BOREL SETS AND BAIRE FUNCTIONS

APPROVED;

Major Professor

1. B Minor Professor

sr. /yyj^L^h rector of the Department of Mathematics

Dean of the Graduate School

BOREL SETS AND BAIRE FUNCTIONS

THESIS

Presented to the Graduate Council of the

North Texas State University in Partial

Fulfillment of the Requirements

For the Degree of

MASTER OF SCIENCE

By

Fred W. Wemple, B.S.

Denton, Texas

January 1970

TABLE OF CONTENTS

Chapter Page

I. INTRODUCTION 1

Preliminary Remarks Definitions Assumed Theorems

II. BOREL SETS 19

III. BAIRE FUNCTIONS . . . . . 57

IV. RELATION BETWEEN BOREL SETS AND BAIRE FUNCTIONS 56

BIBLIOGRAPHY . . 68

iii

CHAPTER I

INTRODUCTION

Preliminary Remarks

One of the decisive advances in the study of analysis

without restriction to continuous functions was made by

Henri Lebesgue in the first years of the twentieth century

with his "Integrale, longueur, aire," Annali dl Matematlca

Pura e Appllcata (1902), based upon his dissertation, in a

successful attempt to eliminate some of the confining inade-

quacies of the Riemann integral and to broaden the class of

functions which can be brought under a satisfactory theory

of integration. His studies were based upon the work of •

Emile Borel on the theory of measure. Borel had already in

1898 in Legons sur la theorie des fonctions presented a

theory of measure for the class of sets now called Borel

sets (5)- The closed sets, open sets, and denumerable

unions of closed and open sets, i.e., sets of types and

G^, are special subclasses of the wider class of sets called

Borel sets. While the classical theory of Lebesgue measure

is more complete,—there are Lebesgue-measurable sets which

are not Borel sets—the system of Borel sets nevertheless

continues to play an important role in the theory of

Lebesgue measure, measure in abstract topological spaces,

etc., as it does in set theory and topology in general.

Many authors define the Borel sets as the members of

the cr-algebra generated by the closed sets. P. R. Halmos

in Measure Theory defines the Borel sets of a locally

compact Hausdorff space as the members of a cr-ring generated

by the compact sets (3). These definitions are equivalent

when applied to Euclidean n-space.

In Arinall di Matematica (1899) R. Baire classified a

collection of functions which not only can be defined in a

manner similar to the Borel sets, but are also related to

them. The exploration of this relationship will be one of

the purposes of this paper. Baire described discontinuous

limits of continuous functions as functions of class 1;

limits of functions of class 1, not themselves of class 1,

as of class 2 ; and so on (l, 2 ) . F. Hausdorff (4) and

others have redefined Baire function subclasses to include

all of the preceding subclasses, i.e., continuous funtions

are contained in the set of all functions of class 1;

continuous functions and functions of class 1 are contained

in the set of functions of class 2 j etc. This definition is

an even closer analogy of the Borel set classification and

is simpler in that each class represents a complete system.

This will be the method adopted in this paper

In Chapter II, the basic properties of Borel sets will

be examined, beginning with the well-known subclasses

containing sets of type F^ and G/ . Finally some important

theorems will be established involving general Borel sets.

Chapter III will consist of a study of the characteristics

of Baire functions, first Baire functions of a general class

and then particular Baire classes 1 and 2, with some reference

to functions which fall into these classes.

The final chapter is a series of theorems which connects

the two classifications. A significant necessary and

sufficient inter-relationship of Borel sets and Baire

functions will be shown.

This paper assumes an elementary knowledge of sets and

the operations union LJ and intersection O , the symbols € ,

( | ) , C Z , etc., and such notations as subset, class of sets,

subclasses, and similar terms. It also assumes an elementary

knowledge of the real number system and its subsystems, the

rational numbers and the irrational numbers.

Definitions

The following definitions will be used throughout

the text:

The symbol R will represent the set of all real numbers.

A point will mean a real number.

A point setf or set, will mean a set of real numbers.

•A closed interval [a,b] is the set of real numbers x

such that a ^ x ^ b , a, b £R.

A closed rav (an interval classification) is either

Ca,

An open interval (a,b) is the set of real numbers x

such that a < x < b, a,b €R.

A n open ray (an interval classification) is either

(a,oo), the set of all real numbers x such that a < x, a£R,

or (-oo,b), the set of all real numbers x such that x < b, b G R«

A semiopen interval is either [a,b) the set of real

numbers x such that a x < b, a,b€R, or (a,b], the set

of real numbers x such that a < x £ b, a,b€R.

A degenerate interval [a, a] is the real number a.

Any open interval containing a point x will be called

a neighborhood of x.

The complement of a set S, denoted by C(S), consists of

those points which are not in S. The complement of a set S

may be with respect to any set T containing S or with respect

to the set of all real, numbers.

The difference of two sets A and B, denoted A ~ B,

is the set of real numbers x such that x€A, x^B.

Sets A and B are dis.joint if A O b = 0,

A set is denumerable if there exists a one-to-one

correspondence between the elements of the set and the set

of positive integers.

A set which is either finite, i.e., there exists a

non-negative integer.n such that the set contains n elements

(n = 0 implies the set is empty set), or is denumerable,

is said to be countable.

A set S is bounded If there exists a real number K > 0

such that for every xGS, |x| £ K.

least upper bound (lub) M of a set S is a real

number such that for every x€S, x 0 there exists an x 6S such that

x > M - £ •

The greatest lower bound (gib) m of a set S is a real

number such that for every x€S, x ^ m (mis a lower bound),

and such that for every £ > 0 there exists an x € S such that

x < m + € •

A sequence £an} of real numbers is a function from the

set of positive integers into the set of real numbers.

A sequence { a ^ is said to converge to f, written

ajj—• f, if for every 8 > 0 there exists a positive Integer

N so that for every n > N, ̂ -£|< £.

T h e length of an interval [a,b] or (a,b) is the

number b - a.

A closed, descending, infinitesimal, interval sequence

{ xn} is a sequence of closed intervals such that for every

n, I n C In-1* 8X1(1 a s n — > 0 0 » th® length of I n goes to 0.

If every neighborhood of € contains an infinite number

of points of a set S, then £ is a limit point of S.

If a neighborhood of f can be found which contains only

points of a set St then £ is an interior point of S,

A set P is said to be closed if F contains all of its

limit points.

A set G is said to be open if every point of G is an

interior point of G.

A set S will be said to be of type if it is the

union of a countable number of closed sets.

A set S will be said to be of type G(x,A) « gib jy»(x,y) | y a} .

A set S is non-dense (nowhere dense) if every interval

contains a subinterval (a subset of the interval which is

itself an interval) J such that no point of S is contained

in J, i.e., I O J = 0.

S is then nondense if and only if for every closed

interval I there is a closed interval J such that no point

of S is in J.

If S and T are sets of real numbers, then S is said to

be dense in T if every X€T is a limit point of S.

A set S is said to be everywhere dense if it is dense

in the set of all real numbers.

A set is said to be of the first category (exhaustible)

if It is the union of a countable number of nondense sets.

If S is riot of the first category it is said to be

of the second category (inexhaustible).

The main interest in functions in this paper will be

in real functions of a real variable. The functions will be

defined on a set S, usually the set of all real numbers

( - 0 0 , 0 0 ) , but sometimes on the closed interval [a,b], the open

interval (a,b), or on some other set of real numbers. The

set S is called the domain of the function. The function f(x)

associates with every £ € S a real number called f(£).

