Table Of ContentBASIC REAL
AND ABSTRACT
ANALYSIS
John F. Randolph
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF ROCHESTER
ACADEMIC PRESS New York and London
COPYRIGHT © 1968, BY ACADEMIC PRESS INC.
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PREFACE
The keen precision toward which mathematics constantly strives is greatly
enhanced by abstraction. Without sharp awareness of the sources of abstraction,
however, there is danger of fostering an impression that abstraction is compli
cation instead of simplification. Hence the minds visualized as maturing on this
presentation are led from the concrete to the abstract rather than the other way
around.
Chapter 1 (Orientation) emerged last and was the hardest to write, since few,
if any, instructors agree on an appropriate starting point for a course which
could follow the two-year calculus sequence. Contact with past mathematical
experience is made, the stage is set, abstractions to come are foretold by a taste
of Boolean algebra, and careful reasoning by Dedekind cuts. Chapter 2 (Sets
and Spaces) begins with the familiar, eases into generalizations, and includes
just enough about transfinite cardinals to prove (in Chapter 5) that some
Lebesgue measurable sets are not Borei sets. The classical material of Chapter 3
(Sequences and Series) shows how readily previous work may be tightened and
extended. Limits inferior and superior are defined with care and their merit
demonstrated by repeated use, not the least of which is 1'Hospital's rule in
Chapter 7. The first three chapters provide sufficient preparation for Chapter 4
(Measure and Integration).
One guiding light for this book was a strong conviction that time is overdue
for Lebesgue theory to come into its own at an early stage without initial high
abstraction. By starting with sets on a line, generalizing length, then sloughing
off all except pertinent properties, Carathéodory's definition of measurability
is revealed as a natural and intuitive requirement. Meaningful and significant
theorems on integration are then proved with minimal effort and never a whisper
of mystical visions. Those who continue to more advanced work will be equipped
to fathom the thought processes that led to the Riesz or Danieli development
of integration, either of which is excellent for a second exposure.
Classical Stieltjes integrals are kept for their role in the procession of ideas
from Riemann to Radon-Nikodym, even though they are subsumed (in Sec. 8-6)
VI Preface
by Lebesgue integrals upon replacing the awkward integrator functions with
measures. Riemann integration takes its place as an antique in the true sense of
the word: not a piece of junk, but a prized relic from a former era honored for
inspiring replacements of greater comfort and utility, and still in working order
for many uses.
Problems are scattered throughout the text and each chapter ends with a set
of problems. The text problems encourage immediate checks on comprehension.
Some of these problems are part of the development and are stated in theorem
form with neither command nor entreaty to work them; a subsequent dis
cussion may justify a step by referring to such a problem as readily as to any
other theorem. The only reference to an end-of-chapter problem is in some
problem at the end of a later chapter.
An "if" in a definition should be interpreted as if and only if, but "if" in a
theorem means if. If a theorem is of the "if · · ·, then · · · " form, then the proof
does not "Let the hypotheses be satisfied," but starts right in as if they were.
No theorem is stated as "Suppose · · ·. Then · · ·." In fact any "suppose" or
"assume" is a signal that a contradiction will be reached.
The student's mathematical preparation is neither ignored nor relied upon.
A proof of the Weierstrass approximation theorem, for example, is based on
simple facts about derivatives of polynomials although derivatives are defined
later. Trigonometric functions are used in examples and problems prior to their
definition and theory in Chapter 7. The disparaging attitude that the student
brings little ability to prove anything from any previous course is offset by
trusting him to see that integration before differentiation (or even continuity)
is realistic a£ this level and that methods yielding numerical answers need not
immediately follow the definition.
Built-in flexibility for a two-semester course may be judged by the variety
of topics tucked in where their omission will not dilute the course, but they are
available for enrichment as time and inclination permit; for example, infinite
products, Jensen's inequalities, and the Stone-Weierstrass theorem. A suggested
hard-core one-semester course includes sections 1.1 to 1.10, 2.1 to 2.8, 3.1
to 3.9, 3.12, 4.1 to 4.5, 5.1, 5.2, 6.1 to 6.8, 7.1 to 7.10, and 8.1 to 8.4.
Rochester, New York J. F. R.
January, 1968
Chapter 1
ORIENTATION
1-1 REAL NUMBERS
Classical mathematical analysis is based directly on the real number system
and abstract analysis consists largely of structures inspired by either specializations
or generalizations of real numbers. The real number system will therefore be
subjected to close scrutiny in Section 1-6. We do not, however, ignore experience
with real numbers and shall consider that antecedents for the early work are :
1. Ordinary arithmetic.
2. The use of < and > so that if a and b are real numbers, then one and only
one of
a < b, a = b, a > b
holds. Also, for c any real number, then
a + c < b + c if and only if a < b,
and whenever c> 0, then
ac < be if and only if a < b,
but whenever c < 0, then
ac < be if and only if a > b.
