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8-1-1963
Borel sets and baire functions
Jones Samuel Smith
Atlanta University
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BOREL SETS AND BAIRE FUNCTIONS
SUBMITTED TO THE FACULTY OF ATLAHTA UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THK DEGREE OF MASTER OF SCIENCE
IN MATHEMATICS
JONES SAMUEL SMITH
SCHOOL OF ARTS AND SCIENCES
ATLAHTA UNIVERSITY
*
AUG-UST 1963
TABLE OF CONTENTS
Chapter
I.
II.
Page
INTRODUCTION
1
PRESENTATION AND ANALYSIS OF DATA REQUIRED
FOR UNDERSTANDING OF BOREL SETS AND BAIRE
FUNCTIONS
,
2
Ordinal Numbers
Transfinite Induction
DualityIntersection
Union
Limits and Convergence
Convergent sequence of functions
Uniform convergent sequence of
functions
Properties of measurable sets and
functions
III.
BOBEL SETS
7
Definitions
Notations of lower types
Propositions and Proofs
Borel sets as measurable sets
IV.
BAIRE FUNCTIONS
13
Definitions
Notations
Propositions and Proofs
Theorems and Proofs
Lemmas and Proofs
V.
RELATION BETWEEN BOREL SETS
FUNCTIONS
.
•
.
AND BAIRE
19
Lemmas and Proofs
Proposition and Proofs
Theorems and Proofs
Baire functions as measurable functions
APPENDIX
25
BIBLIOGRAPHY
26
ii
CHAPTER
I
INTRODUCTION
This
thesis
Variables
at
is
an outgrowth of the course in Real
Atlanta University.
It can easily be under
stood if the reader has a fair knowledge of set theory
and theory of real variables.
The material has been
strongly influenced by the work on set theory by Felix
Housdroff,
the book on Real Functions
and many lectures
given by Dr.
S.
by Gasper Goofman
Saxena and Dr.
L.
Here you will be given some of the basic properties
open sets, closed sets and sets of Fot
called Borel
type fot
sets.
Also,
and G^
Cross.
of
, which are
some properties of functions of
9 which are called Baire functions.
In view of the parallel way in which Borel sets and
Baire functions have been defined and restricted to an
ordinal number,
the relationship between Borel sets and
Baire functions is discussed.
Since all Borel sets are measurable sets,
functions are measurable functions,
and Baire
therefore, some of the
basic properties of measurable sets and measurable functions
are
discussed.
CHAPTER II
PRESENTATION AND ANALYSIS OF DATA REQUIRED
FOR UNDERSTANDING OF BOREL SETS AND BAIRE FUNCTIONS
The ordinal number
number.
is
the smallest
infinite ordinal
It has denumerable cardinal numbers.
sider the set of ordinal numbers which have
or denumerable cardinals numbers,
numbers form a well-ordered set
a first element).
this
If we con
either finite
set of ordinal
(i.e., every suborder has
Thus, the well-ordered setw has an ordi-
nal number which we designate by/l .
Proposition.—For every infinite ordinal number
the set
of ordinal numbers less
set whose ordinal number
Proof.—Let W
number is<X .
segment of W
than
,
is a well-ordered
is
be a well-ordered set whose ordinal
Let /•*<**and let W (ft) - W^ be the initial
whose ordinal number is QC*
This
establishes
a one-one order-preserving correspondence between the ele
ments and its initial segments.
(1:125)
Now/2 is the well-ordered set of all finite or denumerable ordinals Qt .
For every denumerable ordinal number ex
,
the above proposition shows that the initial segment Ta^
of ¥ has ordinal numbero( .
Hence/|>dfor every denumerable
ordinal number^ 5 soil is nondenumerable.
If OC is a nondenumerable ordinal number and W1 is a
well-ordered set with ordinal numbera
similar to any initial segment of W.
> tnen w
is not
Hence,Qt £.il
This shows thatfl is the smallest nondenumerable ordinal
number.
Transfinite Induction.—The method of transfinite
induction for the positive integers is the following:
If a set S of positive integers is such that US and
ifVtiS implies YL+ i±S , it follows that S is the set of all
positive integers.