The set which consists of all such real numbers f(£) is

called the range.

If 4 is a real number, then f(x) is said to be continuous

M jL i f f o r every £ > 0 there is a J > 0 so that if |x - § | < J ,

then |f(x) - f($)| 0 there is a S > 0 such that if x £ E and

|x - §| < cT, then |f(x) - f(f;)| < £.

f(x) is said to be continuous on S relative to E if it

is continuous at every point of S relative to E.

f(x) is not continuous at §(a point of discontinuity)

if there exists ail J > 0 such that if/> 0 there exists an

XQ in the domain of f(x) such that |xQ -£|

8

A function f(x) is said to be bounded on a set S if

there is an M > 0 such that for any x€S, |f(x)| M.

If f(x) is defined on a set S, then f(x) is lower-

s emicontinuous at relative to S if for every £ > 0

there is a/> 0 so that if x€s and Jx then

f(x) > f(§) - £ * f(x) is upper-semicontinuous at g relative

to S if for every £ > 0 there is a > 0 so that if x € S and

|x -£|< then f (x) < f (£) + E .

f(x) is lower-semicontinuous (upper-semicontinuous) on

S relative to S if it is lower-semicontinuous ( upper-

semicontinuous) at every § € S relative to S.

Let f(x) be a bounded real function defined on S = R,

The following interval functions are associated with f(x):

(i) M(l) = lub. jV(x) | x€lj , where I is an open

interval. The function M(l) is called the upper bound of

f(x) in I.

(ii) m(l) » gib. £f(x) J xGl} , the lower bound of

f(x) in I.

(iii) s(l) = M(l) - m(l). The function s(l) is called

the saltus of f (x) in I_.

The following point functions are also associated

with f(x):

(i) M(C) = gib. M(I), for all I such that §€l.

The function M(x) is called the upper boundary function of f (x)•

(ii) m(§) = lub. m(l)# for all I such that £€!• The

function m(x) is called the lower boundary function of f(x).

(iii) s(§) = M(£) - m(£). s(x) is called the saltus

function of f (x).

Let A be a class of functions. The sequence of functions

^fn(x)}" is a function from the set of positive integers

into A. The sequence of functions £fn(x)J l s said to be

convergent at £_ if the sequence £fn($)J is convergent.

The sequence £f n(x)} is said to converge on a set S if

£fn(£)} converges for every £ S S .

The sequence of functions ^ n (x ) } is said to converge

uniformly on a set S if there exists a function f(x) such that

for every £ > 0 there is an N so that for every n > N and

every x € S, | f n(x) - f(x)| < £•

A sequence of functions (fn(x)} is said to converge

uniformly at J?_ if there exists a function f(x) such that

fn(x)—>f (x) and for every £ > 0 there is a 0 and an N

so that if i, n > N and |x - gj < y> f( x)

10

Let f(x) be defined on S. f(x) is differentiable at f £ 8

if there exists a real number denoted by called

"the derivative of f (x) at §, such that for every € > 0 there

is a /> 0 so that if x 6 s and |x - £| < - f (g)|

11

Principle of Transfinite Induction, Let S be a

subset of a well-ordered set A with the following properties:

(i) the first element of A is in S

(ii) the statement every element which precedes some

element a£A is in S, implies a6S.

Then S « A.

Proof. If A - S is non-empty, then there is a first

ordinal P £ A - S. P is not the first element of A since

by (i) the first element of A is in S. Therefore there are

elements preceding P. All ordinals prededing 0 are in A

so, by (ii), £ £ A, a contradiction. Hence S - A = 0

and A = S.

Let A be any well-ordered set and let A denote the

family of well-ordered sets which are similar to A, Then

A is called an ordinal number. The ordinal number of each

of the well-ordered set 0, [l} , [1,2} , ^1,2,?J , . . .

is denoted by 0, 1, 2, 5# . • • respectively and is called

a finite ordinal number. All other ordinal numbers are called

transfinite numbers. The ordinal number of the set of

natural numbers is denoted by The following theorems

regarding ordinal numbers will be accepted without proof.

Theorem. Let s(A) be the set of ordinals less than the

ordinal A* Then A is the ordinal number of s(A) •

Theorem. Let A be any ordinal number. Then A + 1 is

immediate successor of A*

12

Now some of the ordinal numbers are shown according

to their order. First come the finite ordinals 0, 1, 2, 3,

. . . and then comes what is called the first limit ordinal

U> and its successors 1, 2, • . • (!J is the least

number of the second class of ordinal numbers, the least

transfinite number. The set of all ordinal numbers which

are types of denumerable well-ordered sets is called the

second number class, KQ.

Theorem. KQ is nondenumerable.

The first ordinal number following KQ is denoted by fl.

Set A is equivalent to set B, A—B, if there exists a

one-to-one correspondence between the sets A and B,

Let A be any set and lety* denote the family of sets

which are equivalent to A, T h e n i s called the cardinal

number of set S (S has cardinality/*), y " may be denoted

by

If A is equivalent to the interval [091]* then A

has cardinality c.

The symbol W,is used to denote the cardinality of

denumerable sets; and Wi denotes the cardinal number of J i ,

i.e., the cardinal number of the set of all ordinal numbers

a with a < S L

If A is equivalent to a subset of B, A £ B*

A < B means A "B and not A^B.

13

The l e a s t upper bound. of a se t A of ordinal numbers

iS defined to be the ordinal number such tha t

( i ) i f a £ A , then a j u ,

( i i ) every ordinal number P such tha t a c

Theorem. 2 = c .

Theorem. c->fi= c .

Theorem. c»c = c . yr

Theorem. c = c .

Theorem. The c lass of a l l open se ts has cardinal c .

Theorem, c - ^ ^ c»c « c , A

Now suppose E^G O Ga, i = 1, 2, • • • . Then, for

every i, and every a, E^6G a. Since each Ga is a

15

(i) lim (f (x) + gn(x)) = f(x) + g(x); n-*oo

(ii) lim fn(x)'gn(x) = f(x)'g(x); n—-co

(ill) lim kf (x) = kf(x), for any constant kj n—#00

(iv) lim ^ « g'(x)# ^ °* ^ 0 f o r a 1 1 n'

(v) lim |fn(x)| = |f(x)| .

n-*oo

Theorem 1,4. The set of real numbers is nondenumerable•

Theorem 1,5. If A and B are sets, A - B « AOC(B).

Theorem 1,6. CCLJA,^) « ric(iL)j a€A a€A

0 ( O O =LJc(Ar.), where A. is an index set.

acA a€A a

Theorem 1.7. A set H is closed if and only if its

complement is open.

Theorem 1.8. The intersection of a finite collection

of open sets is open.

Theorem 1.9, The union of a finite collection of

closed sets is closed.

Theorem 1,10. The union of any collection of open

sets is open.

Theorem 1.11. The intersection of any collection of

closed sets is closed*

16

Theorem 1.12, If I^* * 3-n* . • • is a

sequence of closed intervals such that

I1 3 • • • 3 ^n-l 3 ^n3 • # •

and lim L(l„) = 0, where I n = [%i*bn] imPlies k(ln) » bn - a ^

n—>00 ^ „ then there exists a unique real number t, such that 5 c ln for

every n, and A *n = fei-n;=l **

Theorem 1.13. If f(x) is continuous on [a,b], and

g(x) is continuous on [a,b] and is never 0, then

(i) f(x) + g(x) is continuous on [a,bj.

