A R K I V FOR MATEMATIK Band 6 nr 8
1.64 12 1 Commutaicated 14 October 1964 by O. FROSTMAN and L. CARLESON
B a i r e s e t s a n d B a i r e m e a s u r e s B y K E N N E T H A . R o s s a n d K A R L STROMBERG
I n t r o d u c t i o n
The p u r p o s e of t h i s p a p e r is t o a s c e r t a i n to w h a t e x t e n t c e r t a i n k n o w n results a b o u t B a i r e sets a n d B a i r e m e a s u r e s on a - c o m p a c t , l o c a l l y c o m p a c t spaces 1 are v a l i d in more g e n e r a l spaces. W e f r e q u e n t l y find t h a t t h e k n o w n results c a r r y over t o p a r a c o m p a c t , l o c a l l y c o m p a c t spaces, hence, in p a r t i c u l a r , to all locally c o m p a c t t o p o l o g i c a l groups. W e also list some a d d i t i o n a l p r o p e r t i e s a b o u t locally c o m p a c t g r o u p s t h a t are n o t v a l i d in all p a r a c o m p a c t , l o c a l l y c o m p a c t spaces.
L e t X be a t o p o l o g i c a l space. A set Z c X is a zero-sct if Z =/-1(0) for some continuous r e a l - v a l u e d f u n c t i o n / on X; we often w r i t e Z(/) f o r / - 1 ( 0 ) . The Baire sets are those subsets of X belonging t o t h e s m a l l e s t a - a l g e b r a containing all zero-sets in X. This definition of B a i r e sets is a p p a r e n t l y due to H e w i t t [6]. The Borel sets are those sets belonging to t h e s m a l l e s t a - a l g e b r a t h a t contains all closed subsets of X. Clearly a Baire set is a l w a y s a B o r e l set. I n m a n y f a m i l i a r spaces, including all m e t r i c spaces, t h e classes of B a i r e sets a n d Borel sets coincide. Our definition of B a i r e sets is con- s i s t e n t w i t h t h a t of H a l m o s [5] w h e n e v e r X is a - c o m p a c t a n d locally compact; see 1.1. All of our results are well k n o w n for a - c o m p a c t , l o c a l l y c o m p a c t spaces. W e h a v e o b s e r v e d a t e n d e n c y a m o n g writers to assume t h a t t h e results are t r u e for g e n e r a l l o c a l l y c o m p a c t spaces; we hope t h a t t h e p r e s e n t p a p e r will show t h e l i m i t a : tions of these a s s u m p t i o n s .
F o r a topological space X , C(X) will d e n o t e t h e f a m i l y of continuous r e a l - v a l u e d functions on X . W e also define Coo(X)= ( / E C ( X ) : / h a s c o m p a c t support}, C+(X)=
{/E C(X):/~> 0}, a n d Cgo(X) = {/E C0o(X):/>/0}.
F o r s t a n d a r d t e r m i n o l o g y a n d results in m e a s u r e t h e o r y , t o p o l o g y , a n d topological groups, we refer t h e r e a d e r to H a l m o s [5], K e l l e y [9], a n d H e w i t t a n d Ross [7], where he will also f i n d references t o t h e original sources.
1. B a i r e sets W e begin w i t h a n e l e m e n t a r y o b s e r v a t i o n .
Proposition 1 . l . I / X is a a-compact, locally compact space, the class o/ Baire sets is equal to the smallest a-ring S o containing all compact Go's. I n o t h e r words, a set is a Baire set as d e f i n e d in t h i s p a p e r if a n d o n l y if it is a B a i r e set as defined in H a l -
m o s [5].
1 All topological spaces in this paper are assumed to be Hausdorff spaces.
K . A. ROSS~ K . STROMBERG~ B a i r e sets a n d B a i r e m e a s u r e s
Proof. Since X is regular a n d LindelSf, it is normal.. Therefore every closed G~
is a zero-set, a n d hence every m e m b e r of So is a Baire set.