3. The integers consist of all natural (whole) numbers, their negatives, and
zero.
4. If x is a given real number, then there is a unique integer denoted by [x] such
that
[x] ^ x < [x] + 1
or equivalently x — 1 < [x] < x.
l
2 Ch. 1 Orientation
5. Any number equal to the quotient of integers (denominator not zero) is a
rational number. All other real numbers are irrational.
The following two theorems illustrate how these facts and their immediate
consequences are considered as starting points of proofs.
Theorem 1. (Archimedean Principle). If a and b are positive numbers, then
there is a positive integer q such that
qa > b.
Proof. Since b\a > 0, then 1 + bja > 1 and the integer q defined by
is such that q ^ 1. Hence, from 4,
\ a) a
and multiplication by the positive number a yields the desired result. |
Corollary. Ifa>0, then there is a positive integer q such that
1
-< a.
q
Theorem 2. Between any two real numbers there is a rational number.
Proof. Let a < b be real numbers. Since b — a and 1 are positive, let q be a
positive integer such that
q(b — a) > 1 and hence qb > qa + 1.
With p = [qa + 1], then (qa + 1) - 1 < p ^ qa + 1 so that
qa < p < qb.
Upon dividing by the positive integer q, the result
a<-<b
<?
shows the rational number pjq between a and b. |
A rigorous development of trigonometric and logarithmic functions is given in
Sections 7-3 and 7-9. In the meantime we may use elementary properties of such
functions for illustrative purposes and in problems, but shall not base any of the
intermediate theory on these functions.
The following problem takes for granted that among a finite set of numbers
there is a smallest number, although finite and infinite sets are not discussed until
1-2 Sets 3
Section 2-2. Thus, if q is a positive integer, if 0 < α < a < · · · < a < 1 and
γ 2 q+l
we let
s = min (a — A,·), then 0 < s < -,
i+l
since
q
1 > û - a = X(a - α,) ^ gs.
g+1 1 i+1
Also, the method of contrapositive proof may be used in the solution of the
following problem prior to the discussion of this method in Section 1-4.
PROBLEM I. (a). If ξ is an irrational number and 0 < c < 1, then there are
integers m and n such that
0 < m + ηξ < c.
Noie: ξ and n may be positive or negative and m may be zero, positive, or
negative.
[Hint: For k = 1, 2, 3, · · · set x = Κξ — [fa*]. Hence 0 ^ x < 1, but from the
k k
irrationality of ξ, 0 < x < 1, and also if / Φ k, then x Φ x. Take # = [1/c] + 1.
k t k
Among the numbers x x , · * ·, x there are at least two which differ by less
l9 2 q+i
than \\q < c. A difference of such a pair may be written in the desired form.]
(b). If ξ is irrational and a < b, then there are integers m and n such that
a < m + ηξ < b.
[Hint: Choose τη and η such that 0 <m + ηξ <b — a and let p =
γ γ 1 χ
[1 + al{m + ηξ)]. Then a <pm± + pn£ < b.]
x γ
(c). If 0 < a < b, then there are integers m and n such that
a < 2m3" < b.
[Hint: Show that ξ = (In 3)/(ln 2) is irrational by assuming it is rational and
getting a contradiction. Choose m and n such that
In <z In 3 In b 1
,—- < m + n —— < ;—-.
In 2 In 2 In 2 J
1-2 SETS
The elements of set theory are considered as in the background. Since notation
and terminology are not completely standard, however, the following expressions
will establish unanimity.
a e A
means that a is an element of the set A ; a φ A means a is not an element of A.
The set whose only elements are a, b, c, d is denoted by
{a, fc, c, d} or {&, c, a, d},
or any order of writing these elements between brackets. Thus {a} denotes the
set whose single element is a.
4 Ch. 1 Orientation
{x | statement about x}
is the set of all elements for which the statement is true. Thus
x = — with n — 2m φ 0
n
and {i} are the same.
The union of sets A and B isf
A <u B = {x\x e A or x e B}
and the intersection is
A n B = {x\x e A and x e B).
The union of A and B will always be designated by using u, but n may be omit
ted and the intersection denoted by AB.
Both A a B and B^> A mean that whenever a e A then also ae B; both are
read "A is a subset of B" or "5 is a superset of A," or "5 covers A." A = B if
and only if both A cz B and B a A.