(ii) f(x)-g(x) is continuous on [a,b].

(iii) kf(x) is continuous on [a,b3, where k is some constant,

(iv) [f(x)J is continuous on [a,b].

(v) • is continuous on [a#b].

(vi) f[g(x)3 is continuous on [a,b], if the range of

g(x) is in [a,b].

Theorem 1.14. f(x) is continuous at § G S if and only

if for every sequence [xn} € S such that xn—*• £ , f (xn)-r»-f (|).

Theorem 1.15. If f(x) is continuous on [a,b], then for

k6R, {x€ta,b3 I f(x) > k} is an open set and {x€[a,b] | f(x) ̂ kj

is closed.

Theorem 1.16. If f(x) is continuous on [a,b], then there

exists a y£[a,b] such that f(y) = lub f(x) on [a,b].

Theorem 1.17. If f(x) is of bounded variation on [a,b],

then there exists a monotone nondecreasing function g(x)

defined on [a,b] and a monotone nonincreasing function h(x)

defined on [a,b] such that f(x) « g(x) + h(x).

17

Theorem 1.18. If G is a non-empty open set of real

numbers, then G is the union of a countable number of non-

overlapping open intervals, where at most one interval

can be in each of the forms (-00, o), (a,oo), or f-oo# b).

Theorem 1,19. If f(x) » lim f„{x), where each f«(x) is . n-*oon

continuous and (fn(x)J is uniformly convergent, then f(x)

is continuous.

Theorem 1.20.A sequence of functions £fn(x)J converges

uniformly on a set S if and only if for every € > 0 there

is a positive integer N so that for every n, m > N and

every xSS, |fm(x) - fn(x)|

CHAPTER BIBLIOGRAPHY

1. Boas, Ralph P. Jr., A Primer of Real Functions, Quinn and Boden Co. Inc., Rahway, New Jersey, Mathematical Association of America, i960.

2. Hahn, Hans, Reele Funktlonen, Leipzig, Akademische Verlagsgesellschaft M. B. H., 1952.

3. Halmos, P. R., Measure Theory, Princeton, New Jersey, D. Van Nostrand Company, Inc., 1950.

4. Hausdorff, Felix, Set Theory, translated by John R. Aumann and others, New York, Chelsea Publishing Co., 1957.

5. Taylor, Angus E., General Theory of Functions and Integration, New York, Blaisdell Publishing Co., 1965.

18

CHAPTER II

BOREL SETS

The Borel sets are all of the sets which may be obtained

by repeatedly applying the operations of union and inter-

section to denumerable numbers of sets in the following

manner. The closed sets are said to be of type FQ. The

unions of denumerable numbers of sets of type FQ are said to

be of type F^. The sets of type F^ are also called sets of

type F^, denoting one of the more important subclasses.

The intersections of denumerable numbers of sets of type F^

are said to be of type Fg. The unions of denumerable numbers

of sets of type Fg are said to be of type F^. The intersections

of denumerable numbers of sets of type F^ are said to be of

"type Fij.« Sets of type Fa, for every a

20

In the above, sets of type F have been defined only

for a

21

A n o t a t i o n o f t e n u s e d f o r t h e B o r e l s e t s o f l o w e r

t y p e s i s F , F f , F

22

(iv) (-OD,b) = [b - 2,b - 1] U [b - 3,b - U . • •

U[b - (n+l),b - |] U . . . i

(a, oo) = [a + l,a + 2] U [a + |,a + 5] U . . .

LJ [ a + a + (n+1) ] • • • •

(v) (-oo,b] = [b - l,b]U[b - 2,b](J . . .

U [b - n,b] U -• • • J

[a#oo) = [a,a + l] O [a#a + 2] O . . * 0[a,a + n] . • . •

(vi) ( - 0 0 , 0 0 ) - [ - 1 , 1 ] Ut-2,23U . . . U [ - n , n ] ( J . . . .

Theorem 2.2. The following intervals are sets of

type G/.

(i) [a,b], including the degenerate interval#

(ii) £a#b), (a*b], (~oo,b], [a#oo),

(iii) (a,b), (-00,b), (a,oo), (-00,00).

Proof, (i) [a,b] = (a - l,b + l) O (a - A,b + . . .

Pl(a-i,b + 1 ) 0 . . . .

(ii) la.b) = (a - l,b) 0 ( a - ̂ » b ) 0 . . . 0 ( a - i,b)P). 2 n

[a,00) is obtained as above by replacing b with °o.

(a,b] = (a,b + l) O (a,b + - J O . . . H(a,b + ̂ )P> . . . .

(-°°,b3 is obtained as above by replacing a with-oo.

(iii) (a,b) - (a,b)n(a,b)P> . . .

foo,b), (a# 00 ) , (-00,00) are shown to be sets of type G/

in the same manner as (a#b).

23

Theorem 2.3. The union of any set of intervals,

closed, open, or serai-open, is a set of type Fr,

Proof. Let G be the union of a set of intervals.

The union of any set of overlapping intervals, closed, open,

or semi-open, is again an interval, either closed, open, or

semi-open. Hence G = O G n where Gj = 0$ i £

i, j = 1, 2, . . • • Each Gn can be mated to some rational

number contained in the union.

Thus the union of any set of intervals of the hypothesis

can be written as the countable union of disjoint intervals,

closed, open, or semi-open.

Each such interval, by Theorem 2.1, is the union of

a denumerable number of closed sets, and the countable union

of a denumerable union of closed sets is a denumerable union

of closed sets. Hence the union of any set of intervals,

closed, open, or semi-open, is a set of type F̂ -.

Lemma 2.4. Let A be a non-empty point set and let

d be a positive number. Define

B « £x€R| y«>(x,A) < d}.

Then A(^B and B is an open set.

Proof. From the definition of ̂ >(x,A), A(2 B follows.

Let xQ £ B. Then yo(xQ,A) < d. There exists a point x^ £ A

such that ̂ (xQ,x1) < d.

Define d - y^(x.0,Xj) « h > 0.

Now let y € J ® (xQ - h, xQ + h). Then | y - xD| < h,

and since |xQ - x̂ J = d - h.

24

I y - Xx| l |y - x

0| + lx0 "

xil < h + (a - h) = d.

Therefore /> ( y ^ ) < d, yo(y,A) < d, and y 6 B. Thus O ,

and hence B is open.

Theorem 2.5. A closed set is a set of type G ^ .

Proof, Let P be a closed set.

Let Gn = (x6R | yo(x,F) < A }, n = 1, 2, . . . .

From Lemma 2.4, Gn is an open set. We show F « G n=l n

and thus F is a G/ set.

Let y £ F. /*(y,F) « 0 and 0 < A, for every n, oo n

Therefore y £ O Gn* n=l

oo Now suppose p 6 O but by way of contradiction,

n=l n

p £ F. Then we look at two cases,

(i) y"(p,F) = 0, and

(ii) />(p,F) > 0.