Theorem.— If S is a set of all ordinals less than a
given ordinalCC , and T is a subset of S such that:
1.
the first ordinal 1, is
in T
2.
fpr every ordinal***, if all ordinals less than
pare in T then $ is also in T, then T = S.
Proof .—If S-T is nonempty, then there is a first
ordinalAs-rby (1) M\ 1
/#*T.
S = T.
but all ordinals less than
Hence, S-T is empty.
Therefore, S-T = 0.
Hence
(1:131)
Duality,—If given any valid statement, and if we
interchange the words open and closed sets, union and
intersection, we get another valid statement.
tffki jOtSrt is a system of sets, their intersection
is the set consisting of those elements belong
ing to all the sets^ , OL^f\>
Thus y. $ (\ s f\ ft^ 7Z x$IU for every
A
*0
If A]_, A2, A3,
of sets
(i.e.,
..., An,
...
is an infinite sequence
a set k& associated with every positive
integer k), their unions /•)*[}$$
is tne set con"
sisting of all the elements belonging to at least one of
the sets A^, k = 1, 2,
Limits>—If A]_,
3?
A2,
•♦•> n,
A3,
...,
...
.
An,
...
is any sequence
of sets, the set of all elements which belong to any infi
nite number of sets in the sequence
su~perior«
is called its limit
and the set of all elements which belong to all
but a finite number of sets in the sequence is
limit inferior.
The limit inferior is always a subset of
the limit superior.
limit superior,
called its
If the limit inferior is equal to
the limit of the sequence of sets
Ag, A3,
said
to exist.
That is
...
if every element which belongs to an infinite
exists
to say, the limit A]_,
is
..., An,
number of sets belongs to all but a finite number of them.
Convergent Sequence of Functions
sequence SJm, £jO {
point Xo
at
a Point.—The
of a function is said to converge at a
for6.>Othere exist a positive N =V( 6, X© )
A sequence of function j^L&M is convergent on a set
if it is convergent at every point on the set, S.
The sequence of function }fm.00hs uniformly convergent
on a set S,
IL
if for a.ny£yO there exist a positive N = N (6)
/|^ 6
for *>«■>A/mr for
5
Properties of Measurable Sets
1.
If S-p S2?
...j Sn is a sequence of set and
S =
then
2.
it
If Gj£ G2'C" •• C Gn
... is a nondecreasing
sequence of open sets and G = \>J Gn, then for
every 6>Othere is an yi (& 3-
3.
If Sj, S2, S3,
...,
Sn,
...
is
number of measurable sets, then
^£&\
a denumerable
S- LJSin*
is measurable.
h*
If S-, and S2 are disjoint measurable sets and
if S = S1
5*
S2
then fll(S)- ^
If S-,, S2j«.-j Sn,
... is a denumerable number
^r 1
then
6,
^)* ^6S)
If Sn, Sn.
-L7
d *
"t*7
.... Sw,
'
n7
... is a denumerable number
^9
of measurable sets and ^s /|5k, > "t^en s is
measurable.
7.
If S^, S2?
•-•} Sn,
...
is a nonincreasing se
quence of measurable sets and S = (\ 5*,
then
m, ^5) x
jK*^vw kvl (^«-)
Properties of Measurable Functions,—If f(x)
measurable,
tions
SL>
is
if for every real number k the following func
are measurable.
1.
- f(x)
2.
cf(x)
3.
f(x) + c
5.
1
, if f (x) j* 0
6.
J f(x)|
7.
f(x) + g(x)
8.
f(x)
9.
f(x) - g(x)
10.
. g(x)
f(x)
i(xT
The proofs
of this
on set
thesis.
,
if g(x)# 0
of the propositions
are beyond
the
scope
They can easily be found in most books
theory.
Definitions
Definition 1.
For every set
S,
**te £5)s 9' £• £• Vp-Cfl) I 6 5 Sj
all open sets
containing S,
of S, where
O £ &Le £S)£ 1-
Definition 2.
of S is
where G varies over
is called the exterior measure
For every set S
the
interior measure
the number
where F is a
closed
Definition 3»
is
the number
set.
If S is measurable, the number
called the Lebesgue measure
of S.