F
Consider a zero-set Z; clearly Z is a G~. We have X = (J ~=1 ~ where each F~ is compact. F o r each n, there is an f~ in C~0(X) such t h a t f , ( F , ) = 1. Defining B ~ =
B
/~1(1) for each n, we obtain compact G~'s such t h a t X = [J ~ffil ~. Thus Z = (.J ~ 1 (Z N B~) is a countable union of compact G0's and Z E S o. Hence every zero-set be- longs to So, and so every Baire set belongs to S 0.
The next three theorems generalize Theorem 51.D, H a l m o s [5].
Theorem 1.2. A compact Baire set F i~ any topological space X is a zero.set.
Proof. E x a c t l y as in Halmos' proof of 51.D, there is a metric space M a n d a con- tinuous mapping T of X onto M such t h a t F = T-I(Mo) for some subset M 0 of M.
Since T ( F ) = M e , M 0 is compact a n d hence closed in M. Therefore M o is the zero- set of some g in C(M). If we d e f i n e / = g o T , t h e n / E C(X) a n d Z(/) = F .
Theorem 1.3. A closed Baire set E in a paracompact, locally compact space X is a zero-set.
Proof. Let ~ consist of all open subsets of X having compact closure, a n d let ~q be a locally finite open refinement of ~ . Since ~ is also a point finite cover, there is an indexed family ~ q = { W v : V G ~ } of open sets Wv such t h a t W { ~ V for all VElq a n d [J v ~ W v = X (see Ex. V, page 171, Kelley [9]). F o r V G ~ , we choose fv i n C~o(X) such t h a t f v ( W ~ ) = 1 a n d / v ( X - V)=0. Finally, we define B v = / ~ l ( 1 ) for V E ~q. Then (J v ~ B v = X and each By is a compact Baire set. Hence each Bv N E is a compact Baire set; using 1.2, it is easy to see t h a t there exist functions gvE C+(X) such t h a t Z ( g v ) = B y N E a n d g v ( X - V ) = 1 . Define g = i n f v ~ g v ; since ~ is locally finite, g is continuous and it is obvious t h a t Z(g) = E.
Examples 3.1 and 3.2 show t h a t the conclusion of Theorem 1.3 might fail if either the hypothesis t h a t X be p a r a c o m p a c t or the hypothesis t h a t X be locally compact is dropped. We have been unable to ascertain whether or n o t the conclusion holds when the hypothesis on X is weakened to "normal, locally compact"; however, see Theorem 1.5 below. E x a m p l e 3.3 shows t h a t the conclusion of Theorem 1.3 does hold in all well-ordered spaces.
The following routine l e m m a will be used in proving Theorem 1.5 and also in Section 2.
L e m m a 1.4. Let X be a normal space, and let Y be a closed Baire set in X . Then a set E c Y is a Baire set in X i / a n d only if it is a Baire set in Y.
Proof. ( ~ ) . Let ~ = ( F c X : F n Y is a Baire set in Y}. I t is easy to verify t h a t every zero-set in X belongs to :~, t h a t F E :~ implies X - F E :~, and t h a t (Fn}~=l c :~
implies (J~%1 FnE:~. Hence every Baire set in X belongs to :~. Thus i f E c Y is a Baire set in X, t h e n E N Y = E is a Baire set in Y.
( ~ ) . Let E = { E ~ Y : E is a Baire set in X}. Clearly ~ is a a-algebra of subsets of Y. To show t h a t ~ contains all Baire sets in Y, we need to show t h a t every zero-set in Y belongs t o E. Suppose t h e n t h a t E is a zero-set in Y a n d t h a t / in C+(Y) is such t h a t Z ( f ) = E. Since X is normal, f can be extended to a function ] in C+(X) b y the Tietze-Urysolm theorem. Then E = Y n ]-1(0) is a Baire set in X and E belongs to ~.
ARKIV FOR MATEMATIK. B d 6 nr 8 T h e o r e m 1.5. A closed a-compact Baire set E in a normal, locally compact space X is a zero-set.