(i) If /*(p,F) - 0, then there exists a point x^G F

such that x± £ p andy^(p,x1) < for if not, either p G F

or 0 / gib {^(p,y) j y G F ^ . By the same reasoning there

exists a point xg ^ x^, x 2 € F, such that

< min. /tiVsXj)} , . . . ,

there exists a point x^ / F, such that

/ ^ ( p ^ min. ̂ (p,Xĵ —i)J , • • . #

Thus there is a sequence of distinct points of F whose

limit is p, Since F is closed, p £ P, a contradiction*

25

(ii) If y^(p,F) > 0, say y&{p,F) =» h£R, then there

exists a positive integer n_ such that i < h. Then 1 nl

p Gn ( >o(x,F) < ~ ] , and hence 1 ' 1

P £ 0 G n > a contradiction.

n=l n

Therefore p € F, and F is a G/ set.

Theorem 2.6. An open set is a set of type Fr.

Proof. Let G be an open set. The complement of G,

C(G), is closed, by Theorem 1.7. Then by Theorem 2.5,

C(G) - ft G . where G is open for every n. xis&x 1

By DeMorgan's theorem

C(C(G)) = C((jG), and n=l

oo G = C(G ), where C(G) is closed for every n.

n=l "

Hence G is a set of type F

26

Suppose G and J are open sets. Then

G - J = GOC(J),

where C(J) is a closed set. By Theorem 2.5, C(j) is a set 00

of type Gj, therefore C(j) = O Gn, where each Gn is an open n=l

set. The intersection of G with each of the open sets Gn is

also an open set. Thus G - J is the denumerable intersection

of open sets, and therefore is a set of type G/.

Now let F and H "be closed sets. Then

P - H - FOC(H),

where C(H) is an open set. By Theorem 2.5# P is a set of

type G/. The proof follows as in the difference of two open sets.

Finally, suppose G and J are open sets. Then

G - J « GOC(J),

where C(j) is a closed set. By Theorem 2.6, G is a set of

type P^. The proof follows as in the proof that the

difference of two closed sets is a set of type F

27

Hence, by transfinite induction, for every a

28

Theorem 2.10, For every a

number of sets of type G follows from Theorem 2,9, as it (X

does, if a is even, for the union of a finite number of sets

of type G and the intersection of a finite number of sets CX

of type Fa.

The remaining cases follow by transfinite induction.

The theorem holds for a = 0, Suppose the theorem is true for K

every 0 < a, and suppose a is odd. Let S « (~) Sn, where Sn n=l A 00

is a set of type for every n. Then S = () S , H=1 m=l

where each S is a set of type Fanm, < a. Hence oo, A oo

s 83 O i ' snm = O Sm' w h e r e> *>y hypothesis, each ^ is m=l n=l m=l

a set of type F , Therefore S is a set of type F . anm x

a

The proofs for S = S , where Sn is a set of type Ga, k n=l

a odd; S = {_) where each S is a set of type F , a even; n=l

x and S = f)S , where each S is a set of type G . a even,

n=l n n a

are similar proofs.

Theorem 2.11. For every ct < D every set of type F

is of type Ga+1 and every set of type Ga is of type Fa+1.

Proof. From Theorem 2.8, every set of type GQ is of

type F1# Let a

29

set of type G^ is of type F ^ , and suppose a is odd. Let

S be a set of type G . Then oo

S = O Sn, where each Sn is of type G™ , a < a. n=l ^ n

By hypothesis, each S is of type F a + 1, with a„+l < a+l. n oo n

Since a+l is even, S = Q ^ s n is of type Fa+1» Accordingly,

by transfinite induction, for every a • • • * Ga, . . . , a < A .

The class of all Borel sets, will be defined

& - U V a < / L

Theorem 2.12. (i) The difference of two sets of type

F a is a set of type F a +j and the difference of two sets of

type Ga is a set of type Ga+^, for a < A .

30

(ii) The difference of two sets of type F a is a set of

type and the difference of two sets of type Ga is a set

of type F a + 1, a

31

n oo Let Tn = S, . Then S =» Tn, where the sets T

k=l n=l

are nondecreasing and. of lower type than F_. Ui

Similarly, if a is odd every set of type Ga is the

intersection of a nonincreasing sequence of sets of lower k

type defining Tn »

n^l

If a is even the proof for Fa and Ga is analogous.

Theorem 2.1k, Let f(x) be any function defined on

[a,b]. The set of points of discontinuity D(f) is a set

of type Ft* .

Proof. Consider the sets Dn(f), n = 1, 2, . . . .

Let £ £ D n ^ *** 811(1 o n ly f o r every neighborhood I of

there are x 6 I, y 6 I such that |f(x) - f(y)*Jj> T h e

following must be shown.

(i) D(f) = 0» n(f). and n=l

(ii) Dn(f) is closed for every n. oo

(i) Let £ C Then there exists an m such that n=l

e e njf)- So for every S> 0 there is an x such that | x - £|< 0 such that

for every /> 0 there is an xQ with |xQ -J|<

22

There exists a positive integer k such that A < £. * 1c Let I be any neighborhood of ? . Then there is ail Xj_G I

such that - f(£)| ;> € > Hence £ GD^f), and

therefore £ £ 0 Dn( f)*

n=l

(ii) Let m be a positive integer and let £ be a limit

point of Dm(f). Then there is a sequence [£n] such that each l i m £n ~

n-*oo

Let I be any neighborhood of £. There is a positive

integer N such that €. I, then I is a neighborhood of

Since Dm(f), there exist x 6 I, y G I such that

|f(x) - f(y)| 2, I is an arbitrary neighborhood of £ ,

therefore £ 6 Dm(f);Whence Dm(f) is closed.

Theorem 2.15* A set of type is either of the

first category or contains an interval.

Proof. Let S be a set of type • Then S = S ,

n=l

where each Sn of a closed set. If the sets Sn are all non-

dense (nowhere dense) then S is of the first category.

If one of the Sn, say sm> is not non-dense, then there

exists an open interval I such that for every open interval

J Cli ^ Since Sm is closed, Sm contains its

limit points and hence, I C sm* and S contains an interval.

53

The set of real numbers is inexhaustible

(of the second category).

Proof. Let S be a set of the first category. Then

s = s x U s 2 U . , . , U S n U . . . ,

where S n is non-dense for every n. Let I be a closed interval.

Since is non-dense, there is a closed interval I^C!l of

length less than 1 such that = 0, There is a closed

subinterval Ig C of length less than such that I 2 O S 2 1=1 0*

Continuing, . . . , there is a closed interval I n C *n-l of

length less than such that i n n s n = 0 . . . .

Thus there exists a descending infinitesimal sequence

of closed intervals

*̂ 1 3 ^ 2 ^ • • • 3-^n-l D ^ n D * • • •

By Theorem 1.12, there exists a unique real number £ such

that tt 3 = Since I n O s n *= 0, ^ Sn, Therefore

S cannot contain all of the real numbers, and the set of

real numbers is inexhaustible.

Theorem 2.17. The set of irrational numbers is not a

set of type F*-.

Proof. Let $ be the set of irrational numbers. J) does

not contain an interval; therefore, by Theorem 2.17, if J- were

of type Ff then $ would be of the first category. This

implies that R =

34

Corollary 2.18. There Is no function f(x) which has

the set of irrational numbers as its set of points of

discontinuity.

Theorem 2.19. The class of Borel sets $. is the

smallest

35

The cases for G are similar. a

By transfinite induction every Borel set is an element

of C • Hence & Q C •

To show ̂ is a ^algebra, let S be a Borel set.

£i = U F » therefore S is a set of type F , for some a, a

56

Theorem 2.22, The Borel sets are c in number.

Proof. The class of all open sets has cardinal c.