CHAPTER III
BOHEL SETS
The Borel sets
are all the sets which can be obtained
from the open and closed sets by repeatedly applying the
operations of intersection and union to denumerable numbers
of sets.
The closed sets
are called of type Fo.
union of a denumerable numbers
Let F^ be
of sets "of type Fo."
the
Let
F2 be the intersection of a denumerable numbers of sets
"of type F-p"
Let F3 be the union of a denumerable numbers
of sets "of type F2»"
Continuing in a similar manner9 we
can define a type of 1^. for every Q(4. £\
.
Let oc and fi be ordinal numbers*, fl , where IJ = «J is
the beginning number of the classZQ*»J
•
We again distin
guish between even ordinals 3.°^ , and odd ordinals
20f+ 1.
Hence, we have the following rule
The sets F are
The sets 1^
fJt
if PC
the sets
systems
are the unions of sequence of sets
is odd and the intersection ifPt is even,
and <sc >ft .
(2:99)
Thus we see by induction we will
F2,
in the
obtain the
...j which can be written as Fj, J.
, F-
sets FQ, Fj,
, ^$r
..
Hence we get F^ as intersections of sequence of preceding
8
sets, theRt+i
as unions of sequences of sets of ftj. , and
so forth.
A slightly different way of generating G is derived
from the above by interchanging the roles played by union
and intersection, so that the system Gfi of sets G are de
fined as
follows:
"The set G are the open sets in the system
The set Got are the intersection ifft is odd and
the union if Qt is even, of sequences of sets
Gs
, where
^
All the sets G^ also form the smallest Borel system
G.
The G^+| are the intersection or union (depending on
whether Qc is even or odd) of sequence of set Got
by induction, we obtain the sets GQ, Gl9G2,
be written as G]_, Gg
Proposition 1;
,
G6»
.
Hence,
••- which can
, %gf
The complement of every set of type
F* is of type G* and the complement of every set of type
Get is of type K* for every at/. A .
Prodf; The proposition holds foroc-aO.
holds for every^OC and suppose Of is even.
OP
set of type Fa , then 5 = /|5vo
Fc^,<xn< ex
by hypothesist($^
Suppose it
Let S be any
> where Sn is of type
is of type G^^ .
But
/r\ l)c£5*)
j so that c<s) ls of tvPe Gor • Hence
c Jfc*\' is of type G* . Q.E. D. (1:135)
Part II:
Foi .
The complement of every set of G* is of type
Suppose c< is even.
Let S be any set of type G<*
then S =\J
S , where Sn is of type G^ , ^-rL<^
by hypothesis c(OnJ is of type ^
so c(S) is of type 0* .
\.
But
. But
c(S) = f)
Sn,
Hencec/(S*\ is of type P^ , Q.E.D.
The cases where Ot is odd can be proved by the princi
ple
of duality.
Prop6fei£ion;2; If**/] is odd, the union of a denumerable numbers of sets of F^ is of type F3 and the intersec
tion of a denumerable number of type Go< is of type Gcr
00
Proof: Let S = \J Sn where each S
then Sn = W
is of type F<y
Sm where each SM is of type F^^,
Hence S = T*j 1 I gnm
, as the union of a denumerable
number of sets of type ?ctnm, **«n <** is of type Fq, .
fore, 1^ is of type E*
Part II:
-
. Q.E.D.
There
(1:135)
The intersection of a denumerable numbers
of sets of type G* is of type G* .
Proof:
00
Let Of be an odd ordinal number and let S =f| Sn
where each Sn is of type GQf, then Sn = ( ]
snm ls of *yPe Gotnm> ocwox*
Hence S = f]
Its/
S
S
where each
() Snin, as
flfcs/
the intersection of a denumerable number of sets of type
G^nm'«Utm^ is of *yPe G^ •
Therefore G* is of type G«t ♦
Q.E.D.
The proof for even ordinal numbers can be proved by
the principle of duality.
Proposition ^: For every Qt</1, the union and inter
section of any finite number of sets of type S^
(of type G^
10
is of type Ij^
Proof:
(of type G^).