Proo[. Since E is a-compact, we can find a sequence {U=}~=I of open sets such that E c 0 n~lUn and each U~ is compact. Using normality twice, we can choose open sets V and W such that E c W c W - ~ V c V - c (J~=lUn. Choose a function k in C+(X) for which k ( V - ) = O and k(X-[J=~_-IU,)=I, and define Y=k-l(O). Then Y
oo oo U -
is a closed Baire set and V - ~ Y ~ [.J~=IU~. Also Y=(.J~=I( ~ f] Y) is a-compact and normal, and hence paracompact. By 1.4, E is a closed Baire set in Y. Hence by 1.3, there is an )t in C+(Y) such that Z(/)= E. We m a y assume that 0 <~/(x)<~ 1 for xE Y. Since X is normal, Urysohn's lemma shows that there is a g in C+(X) such that g(E)=O and g ( X - W ) = 1 . Define h on X by the rule:
h(x)=sup ([(x), g(x)) for x e Y,
=1 for x ~ Y .
Clearly h is continuous on the open sets V and X - W-. Hence h belongs to C+(X);
it is obvious that Z(h) = E.
Example 3.1 shows that the hypothesis in Theorem 1.5 that X be normal is needed.
We next observe two interesting facts about Baire sets that hold for some impor- tant, but special, spaces. Example 3.4 shows that both conclusions may fail in a sufficiently bad compact space.
T h e o r e m 1.6. I[ X is a locally compact group or an arbitrary product o/ separable metric spaces, then the closure o / a Baire set is a Baire set and the closure o / a n y open set is a Baire set.
We prove this theorem after proving Theorem 2.5.
2 . B a i r e m e a s u r e s a n d f u n c t i o n s
The next theorem generalizes Theorem 52.G, Halmos [5]. A Baire measure is simply a measure defined on the Baire sets which assigns finite measure to each compact Baire set.
T h e o r e m 2.1. I] X is lutracompact and locally compact and i / c a r d (X) is less than the first inaccessible cardinal, then every a-finite Baire measure ix on X is regular; i.e., /or each Baire set E we have
ix(E) = s u p (ix(C):Cc E and C is a compact Baire set}
=in/{ix(U): U D E and U is an open Baire set}.
Proo]. Since IX is a-finite, there exists a sequence (IX~}~=x of finite Baire measures such that ix(E)=~%lixn(E) for each Baire set E.
By Th~or~me 5 of chap. I, w 10, No. 12 of Bourbaki [1], we have X ~ L]~AX~
where the Xa's are pairwise disjoint open a-compact sets. 1 For each positive integer n, we define v, on the set of all subsets of A by the rule:
vn(r)=ix,(l.JX~) for F c A .
AEI'
The authors are indebted to ]Professor E. A. Michael for calling this theorem to their attention.
K. A. ROSS~ K. STROMBERG~ B a i r e sets and B a i r e m e a s u r e s
Note t h a t every set U ~rX~ is a Baire set since it is open and closed. I t is evident t h a t each v~ is a finite measure on the set of all subsets of A a n d t h a t card(A) is less t h a n the first inaccessible cardinal. N o w Theorem (A) of U l a m [16] shows t h a t for each n there exists a countable subset A,, of A such t h a t v~(A~)=v~(A). N o w let A 0 = U . = l A ~, and let Y = U~AoXz. Since A 0 is countable, we see t h a t Y is a-com-
pact. Also
n = l n - 1 n - 1
(The existence of a a - c o m p a c t set Y such t h a t / z ( X - Y) = 0 also follows directly from R u b i n ' s theorem announced in [14].)
N o w consider a n y Baire set E in X. Theorem 52.G of Halmos [5] applies to p restricted to Y; t h a t is, # is regular on Y. Since Y is open a n d closed, Y is a Baire set in X and so E N Y is a Baire set in X. B y 1.4, E N Y is a Baire set in Y. N o w applying these facts, we have
/~(E) = # ( E N Y)
= i n f { # ( U ) : U ~ E N Y, U is an open Baire set in Y}
= i n f {#(UU ( X - Y ) ) : U ~ E N Y, U is an open Baire set in Y}
>~inf (/z(V): V ~ E , V is an open Baire set in X}
~>~(E),
and
/~(E) =/~(E N Y)
= sup (ju(C):C~ E, C is a compact Baire set in Y}
sup (/~(C):Cc E, C is a compact Baire set in X}
~<#(E).
These relations show t h a t ~ is regular on X.