So there are at least c Borel sets. Since by Theorem 2.11,

every set of type Ga is also a set of type Pa-fi* ^ will

suffice to show that there are c sets in U OKA

Suppose a < A. and for every 3 < a there are c or fewer

sets which are of type but not of lower type than Pp.

Then the class of all sets of type less than a has at most

cardinal c*K« = c, But every set of type P a is either the

union or the intersection of a denumerable number of sets

of lower types so that the number of sets of type F is no X.

more than c = c.

By transfinite induction this holds for every a

CHAPTER III

BAIRE FUNCTIONS

The Baire functions are a class of functions related

to the Borel sets. The continuous functions are said to be

of type f 0 (Baire class 0). Functions which are limits of

convergent sequences of continuous functions are of type f^

(Baire class l). For every a < A if the functions of type

fp have been defined for every P < a, then the functions of

type f (Baire class a) are limits of convergent sequences a

of functions of types f3 < a, By transfinite induction, this

defines Baire functions for all classes a

Just as with the classes of Borel sets, any attempt to

define functions of type as a limit of convergent sequences

as above is unsuccessful.

Theorem 3*1. If f(x) is a function of type Baire class a,

a < / i , then so is kf(x), where k is an arbitrary constant.

Proof. The theorem holds for a « 0. Suppose a

38

Corollary 3.2. If f(x) Is a function of type Baire

class a, a < fl , then so is - f(x).

Theorem 3.3. If f(x) and g(x) are functions of type

Baire class a

59

Then f(x) = lim f (x), where, for every n, f_(x) is of type n—oo n

fa > an < a* n 2

For each n, by Theorem 3.3* fn(x)«fn(x) = f n (x) is a

function of class an. Since gn(x) « ~ is continuous for

every n, gn(x) is a function of type fQ, and. hence g^x) is

of class a^. Then by Theorem 3.3,

fn2(x) + SnM = f„2(x) + i

is of class an, for every n, and by hypothesis, since

fn2(x) + -i- is never 0, so is — Applying Theorem 3.3

n n f 2(x) + i n n

once more,

- 7 \ 1 f" ( x )

" X fn W + H ~ f n S M + n

is a function of f , for every n. Then n

fn(x) f(x) 1

n-»oofn (x) + i f (x) f(x)

Since the denominator is not 0, the limit exists. Hence

p£xy is a function of type fa.

By transfinite induction the theorem holds for all a

4o

Theorem 3.7. If f (x) is a function of typê Baire

class a then so is |f(x)| .

Proof. The theorem is true for functions of type fQ.

Suppose for a

41

by Theorem 3.1, |[f(x) + g(x)J is of type fa; by Theorem 3.7,

|f(x) + g(x)| is of type fa; and applying Theorem 3.1 again,

•||f(x) + g(x)| is of type fa; by Corollary 2.4, f(x) - g(x)

is of type fQ; and Theorem 3.1 and Theorem 3.7, i|f(x) - g(x)|

if of type fa. Hence the functions

max. {f(x), g(x)j «I[f(x) + g(x)] +^|f(x) + g(x)| and

min. {f(x), g(x)} =l[f.(x) + g(x)] - ̂ |f(x) - g(x)|

are of type f . ^ a

Lemma 3>11. If f(x) is a function of type Baire class a

and |f(x)| £ k for every x, and k a positive real number,

then there exists a sequence of functions (fn(x)} such that

f(x) « lim fn(x), where each fn(x) is of type Baire class n-*oo

otn < a, and |fn(x)|

42

Theorem 3*12. The limit of a uniformly convergent

sequence of functions of type Baire class a < ft is of type

Baire class a.

Proof. Let £fn(x)J "be a uniformly convergent sequence

of functions of type fa and let f(x) = lim fn(x). Since

n—»oo

£fn(x)J converges uniformly, it follows by Theorem 1.20,

there exists a sequence of positive integers m-j_ < m£ < . . •

< UQ < • • • such that

] *msM " | < |

I f m , W " fm,(x> I < f

Hence for every x there is a convergent series of positive Sm

numbers ~ such that for every n and x, n=l Z1

I f„ (x) - (x) I < i-. • mn+l %i '

*3

for every n, g (x)—»[f (x) - f (x)] and. n m. n li mn+l

for every x and every i.

Now consider the sequence ̂ hn(x)J

hl(x) * ST (X) 1

hQ(x) - g, (x) + g0 (x)

(x) "n

where

h, (x) - g, (x) + g (x) + . . . + g (x) 1 Ai dl H

For every i, h* (x) is of lower type than f . a QQ Let £>0. There exists a positive integer N such that

J"*. ~ . Then, for every x, n=N+l 2 3

f(x) - fm (x) - H [ f m (x) - fm (x)]| < i. 1 n=l n+1 n 1 ->

Let x = xQ. There exists a positive integer N' such that

for every n < N and i > N',

lCfm - fm (x )3 - Sn (XJ < IT'

1 n+1 0 n ° ni ° 5N

Let i > max. (n, N'} . Then | f (xQ) - fm (xQ) - hi(x0)j

f(x°> • VXo)- - v*o)] V Ifm . " U s ,

n=l n+1 ®n ° n=l ni

44

N . ,,

l f K > - fmi(xo> -

+ - V x ° ) ] - g » M ~ n i i \ ( x ° }

* l' - \(*o> - i [ S + l ( X o ) " W 1 1

+ | XI [fm (XQ) "

fm 0>. I n=l n+1 • n 1

+ 1 - „ i i S ( x o ) i

i |' - V J t o ) - - V ' o "

+

w Jwi w n«l nri-hi a

< i + 4 + | - £ .

Thus lim h. (x) =» [f(x) - fm (x)]. n—*oo 1

[f(x) - fm (x)] is therefore of type fa. Hence, by

Theorem 3.5, f(x) = Cf(x) - f ^ x ) ] + f ^ U ) is also of

type f0.

Lemma 3.13. If a convergent sequence [fn(x)} of

continuous functions converges uniformly at a point then

f(x) = lim f (x) is continuous at f. n—oo n

Proof. Let £ > 0. Since (fn(x)} is uniformly convergent

at f, there is a ^ > 0 and a positive Integer N such that

if |x - ?]<

45

fN(x) is continuous at § , so there is a 0 such

that if | x -i | < /2, then |fN(x) - fH(f) | < |.

Let 0

2

such that for x 6 (xQ - /,xQ +cf)C^[a,b], | f (x) - f(x ) | < - .

Since xQ is the limit of {xn} $ there exists a point

of ( x j , say xm, such that xm G (xQ - /,xq +

Jj-6

Theorem 3.15. If a sequence {fn(x)J of continuous

functions converges to f(x) on an open interval (a,b), it

converges unifomly at some point £ £ (a,b).

Proof. If x G (a,b), there is a positive integer N

such that if n,ra < N, | fn(x) - ^m(x)| £> £• * where £ is any-

positive real number.

Let En - £x € (a,b) : | fn(x) - fm(x)| & €, for every n, m > nJ .

Then oo

(ajb) E-«r. N*=l "

By Lemma 3.14, for every n, m the set of points

such that |f (x) - f (x) | 0 there is a

positive integer N and a closed interval [a',b«]C(a,b) such

that for every x€[a',b«3 and > m,n > N, | fn(x) fm(x) | _£ £.

Then there is an [a^b-JC(a,b) and an such that if

n,m > and x6[a1,b13, | fn(x) - f mU)| £ 1.