Leto( be an even ordinal number and let Sn be
the sets of type F^ , then S = [\ (\ Srm where Srm are
sets of type F^ j^, e^
<a >
then S is the intersection of
a denumerable number of sets of type Ffltn,
type F^ .
Part
II;
is of
.
Q.E.D*
Letocbe an even ordinal number and let S be
#t
sets of type G« , then 5: (j
Snm is of type G^ nm,
ocnm.<.£X
00
[) Sk*k
-
where each
» then s is the }m^on of a
denumerable number of sets of type G^n,
(c**.^ oc
Therefore a finite number of sets
Therefore a finite number of sets
G^ •
)
Therefore the intersection of a finite number of
sets of the type F^ are of type Foe
type Goc
(or*. < «
) ^s °f
of type G^
of type G^ are of type
Q.E.D.
Proposition U:
is of type G^ + 1
For every ^/]/every set of type F^
and every set of type G^ is of type
Proof: The proposition holds for fl<Qc •
Given
(Every
^ type is a G<< type and every Gqc type is an F^
type).
Note:
Every closed set is the intersection of a
sequence of open sets;
quence of closed sets.
type Gi,
then S is
every open set
Since GQ is
Is the union of a se
of type F^ let S be of
the union of a finite or denumerable
number of open sets.
But every open set is
finite or denumerable number of closed sets.
the union of a
Hence S is
.
11
the union of a finite or denumerable number of closed sets
so S is of type F^.
Gfi is of type F^,
Let<**Aand /3<Otthen every set of
•
Let S be of type G«t then S = /|
where each Sn is of type Gotn, ock.<oc
is of Jtype F-*n + 1 with
<xrL+i£<Ki't 9
S = /] Sn is of type F«* + x.
Q.E.D.
.
S]
By hypothesis Sn
since 1 + ot
is even
Hence G* is of type F<j< + 1.
(1:136)
Part II: Every set of type Fot is of type G^+j.
Proof:
Since every set of type Gi is of type Fo.
S be sets of type F-± then S is closed.
Let
Hence S is the inter
section of a finite or denumerable number of closed sets.
But every closed set is the intersection of a denumerable
number of open sets.
Hence S is the intersection of a finite
or denumerable number of open sets,
so S is of type G^.
Let
<¥<. Jj ij3>*-<< 9 tken every set of F^ is of type G^ + ^.
Let S be of type F^ then S = IL/sn> where each Sn is of type
F^n
o<h.<Q£
.
By hypothesis Sn is of type G^n+j with
since <*+•/ is even, S = \J^»v is of type G
Hence Foe is of type Got + i.
Proposition 5;
Q.E.D.
The Borel sets form the smallest system
of set such that
1.
all closed sets
are in the system,
2.
the union of any denumerable number of sets in
the system is in the system,
3.
The intersection of any denumerable number of sets
in the system is in the system.
12
Proof:
(3).
Let S be any system which satisfies
Then every set of type P^.is in S.
transfinite induction and properties
(1),
(2),
The method of
(2) and (3) assure
that, for every *(/.£l , every set of type Foe is in S. (1:137)
Proposition 6; All Borel sets are measurable.
Proofs The closed and open sets are measurable.
[ 4 **.
and suppose every / <oc 9 every set of type f/s
G£ is measurable.
and
Since the union of a denumerable number
of measurable sets is measurable, then S = \j
measurable.
Let
Sn is
«»/
Also every set of type Foe and of type G«(
is
measurable, since the intersection of a denumerable number
act
of measurable sets S = f\ Sn, then S is measurable.
CHAPTER IV
BAIRE FUNCTIONS
The Baire Functions are a class of functions similar
to the Borel sets.
The continuous functions are said to
be of type f0 (Baire Class o).
Functions which are limits
of convergent sequence of continuous functions are of type
fx ( Baire Class 1), for every ©t<H.
are of type f
If the functions
have been defined for every £<0c , then the
functions of type f^ (Baire Class Ot) are limits of conver
gent sequences of functions of type^Ct.
Baire has introduced an important classification of
functions as follow:
Let f(x ) be defined over U ; f and U are bounded or
unbounded.
If f is continuous in U , we say the class is
0 in U and write class fro, mod U
'
where each fn being of class o in U.