The empty set is denoted by 0. For any set A,
0aA, Au 0 = A, and A n 0 = A0 = 0.
No symbol will be reserved for the universal set. If in some discourse the
universal set is denoted by /, then
A <= /, Au 1 = 1, AI = A,
and the complement of A is denoted by Ac where
Ac = {x | x e I, but x φ A}.
Even though the universal set is not denoted, the set BAC means the set of all
points of B not in A and the alternate notation B — A may be used:
B - A = B n Ac = {x \ x e B but x φ A} = BAC.
The following laws hold :
Commutative. A u B = B u A, A n B = B n A.
Associative. (A u B) u C = A u (B u C), (AB)C = A(BC).
Distributive. A n (B u C) = (A n B) u (A n C)
= ABu AC;
Au(BnC) = (Av B)n(Au C)
= (Au B)(A u C).
de Morgan's laws. (A u B)c = Ac n Bc;
(A n B)c = ACKJ Bc.
t The "or" is the "inclusive or," meaning "and/or."
1-2 Sets 5
A crutch for visualizing set operations is the Venn diagram scheme. The points
of the plane are identified with the universal set and the interiors of simple closed
curves with sets A, B, C, D, etc.
A OB (AUB)ClCc {AUB)nCcnD
FIG. 1
Definition 1. Let & be a set, of at least two elements, having two binary opera
tions v and A such that whenever a e 08 and b e 08, then also a v b e & and
a A b e 08. Then 0&, together with these two operations, is said to be a Boolean
algebra if the following conditions hold for all a e 08, b e 08, and ceJ:
(i) a v b = b v a a A b = b A a.
(ii) (a v b) v c = a v (b v c) (a A b) A c = a A (b A c).
(iii) a A (ft v c) = (a A ft) v {a A c) and
a v (ft Λ c) = (a v b) A {a v c).
(iv) There are elements of '08 denoted by 0 and 1 such that
a v 0 = a and a A 1 = a for all a e 08.
(v) Corresponding to each element a there is at least one element a! such that
a v a' = 1 and a A a' = 0.
[Note: In some situations we read v as "join," Λ as "meet," and the prime
as "complement." If, however, the setting is in logic, v is read as "or," Λ as
"and," and a' as "not a."]
Select any nonempty set /. Let J* be the collection of all subsets of /. Hence
A e 08 means that A a I. Then 08 has at least two elements, / and 0. By inter
preting v to mean u, Λ to mean n, 1 to mean /, 0 to mean 0, and A' to mean
Ac, it is seen that 08 is a Boolean algebra.
PROBLEM I. Let 0ß have the two integers 0 and 1 as its only elements. For
a e 08 and b e 08 interpret
a v ft to mean a + ft + ab (mod 2),
a A ft to mean ab.
Show that 08 is a Boolean algebra. [Note that if a v ft means a + ft (mod 2), then
the second part of (iii) does not hold when a = ft = 1 and c = 0.]
6 Ch. 1 Orientation
Theorem 1. Let (ß, v, A) be a Boolean algebra. Then:
1. 0 and 1 are unique.
2. Each ae& has a unique complement.
3. V = 0and0'= 1.
Proof. We are given that a v 0 = a for all a e @t. Let Ö be any element of J*
having the property that
a v Ö = a for all a e @ì.
In particular we then have both
Ö v 0 = Ô and 0 v Ö = 0.
Hence, from the commutativity expressed by the first equation in (i),
Ö = Öv0 = 0vÖ = 0.
It is in this sense that the element 0 is said to be unique.
Let I e J* be such that a A I = a for all a e $&. Putting a = 1 here and a = Ï in
(iv) we have
1 = 1ΛΤ = ΤΛ1=Τ.
Given any ûeJ, let a* e & be such that
a v a* = 1 and a A a* = 0.
[There is at least one element having this property, namely, a'.] Then
a* = a* v 0 by (iv)
= a* v (a A a') by (v)
= (a* v a) A (a* v a') by second equation (iii)
= (a v a*) A (a* v a') by (i)
= 1 Λ (a* v A') = a* v a' by (iv).
On the other hand, a' = a' v 0 = a' v (a A a*) = etc., as above, leads to
a' = a' v a*. Thus
a* = a* v a' = a' v a* = a'.
Hence each a e^ has a' as its unique complement.
As to 3, we know that Γ, the complement of 1, is the unique element of J^
such that 1 v Γ = 1. But the first equation of (iv) with a = 1 is
1 v0=l,
which says that 0 is a complement of 1. Thus 0 = Γ.
In a similar way, 1=0'. |