47

There is an [a^jbg] (alJk^) 8X1 ^2 s u c^ ^ o r

every n,m > Ng arid x 6 [a2,bg], |fn(x) - fm(x) | < "£•

• • *

There is an [a^b.^] (2 a n d 8X1 Ni s u c h tha-t f o r

every n,m > and xGCa^jb^],|fn(x) " ^m(x) | < J-

• # •

Thus is obtained a sequence of intervals

(a,b) 3[a1,b1] 3 (a^b^ 3 • • • 3 Ca.pb.jJ 3(a i,b i)

3[ai-1,bi_13 3 • • •

There is a f 6 O [a^b^] so that § £ O (a-pb^). i s = l i —1 •

Let £> 0. There is a positive integer j such that

— < £. For every x 6 (a^jb^) and n,m > N^,

l f n M " fm^x^ I "J < £*

^ 6 (aj>bj), therefore fn(x) converges uniformly at $.

Theorem 3.16. If ff (x)] is a convergent sequence of

continuous functions whose limit is f(x), then f(x) has a

point of continuityift every interval.

Proof. Let (a,b) be an open interval. By Theorem 5.15#

there is a £ £(a,b) such that (f (x)j converges uniformly

at f. By Lemma 3.13^ f(x) is continuous at £.

Theorem 3.17. The set of points of discontinuity of a

function of type Baire class 1 is of the first category.

Proof. Let f(x) be a function of Baire class 1.

From Theorem 3» 16» the set of points of continuity of f(x)

is everywhere dense. On the other hand; Theorem 2.16 states

48

that the points of discontinuity of f(x) form a set of type

, which, since its complement is everywhere dense, is of

the first category.

Theorem 3.18. If f(x) has a denumerable number of

points of discontinuity, then f(x) is a function of type

Baire class 1.

Proof. Let f(x) be a function such that D(f), the set

of points of discontinuity, is denumerable. Let the points

of D(f) be enumerated x^, x2, . . . , xn, . . . . Let 0.

Define g^(x) = f(x) if x 6 CtCx-̂ -

49

be linear on [x^ - /n,x^], [x-̂ x.̂ + ' ̂ x2 " *

[xgjxg + ef^l» « • • » C*n *" ^xn,xn ^n^ *

f(x) = lim gn(x) and g (x) is continuous for every n. n-»oo n

Theorem 3.19. If f(x) has a finite number of points

of discontinuity, then f(x) is a function of type Baire

class 1.

Proof. The proof is similar to the proof of Theorem J.18,

but somewhat easier.

Theorem 3.20. Dirichlet's function is a function of

Baire class 2.

Proof. The Dirichlet function is defined on [0,1] as

[1, if x is rational, and f(x) » <

(0, if x is irrational.

It is discontinuous everywhere, and hence, by Theorem 3.17,

cannot be a function of the first class.

The rational numbers in [0,1] can be placed in sequence,

r-j., r2, r^, . . . . Define

{1 for x « rv, k « 1, 2, . . ., n 0 for every other x £ [0,1]. gn(x) has a finite number of points of discontinuity and

hence, by Theorem 3.19 is a function of the first class.

But lim g (x) » f(x). n-*oo n

Hence f(x) is of class fg.

50

Theorem 3.21. A bounded monotone function has at most

a denumerable set of points of discontinuity.

Proof. Suppose f(x) is a bounded monotonically

nondecreasing function. Let M be a positive integer such

that |f(x)| < M, for every x. Let Sn « £x € R | s(x) > — } .

Suppose S n is infinite. Consider 2nM points in Sn.

There are 2nM disjoint intervals in each of which s(x) >

In each of these intervals, Ij, J = 1, 2, . . . , 2nM, there

are Xj and y^ with Xj < y^ such that f(yj) -

2nM Then f(y2nM) - f(x1) 2 I I (yj) - > 2nM'~ - 2M.

a contradiction. Therefore S is finite, so the set n

oo s - u s„

n=*l

of points of discontinuity of f(x) is finite or denumerable.

The proof for nonincreasing functions is similar.

Corollary 3.22. Monotone functions (and the sum and

difference of any two monotone functions) are functions of

Baire class 1.

Proof. The proof follows from Theorem 3.18. The sum

and difference follow from Theorem 3.3 and Corollary J>A,

Corollary 3.23. Every function of bounded variation

is a function of Baire class 1.

Proof. Follows from Theorem 1.16.

51

Theorem 5.24. Every bounded lower-semicontinuous

function f(x) is the limit of a nondecreasing sequence of

continuous functions.

Proof, Let f(x) be a bounded lower-semicontinuous

function. Therefore there exists a real number M > 0

such that |f(x)| < M, for all x. For every n» 1, 2, . . . ,

define

Sn(x) » gib [f(y) + n|x - y|],

where varies over the set of all real numbers.

For x^, x2 ̂

gn(xi) • elb tf(y) + - y| 3

£ gib [f(y) + n |xx - Xgj + n|x2 - y| ]

m Sn(x2> + nlxl * x2l

Interchanging x^ and Xg,

g„(*2> ̂ «n(xl> + n K " X2I

Then gn(xx) - gn(*2) 1 n|xl " xa|•

Let £">0. There is a / »••-£. such that if j x^ - x2 < (f»

then |gn(x1) - gn(xg)| < ۥ

Therefore g^x) is uniformly continuous, for all n.

Now gn+i(

x) - glb Cf(y) + nlx - y | + |x - y|]

2 glb tf(y) + n|x - y |]

- gn(x).

Hence gn+i(*) ̂ gn(x), for every n, and gn(x) is nondecreasing.

52

Finally, for every n,

g (x) - gib [f(y) + nix - y|], then

II 1

gn(x) ̂ + nl x - yIJ * f o r a 1 1 y*

Let y = xj then gn(x) £. f(x), for every n. Hence |gn(x)} is bounded above, the lim g (x) exists and lim g (x) 0. Let x Q 6 R. Let k = f( x c) " £'• Since

f(x) is lower-semicontinuous, there exists a 0 such that

if Ix - y I < / , then f(y) > f(x ) - V - k. « Q o

Then gib [f(y) + n|x - y|] 2. where y is restricted to

those numbers for which | x - y|< - M + n/,

There exists a positive integer N such that if n > N,

- M + n j > k. Then for every n > N, gn(x0) ̂ Therefore lim gn(x0) 2. k » f(x ) - Since f is arbitrarily small, n-*oo

iif00gn(x°) ̂ f(xo'-

xQ was arbitrary, therefore 2 f(x), for all x.

The two relations imply lim g„(x) • f(x), and the n-*co

theorem is proved.

Theorem 3.25. Every bounded upper-semicontinuous

function f(x) is the limit of a nonincreasing sequence of

continuous functions.

Proof. The proof is similar to the previous proof,

redefining g (x) « lub [f(y) - nlx - yl 3. n '

Corollary 3.26. A bounded semicontinuous function

is of Baire class 1.

55

Theorem 3.27. Every derivative function f'(x) is of

Baire class 1.

Proof. Since f'(x) exists, f(x) is a continuous

function. For every positive integer n, define the function

gn(x) = n[f (x + i) - f(x)].

Then g (x) is continuous, and for every x, f'(x) = lim g (x).

n-»oo n

Lemma 3.28. There are c continuous functions.