1 if f does not lie in the class
then f (x) = lira f_
n-><*> n
(x)
We say its class is
o mod.
Let the series F(x) =£fn(x) converge in U, each term
fn being continuous in U.
we see F is of class o,
or not continuous in U.
Since f (jt) = Jhsr*- iZvlC^)
or 1, according as F is
continuous
A similar remark holds for infinite
products.
13
Ik
The derivatives of a function f(x)
tions of class o or 1.
Let f(x) have a unique differential
coefficient g(x) at each point of U.
unbounded.
To fix the ideas,
ential coeficient
give rise to func
exists.
Both f and U may be
suppose the right-hand differ
Let hj_ y
h2>»»
= o.
Then
is a continuous function of x in U, but g(x) = lim g
(x)
exist at each x in U by hypothesis.
Proposition 1:
two functions
Proofs
For every <*<Qj the sum and product of
of type
fq is
of type fot •
The proposition holds
for 0(^0.
and the proposition holds for all jS^a m
g(x) be the two functions of type fc*. .
Suppose
Let f(x) and
Then f(x) = lim f(x)
and g(x) _ lim gn(x) where, for every n, fn(x) is of type
fo<n9 O(n.<CC > and ^n^ is of tyP®/X^Ot 5 so that
fn(x) + gn(x) and fn(x) . gn(x) are of type Y^_ = max
m^sfiu )•
since f(x) + g(x) = !*.% [^OO^kCx)] and
fn(x) . g_(x) = lim (fn00 . gn(x)) , it follows then
that f(x) + g(x) and f(x)
.
g(x) are of type foe .
Q.E.D.
(1:137)
Proposition 2:
If for every ^co/f(x) is of type fe^
and f(x) ^o, then A i is of type fq and - f(x) is of type
Proof;
For everyCt<Lz > f(x) #o and the proposition
holds for evexy/&<C Oc •
where
each .»
Let } . be of type foi , then ^i
n
i^\
is
i
of
f type
t
-s
.
Let or
= 1.
15
then each
_!
such that
Jl
is of type f« .
FTx7
fn(x)
Part II;
is of type
f,y
QCn
.
Now f(x) = lim
«^
Q.E.D.
n
If f(x) is of type f<* for everyxx<\L and the
proposition holds for every/^rot , then let - f(x) be of
type f<*
.
Hence, - f (x) = lim - f (x), where each - fR(x)
is of type - fo(n,«iio». Hence (-1) f_f(x) = lim - fn(x)7 =
ff(x) = lim fn(x)"| .
Therefore - f(x) is of type f^
.
Q.E.D.
Proposition 3:
then
For every attQ if f(x) is of type f<x ,
(f(x)| is of type f* .
Proof;
f0 .
The proposition holds for functions of type
SupposeCuQ.
fi<OL*
It holds for all functions of type
Let /fCfl/ be of type foe .
Thenjf(x)|= lim|fn(x)|,
where each fn(x) is of type fatn»a*^a> but then each
function /fn(x)| is of type f^ n.
so then
f(x)
is of type fo<
Each /f(x)/ = lim fcn(x)\
(by transfinite induction).
Q.E.D.
Proposition h:
For every^Q,
if f (x) and g(x) are
of type f<x^, then max (f(x), g(x)) and min (f(x), g(x) )
are
of type f «* .
Proof; The functions f(x) + g(x) and |f(x) - g(x)| are
of type f« .
Hence the functions max
f(x), g(x)
(f (x) + g(x)) + i | f(x) - g(x)| and min
= i
f(x), g(x) = i
(f (x) + g(x)) - i I f(x) - g(x)| are of type f* . Q.S.D.
(1:138)
16
Theorem 1:
For every^Q, the limit of a uniformly
convergent sequence of functions of type f^ is of type f^
.
The proof of this theorem depends on the following
lemma.
Lemmas
For every<y<*Q, if f(x) is of type f<* and
|f(x)J£ k for every x where k>0, then f(x) = lim f (x),
where each fn(x) is of type focn,<*,,«*, and /fn(x)/< k for
every
x.
