller, and universally
adopted in science, of the chest, falsetto and head registers. I
employ the same distinctions—a fact which it seems worth while to
mention, simply because every teacher and school have their own
terminology, and instead of falsetto we have fistel, throat, and
middle or neck voice, c. These denominations of the same registers

have thus far only increased the obscurity prevailing in the art of
singing.
MY OWN OBSERVATIONS WITH THE
LARYNGOSCOPE
In giving an account of my own observations with the
laryngoscope, I premise that laryngoscopy has of late attracted
much attention among the learned, and that Czermak, Turk, Merkel,
Lewin, Bataille, c., have published a 51 series of valuable
observations, all of which, however, with the exception of Bataille’s,
were made in the interest of science, for pathological purposes
especially. My aim, in the employment of the laryngoscope, has been
directed exclusively to the discovery of the natural limits of the
different registers of the human voice; and although I have thus
been able to observe many other interesting processes, it would not
at all accord with the design of this book to communicate
observations which have no direct relation to the culture of the voice
in singing, and which come better from men of science than from a
teacher of vocal music.
In using the laryngoscope while the breath is quietly drawn, I saw,
as Garcia did, the whole larynx wide open, so that one could easily
introduce a finger into it, and the rings of the trachea were plainly
visible.

a. Arytenoid cartilages.
b. Epiglottis.
c. Trachea. 2
d. Vocal chords.
52 When those who had become accustomed to the introduction of
the instrument sang, at my request, a, as pronounced in the English
word man, in a deep tone, the epiglottis rose, the tongue formed a
cavity from within forwards, and thus rendered it easy to see into
the larynx. So soon as the a, as in father, was sung, the cover
quickly fell, the tongue rose, and prevented all observation of the
organ of singing. The other vowels are still less favorable to
observation, because they do not admit of any such wide opening of
the mouth. Strong tones also are unfavorable to observation, as
Garcia also remarked; and this is very natural, because strong and
sonorous tones require greater exertions of the singing organ, and,
above all things, the right position of those parts of the larynx and
mouth which serve as a resonance apparatus in the formation of
sound. In order to be able to see perfectly the whole glottis, all this
resonance apparatus must be drawn back as far as possible, and the
rim of the larynx must be tolerably flat. Thus only faint and weak
sounds are favorable to observation.

53 THE CHEST REGISTER
When the vowel a, as in man, was sung, I could, after long-
continued practice, plainly see how the arytenoid cartilages quickly
rose with their summits in their mucous membranous case and
approached to mutual contact. In like manner, the chordæ vocales,
or inferior vocal chords, approached each other so closely that
scarcely any space between them was observable. The superior or
false vocal ligaments formed the ellipse described by Garcia in the
upper part of the glottis.
Representation in the mirror of the vocal organ in giving out sound.
a. Superior or false vocal ligaments, or chords.
b. Epiglottis.
c. Inferior or true vocal ligaments.
d. Arytenoid cartilages.
e. Capitula Santorini.
When, in using the laryngoscope upon myself, I slowly sang the
ascending scale, this movement of the vocal chords and arytenoid
cartilages was repeated at every tone. They separated and appeared
to retreat, in order to close again anew, and to rise somewhat more
than before. This movement of the arytenoid cartilages may 54 best

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