Proof. Any continuous function can be determined by its

values at the rational numbers in its domain, i.e., given

any continuous function f(x), the value of f(£), for any

g £ D c a n be determined by a sequence °** rational

numbers in D̂ , such that x n *c/fi KX. Vf.

c » (2 « 2 » 2 = c.

But there are at least c continuous, since f(x) » k,

for every k 6 R, is continuous.

Hence there are c continuous functions.

Theorem 3.29. If a

54

at least c functions of Baire class a. Suppose a < A ,

and for every 0 < a, there are c functions of type f^. Then

the class of all functions whose type less than a has at most

cardinal c*X= c. Every function of type f is the limit Of

of a sequence of functions of lower type than f^. Since there

are c = c sequences of these functions of lower class, there

are no more than c functions of type f .

Hence there are exactly c functions of class f_. Qi

By transfinite induction, this holds for every a

55

Corollary 3.32. A convergent infinite series of Baire

functions is a Baire function.

Proof. The sum is the limit of the sequence of partial

sums. Each element of the sequence is also a Baire function,

Corollary 3.33. A convergent infinite product of

Baire functions is a Baire function.

CHAPTER IV

RELATION BETWEEN BOREL SETS

AND BAIRE FUNCTIONS

One might expect a connection to exist between the

Borel sets and the Baire functions in view of the parallel

way in which they have been defined. The methods of the

proofs necessitate restriction to finite ordinals, and it

will be shown that certain sets associated with the functions

of finite Baire type are of finite Borel type, and conversely

that if certain sets associated with a function are all of

the same finite Borel type then the function is of finite

Baire type. Precisely, Theorems 4.7 and ^10 together state

that a function f(x) is of type f , where a is an even integer, U

if and only if for every real number k, the set

% "* (x ^ R I ^ k} is a set of type G , and the set r a

E2 • {x € R | f (x) 2 k}

is a set of type Fa; and where a is an odd positive integer,

the set E^ is a set of type Fq and Eg is a set of type Ga.

This is a generalization (and different proof) of a

theorem by Lebesgue for sets and functions of class 1 (l).

In order to carry out the following theorems it is

convenient to define sets of type AQ and B^.

56

57

For every a

58

Lemma 4.2. S is a set of type if and only if C(S) r"' "' " ' 1111 "l Ct

is a set of type A. .

Proof. Let S be a set of type Ba. Then there exists a

function of type f and a real number k such that a

S = {x 6 R | f(x) 2 k ] • A l s o s » € R | - f(x) -kJ . Therefore there exists a

real function - f(x), which by Corollary 3.2, is a function

of type fQ, and a real number -k such that ^

C(S) » ( x £ R J - f(x) > -k}.

Hence C(s) is a set of type Aa.

Now suppose C(s) is a set of type AQ. Then there exists

a function of type fa and a real number k such that

C(s} » ( x 6 R | f (X) > k j . Hence

C[C(S)] = S ® G R | f(x) ̂ k j . Also

S = £x € R J - f(x) 2. -k J . Therefore

there exists a real function - f(x), which by Corollary 3.2

is a function of type fa, and a real number -k such that S « £x £ R | - f (x) ;> -k J , and hence

S is of type B . vX

Theorem 4.3. For every finite odd ordinal a, every set

of type Aa is of type F and every set of type B is of type ct

Ga. For every finite even ordinal a, every set of type AQ is

of type Ga and every set of type B is of type F„. vX

Proof. The theorem holds for a » 0. Suppose it holds

for every 0 < a, and suppose a is odd. Let S be any set of

59

type Aa. Then there is a function of type fo and a real

number k such that S = £x £• R J f (x) > k J .

f(x) = lim f (x), where the f (x) are Baire functions of v ' n n-oo

lower type than fa.

By Lemma 4.1*

S = U U O { x 6 R 1 fn(x) ̂ k + £ }* m=l r=l n=r

But by hypothesis, each [x € R | fn(x) ^ + -̂s f̂i?e -^a-1.

From Theorem 2.9, since a - 1 is even, the intersection of

a denumerable number of sets of type F a - 1 is of type F a - 1. 00

Then the denumerable union, (j , is a set of type FQ. Since r=l

S is the union of a denumerable number of sets of type FQ,

bu Theorem 2.9 again, S is of type F^.

Now suppose S is a set of type Ba. There is a function

f (x) of type f and a real number k such that ot

S a ( x £ K I f (x) 2 k } ' B u t a l S°

S « { x 6 R | - f(x) ̂ -*} , so that

C(s) , ® { x 6 R | - f (X) > -k J .

By Corollary 35.2, - f(x) is a function of class fQ. Then by

definition C(s) is a set of type Aa. From above, C(s) is

then a set of type Fa. Then, by Theorem 2.8, S is a set of

type Ga.

If a is a finite even ordinal, the proof is similar.

The proof is thus achieved by means of finite induction.

60

Lemma 4.4. For every finite ordinal a, if S is a set

of type Aa or Ba, there is a function f(x) of type fa+i such

that f(x) = 1, for every x 6 S and f(x) » 0, for every x £ c ( s ) .

Proof. Suppose S is of type Aa. Then there is a function

g(x) of type fa such that S = |x £ R | g(x) > 0 } .

Let h(x) - max. [g(x), o } . Then h(x) is also of type fQ.

For every positive integer n, let fn(x) = min.^nh(x), ij .

The functions fn(x) are all of type fa. The sequence (fn(x)j

converges everywhere and lim = 1, for x 6 S, and n—*oo

lim f_(x) =0, for x 6 C(S), and n-*oo

lim fn(x) is of type fa+i.

n-*oo

Suppose S is of type Ba. Then by Lemma 4.2, C(S) is

of type Aa. Hence there is a function f(x) of type f ^ such

that f(x) » 1, for x £ C(S) and f(x) = 0 for x £ S. The function 1 - f(x) is of type fa+i and satisfies

the conclusion.

Theorem 4.5. For every finite even ordinal a, every

set of type Ga is of type and every set of type Fa is of

type B . For every finite odd ordinal a, every set of type F ct

is of type Aa and every set of type Qa is of type Ba.

Proof. Let G be an open set. By. Theorem 1.18, G is

the countable union of disjoint open intervals, 1-2* • • • *

In, . . . , where « (ai,bi), i » 1, 2, 3, . . . . Let k £ R

and let x1 be some point in 3^. Define f(x) as the following.

62

Theorem 4.6. For finite odd ordinals, a set S is of

type A if and only if it is of type F . and S is of type B a ex oc

if and only if it is of type G . For finite even ordinals, (X

a set S is of type Aa if and only if it is of type Ga, and

S is of type B if and only if it is of type Fa. Or

Proof. Theorems 4.3 and 4.5,

Theorem 4.7. If f(x) is any function of type fa, where

a is a finite ordinal number, then for every number k,

the sets

| x G R I f(x) > kj and

(x e R I f (x) 2 kj-

are of type F a and Ga respectively, if a is odd, and of type

G and F respectively, if a is even. C& \X

Proof. Follows from Theorem 4.6.

Theorem 4.8. If a is a finite ordinal and S and T are

disjoint sets of type Ba, then there is a function g(x) of

type f such that g(x) » 1 on S, g(x) = 0 on T, and OL 0 < s(x) < 1 elsewhere.

Proof. There is an f-,(x) of type f such that ot

S = (x e E I f^x) £ o } and an f2(x) of type fQ such that

T - {x 6 R | f2(x) 0 on C(S), gg(x) » 0 on T, and g2(x) > 0 on C(T).