Proof; Since f(x) is of type f« , there is a sequence
|gn(x)j9 the limit of gn(x) = f(x), where gn(x) is of
type fOtn, <*»t<ac •
hn(x), - kj .
type Of
Let hn(x) = min fg (x)5 k7 and fn(x) =
By the above Proposition^ fn(x) is of
; therefore, f(x) = lim f^Cx).
Proof of Theorem:
If the proposition holds forCfc-C7,
suppose it holds for every^cx,
Let ?fn(x)f be a uniformly
convergent sequence of functiongof type f<* and let f(x) =
Now
where the series converges uniformly to f(x).
is a convergent series >K^,
Hence there
of positive numbers such that
for every n and x, |fn *^'(x) _ fn(x)J^k.
For every n the function fn +1(x) - fn(x) if of type
fc* .
Hence there is a sequence ff
) w
(x)^ of functions of
f
lower type f^ , which converges to fn + x(x) - fn(x), such
that for every m and x
...f
I fnm W I 5 kn»
Consider the
(x).#. for every n, where gn(x) is of lower type
than f «,
•
We shall show that f(x) =
lim £$mUh
fc>o7 a positive N such that £__ ^n<% •
every x,
| f(x) - fx(x) -X
Hence for
fn+
andm> N<, I fn+ X(x) - fn(x) l^B').
^
Then | fU) - g^UJJ < |f(x) - fx(x) -
(ls 139)
Theorem 2s
there
is
a
Let
set
******
For anyo<4i-^» if f(x) is of class
S whose
complement
is
of the
first
cate
gory, such that f(x) is continuoaa on S relative bo &•
Proof:
Sn whose
fn(x)
Leto(^L/and suppose it holds for every
Let f(x) be of typedf^^of
•
Goraplement is of the
category such that
first
^very n is of type
is contin^s on Sn relative to Sn.
plemenfc of S- /
S
3ufc
the coxn
is of the first category and
£ (x) is continuous on S relative to 5 for every n* <^,K.
Theorem 3:
Proof:
a continuous
The Baire
There are
fanotions are c in number.
c continuous
functions because
function is completely determined by its
values at the rationals.
Hence there ared^ ^y« j7 i.
18
continuous functions.
in number.
So the functions of type f0 are c
Suppose Qf<fl.and for every/S^Qt there are c
or fewer functions of tyrce f^
whose type is less than
.
The class
of all functions
has cardinal number c ^Jft
= c,
But every function of type f<* is the limit of a sequence
of function of lower type so that the number of function
of type ft* is no more than c * = c.
By induction this holds
for everyo^Q hence there is no more than c *J$b = c Baire
functions.
But there are at least c Baire functions, since
the functions f(x) = a are Baire functions.
Theorem kz Every Baire function is measurable.
Proof: If f(x) is a Baire function, then for every
real number k the set S = E[f(x)> kj
so is measurable.
Q.E.D.
is a Borel set and
CHAPTER V
RELATION BETWEEN BOREL SETS AND BAIRE FUNCTIONS
Definition 1.
f (x) is said to be continuous on S if
it is continuous at every point of S.
Theorem 1, (a):
f(x)
is continuous on U if and only
if for every k, the sets S [f (x) > k | and E If (x) c M are
open sets.
Proof; Suppose f(x) is continuous on U.
Let f£ U. 3
^(f)>^there is an £yo
$ 4C0~€'?^ ' Sln°e f(x)
whenever
-
is continuous at jf
^% f ^
, there is a
Jf(x) - f (£)/<.£
S>O
if / x - i" /<£ 8
Hence there is a neighborhood
fOP 6Very ¥o*2 > f ^X^ > f ty " ^ > k
E ff(x) > k7 and E /f (x) <: k7 are open sets.
Suppose the sets E |_f(x)>k7and S jf(x)<kj
open for every k.
E £f(x) >
Let f ^ (0) and let £> o
f{\) -gj is open,
-
are
Then, since
There is a neighborhood I
off such that for every xQ-£ I
f(x)> f(£)-£ •
since E £f(x)>f (f) + £| is open.
Moreover,
There is a neighborhood
of I2 of Jsuch that for every xQ% I2, f(x)^f(^*) +4. I =
I±f\ I2 is a neighborhood of C .
For every xo$ I, I f (x) -
f (^)J<-£ so that f(x) is continuous at
19
20
Theorem 1
(b);
f (x) is continuous on U if and only
if for every k, the sets Eff(x)£kJ and E[ f(x)<kj are
closed.
Proof:
set.
Since the complement of an open set is a closed
If f (x) is continuous then sets E [f (x) > kj and E
[f(x)<kj are open sets (Theorem 1 (a) ).
The complements
of two sets E ff (x) >kj and E ( f (x) <kj are the sets
E ff (x) *kjand E ff (x)j k] respectively.
Therefore, the
sets E[f(x)2 kjand E [f(x)£ kj are closed sets.
Conversely,
if E [f (x)i kj and E I f (x) ■£ kj are closed sets, then sets
E gf(x)>kjand E[f(x)<kJ are open, therefore f(x) is con
tinuous,
Q.E.D.
There is a connection existing between Borel sets and
Baire functions because of the parallel way in which they
have been constructed as well as defined.
The sets associated
with functions of finite Baire type are of finite Borel type
and conversely that if sets associated with a function are
all of the same finite Borel type then the function is of
finite Baire type.
That is to say that a function f(x) is
of type f,, iff for every real number k the sets E[f(x)> kj
and E [f (x)< k] are of type G^ and F* respectively, if (X
is an even positive integer and of type F^ and C-,, respectively,
if C* is an odd positive
integer.
For convenience let sets of type Ac* and B* are defined.
For every C^Q, , a set S will be of type A^ if there is a
function f(x) of type f^ and a real number k such that
21
S = E[f(x)>kJ and a set, S will be of type B^ if there
is a function f(x) of type f^ and a real positive number
k such that S = E [f(x)£kj .
Lemma 2:
Proof;
If f(x) = lim fn(x) then
Suppose f (x) > k.
There is an r
£ for every n
Then /
an m
9
f (x) >k +
r, fn(x);>k + -nr-«
Hence if
xoi B ff(x)?k] then
Suppose
that
Then there is an m
There is also an r
Thus
This
y. i
shows
that
0
Proposition Is
of type Aais
FK
if
For every finite ordinal
of type
is even.
set of type B^ is
Every
G^and every set of type
set
of
type A<* is
of type G,, if Q( is
odd.
,
every set
B^ is of type
of type F,
and
every
22
Proof:
The proposition holds forQr = 0.
holds for everv^<LO(.
type Ac( •
Supposed is odd.
Suppose it
Let S be a set of
There is then a function f (x) of type fac and a
positive real number k3 S =
sf f(x)> kj •
Now f(x) =
lim f-(x) where the f«(x) are Baire functions of lower type
than f^ .
By Lemma 2,
C
But, by assumption, each E |fn(x) £ k + -i—J is of type
F^ - 1.
Sinceo(-
1
is
even,
the
intersection of a denumer
able number of sets
of
type F^ - i is
S is the union of a denumerable number
is of type F^ ,
of type ^- l.
of sets
Hence
of type F«
(ltl*H)
Suppose S is of type B^ .
There is a function f(x)
of type f^ and a real number k $ E If (x)^ kj
S = E^fCxJ^ kj . But S = E [- f(x)i-k] .
.
Hence
c(S) = B
F- f(x)^kl , so that c(S) is of type F^ , whence S is of
type G<x
.
If«tfis
Lemma 3z
a finite even ordinal the proof is
For every finite ordinal
,
if S is
similar •
the
set
of type A^ or Bw there is a function $(x) of type ^+i
f(x) = 1 for every X
Proof;
S and f(x) = 0 for every
Suppose S is
function g(x) of type f*
max Jg(x), OJ
.
of type A* .
Then there
S = E |g(x) > Oj .
Let h(x) =
Then h(x) is also of type f«* .
positive integer n let fn(x) = min I nh(x), lj .
tion fn-'W are of type f^ .
The sequence
is a
fn(x)
For every
The func
converges
23
everywhere and lim fn(x) = 1 for x£S and 0 for x $ S and
is
of type f +j.
Suppose S is of type B^ .