The function g-^x) + S2(x) l s n e v e r °-

62

Theorem 4,6. For finite odd ordinals, a set S is of

type Ao if and only if it is of type Fa, and S is of type Ba

if and only if it is of type Ga. For finite even ordinals,

a set S is of type Aa if and only if it is of type Gra, and

S is of type B if and only if it is of type Fa. cit

Proof. Theorems 4.3 and 4.5.

Theorem 4.7. If f(x) is any function of type fa, where

a is a finite ordinal number, then for every number k,

the sets

| x 6 R I f(x) > k J and

{x € R I f (x) 2. k}

are of type F a and Ga respectively, if a is odd, and of type

Gr̂ and F^ respectively, if a is even.

Proof. Follows from Theorem 4.6.

Theorem 4.8. If a is a finite ordinal and S and T are

disjoint sets of type Ba, then there is a function g(x) of

type f such that g(x) = 1 on S, g(x) = 0 on T, and a

0 < g(x) < 1 elsewhere.

Proof. There is an f-^(x) of type f such that

S = (x 6 E | f^(x) 0 on C(S), g2(x) = 0 on T, and g2(x) > 0 on C(T).

The function g-^W + g2(x) is never 0.

63

Let . , & 2 ^ g(x) =

gx(x) + g2(x)

Then g(x) = 1 for every x 6 S and g(x) = 0 for every x € T,

and 0 < g(x) < 1 for every other x, and g(x) is of type f .

Lemma 4.9. If f(x) is a continuous function and g(x)

is a function of type f . then f[g(x)] is a function of G»

type fa.

Proof. The theorem is true for a « 0. Suppose a for n

every n. Therefore f[g(x)] is a function of type *«•

Theorem 4.10. If a is a finite odd ordinal and f(x) is

such that for every real number k, the sets

A » £x 6 R | f(x) > kj and

B - [x £ R I f (x) ̂ k J

are Of type F^ and respectively, then f( x) is of type f . Cu v* U

If a is a finite even ordinal and the sets A and B are

of type Gra and Fa, respectively, then f(x) is of type fa.

Proof. If a is a finite odd ordinal and f(x) is such

that for every real number k, A is a set of type F a and B

is of type G

64

B is a set of type Ba. If a is a finite even ordinal and

f(x) is such that for every real number k, A is a set of type

G and B is a set of type F . then by Theorem 4.7, A is a a

again a set of type A and B is a set of type B . Thus the 0£

proof will follow "by considering the case where a is a

finite ordinal and f(x) is such that for every real number k,

A and B are sets of type A# and B respectively. ct

For every real number k, suppose the sets

A = 6 R |f(x) > kj and

B - (x 6 R |f(x) ̂ k}

are of type A and B , respectively, where a is a finite Cfc CX

ordinal. Then by Theorem 4.7, the sets

£x € R | f (x) £ k}

are *\of type Ba, since they are complements of sets of type A&

Suppose, for convenience, 0 < f(x) < 1 for every x. Let

n be a positive integer. For every m = 0, 1, 2, . . . , n - 1,

the sets E^ = (x €. R | f (x) £ H ̂ and

E 2 - (x e H I f(*) i are of type Ba. Hence, by Lemma 4.8, there is a gm(x) of

type f such that gm(x) = 0 for x € Eg and 6m(x) ® 1 f o r

x € E^, and 0 < gm(x) < 1 for all other x.

Let gn(x) = ̂ [g0(x) + g-^x) + . . . + gn-1(x)].

Suppose ~ < f (x) < B±i. Then n . n

g0(x) - g2(x) = . . . - g ^ C x ) - 1, 0 £ g^x) ^ "1, and

65

gr(x) =» 0, for every r > m. Hence

I f(x) - gn(x)| < i n n

for every x. Moreover, gn(x), as the sum of a finite number

of functions of type fa is itself of type fa. It follows

that f(x), which has been shown to be the limit of a

uniformly convergent sequence of functions of type f , is of V

type fQ.

It was assumed that 0 < f(x) < 1. The proof can be

extended to all real functions defined on R such that

L < f (x) < M, for integers L and M, L < M, by altering the

definitions of E^ and Eg so that for every m « 0, 1, 2, . . . ,

n - 1,

E1 88 { x ^ 11 I f(x) & + M ^(m)) and

Eg = jx € R | f (x) ̂ L + M-~ ^(m + l)} .

Then define

gn(x) = M ~ L[gQ(x) + gx(x) + . . . + gn_1(x)] - L.

To obtain the general case, let h(x) » arctan f(x). *77** 7T***

h(x) is bounded. For - — < p < ~ , p€.R*

C = ( x € R | h(x) > p] » |x£ R | f (x) > tan p] and

D = [x G R | h(x) 2. P] = {x €. R |f(x) tan p] .

If p i - r. then G = L = Rj and R is a of both types

Fq and If p ̂ ^ then C = D = 0 is a set of both types Fo a n d V

66

Therefore, for every real number p, C and D a*e sets

of type Aa and Ba, respectively.

Applying the case for bounded functions, h(x) is a

function of type fQ.

By Lemma 4 . 9 , tan h(x) = f(x) is of type f „ .

To add to the understanding of the Borel sets and

Baire functions and conclude this paper, several theorems

are mentioned whose backgrounds and proofs are too extensive

to be included in this paper, and note is made of some of

the work in this area. Hausdorff (g) showed that there

exist Borel sets of type Fa, a

CHAPTER BIBLIOGRAPHY

1. Goffman, Casper, Real Functions, New York, Rinehart • and Co., Inc., 1953.

2. Hausdorff, Felix, Set Theory, translated by John R. Aumann and others, New York, Chelsea Publishing Co., 1957.

3. Natanson, I. P., Theory of Functions of a Real Variable, Vol. II, translated by Leo Z. Boron, New York, Frederick Ungar Publishing Co., I960.

4. Taylor, Angus E. General Theory of Functions and Integration, New York, Blalsdell Publishing Co., 19&5.

67

BIBLIOGRAPHY

Boas, Ralph P. Jr., A Primer of Real Functions, Quinn and Boden Co., Inc., Rahvray, New Jersey, Mathematical Association of America, i960.

Goffman, Casper, Real Functions, New York, Rinehart and Co., Inc., 1955-

Hahn, Hans, Reele Funktionen, Leipzig, Akademische Verlags-gesellschaft M. B. H., 1952.

Hahn, Hans and Arthur Rosenthal, Set Functions, Albuquerque, New Mexico, The University of New Mexico Press, 19^8.

Halmos, P. R., Measure Theory, Princeton, New Jersey, D. Van Nostrand Company, Inc., 1950.

Hausdorff, Felix, Set Theory, translated by John R. Aumann and others, New York, Chelsea Publishing Co., 1957.

Natanson, I. P., Theory of Functions of a Real Variable, Vol. I, translated by Leo F. Boron, New York, Frederick Ungar Publishing Co., 1955.

Natanson, I. P., Theory of Functions of a Real Variable, Vol. II, translated by Leo F. Boron, New York, Frederick Ungar Publishing Co., i960.

Sierpinski, Waclaw, General Topology, translated by Cecilia Krieger, Toronto, University of Toronto Press, 1952.

Taylor, Angus E., General Theory of Functions and Integration, New York, Blaisdell Publishing Co., 1965.

68