Then c(S) is of type A^ .
Hence there is a function f(x) of type fw+i
x £ S and f(x) = 0 for xi S.
tvPe £<+ 1
of type Go, is
of type
and every set of
Every set of type
type
Gft is
Then S
f or Of^o-
every set of type K _j is
x 4 Sn and fn.(x) = 0 for every x*
the
OC
Suppose or is
Let S be
IfOcis
by Lemma 3j
CSV
even and
Proofs
of G^ is
But
Similarly, every
(1:1^2)
the proposition holds
for
of type
of type A^ •
The proposition holds foroc- 1 and
holds for<*= 0.
a func
Hence,
- 1, want to show that every set of type F^ is
B^ and every set
the set
series converges uniformly.
set of type G^ is of type B« .
II;
odd.
> ^n^ = -in for everv
S = e| f(x)^Oi , so S is of type Aa .
Part
of type
Since by assumption
of type B^.j,
tion fn(x) of type f^ for every n
of type f^ since
is
=L/Sn where each S_ is of type $* and
the sequence is j is nondecreasing.
is
Fo,
of type %, if of is
the proposition holds for«- 1.
of type Fw •
every set
A« and every set of type F* is of
Proof: The proposition holds
odd and
(IjIU-2)
For every finite ordinalOc,
type IW ifocis even.
A
The function 1 - f(x) is of
and is 1 on S and 0 on c(S).
Proposition 2:
f(x) = 1 for
Let S be set of type G^ .
suppose
it
Then S = \J
where each Sn is of type G^ and the sequence (snj is nondecreasing.
Since every set of type G^ is
Lemma 3, a function fn(x) of type f
of type A«
,
by
for every n ^ fn(x) =
for every x$ Sn and fn(x) = 0 for every x £
S.
Hence
Us/
is
of type i^, since
the series
converges uniformly.
S = E[~f(x)>o7 so S is of type A^ ♦
of type A
•
Similarly,
every set
But
*
Hence set of G^ is
of Fw
is
of type B^ .
Q.E.D.
From the previous proven propositions,
theorems
are
the following
obtained 1
Theorem ht For finite odd ordinals, a set is of type
A* iff it
is
of Ew
,
and
is of type B« iff it is
For finite even ordinals a set is
type G«( and is
Theorem 5;
Q(
of type
Bo, iff it
If f(x)
is a finite ordinal,
of Gm .
of type kn iff it is of
is of type F^
.
is any function of type f«
then for
, where
every real number k the
set E [f (x)> k] and E [f (x)z k] are of type F^ and G*
respectively if
if ocis
is
odd and of type (^
and F^respectively
even.
This is proved by Theorem h.
(lslH-3)
APPENDIX
SYMBOLS
\ J
Omega
iff
if and only if
greater than or equal to
less
than or equal to
not equal
(1:25)
reference
2
contained in
^
such that
Zj
there exist
epsilon
Aleph
zero
union
delta
/
*X
fecoj
i
intersection
alpha
sequence
beta
lim
limit
summation
infinity
of functions
BIBLIOGRAPHY
1.
Goffman, Casper.
and Company,
2.
Hausdorff, Felix.
Set Theory.
Chelsea Publishing Company.
3.
Graves, L. M.
The Theory of Functions of Heal Variables
2nd ed.t Hew York: McGraw-Hill, 1956.
4.
Hartman,
S.
Real
Inc.,
Functions.
1953-
and Mikusinski J.
Measure and Integration.
Pergamon Press,
1961,
Hew York:
Hew York*
Rinehart
Hew York:
The Theory of Lebesque
Vol.
15,
New Yorks
5.
Pierpont, James.
The Theory of Functions of Real
Variables, Vol. II, Hew York, Hew Yorks
Ginn and
Company.
6.
Halmas,
7.
Natanson, I. P.
Theory of Functions of Real Variables.
Hew Yorks
Frederick Ugar Publishing Co., 1955.
Paul Richart,
Variables.
The Theory of Functions of Real
Kew YorlEl
Dover Publication, 1927.
Unpublished Materials
Lectures
Cross.
given In
class by D.
26
S.
Saxena and Dr.
L.