E x a m p l e 3.5 shows t h a t the assumption t h a t ~ be a-finite in Theorem 2.1 is needed. E x a m p l e 3.6 shows t h a t the hypothesis on X in Theorem 2.1 cannot be weakened to "normal and locally c o m p a c t " ; and E x a m p l e 3.7 shows t h a t the hy- pothesis t h a t X be locally compact cannot be dropped.
A Baire ]unction on a space X is a function J into a topological space Y such t h a t / - I ( U ) is a Baire set for every open set U in Y. The next theore,~a ~how~,~ Chat the definition of Baire set used in t;b.i~ p~per is a good one. I n particular', it general- izes Exercise 51.6 of Halmos [5].
Theorem 2.2. Let X be any topolooeal space and let 73 be on X containing all real-valued co~.'~tinuous ]unctions and poin~wis,~ convergent sequence o/]unctions in, it. Then B real-valued Baire ]unctions on X .
the s,mal~s~ cbz~.~ of/unction,~
containing th~ limit o/every consists o] exactly all o/ the
Pr~;~oL Let ~)0 consist o5 ~!..[ continuous iunctions in B. F o r every ordinal a > 0 ttm~ i~ ~ess the~:,l the first uncoc.r~,-~ble ordins,1 ~ , let B~ consist of all functions which ar~ pciu~wise ~imits of seq~ie~ces of ~unctions in U~<~ ~ . T h e n B = U~<aB~. E a c h
ARKIV FOR MATEMATIK. B d 6 n r 8 family B~ is closed under finite suprema and finite linear sums, and hence the same remarks apply to B. I t follows that B is closed under countable suprema.
Now let Z be a zero-set in X; Z =Z(]) for some ]E C+(X). Then the characteristic function ~r of Z is equal to 1 - s u p ~ n.inf(], 1/n), and hence belongs to Y. If ~ =
{E:zEEB}, then ~ contains all zero-sets and is closed under complements and countable unions. Thus if E is a Baire set, then E E E and XE belongs to B.
Finally, every Baire function belongs to B because it is a limit of linear combinations of characteristic functions of Baire sets. That every function in B = I.J~<nB~ is a Baire function follows by transfinite induction on :r 1 ~< ~ < Y~ (clearly every function in B0 is a Baire function).
Theorems 2.3 and 2.5 deal with the question: Given a continuous (respectively, Baire) function g of a topological space X into a metric space Y, do there exist a continuous open mapping ~ of X onto a metric space M and a continuous (respec- tively, Baire) function / on M such that g = / o r ? I n other words, is g essentially a function defined on a metric space?
Note that if the requirement that ~ be open is dropped, then the question becomes trivial. I n this case, M can be taken as g(X) which is a subset of Y, ~ can be taken as g, and ] can be taken as the identity map on M. Example 3.8 shows that the orig- inal question does not have an affirmative answer for all compact spaces.
Theorem 2.3, Let X be a product o/separable metric spaces
Xa,
a E A ; and let g be a continuous {respectively, Baire) ]unction o] X into a separable metric space Y. Then there is a countable subset C o ] A such that g is qssentially de]incd on the metric space M =IIaEc Xa. The mapping v is the ordinary projection o] X onto M .Proo]. For g continuous, this theorem is proved by Corson and Isbell [2], Theorem 2.1. The crucial step of the proof is to show that g is determined by the indices in a countable set C:
x, y E X and Xa=y a for a E C implies g(x)=g(y);
see, for example, Theorem 4 of [13].
The proof for a Baire function g is the same once it is established that such a g is determined by a countable set of indices. This can be proved for all Baire functions by applying transfinite induction to the sequence B~, 1 < cr < ~ , defined in the proof of 2.2.
Lemma 2.4. Let X be a ~-compact, locally compact group. 1] E is a Baire set in X , then there exists a compact normal subgroup N o] X such that E = E N and X / N is metrizable.
Proof. Halmos [5], Theorem 64.G, states that for an appropriate compact normal subgroup N, E = E N and X / N is separable. The term separable used in [5] means that there is a countable base for the topology of X / N . Thus Urysohn's metrization theorem {[9], page 125) shows that X / N is metrizable.
Theorem 2.5. Let X be a or.compact, locally compact group, and let g be a continuous {respectively , Baire) ]unction on X into a separable metric space Y. Then there is a compact normal subgroup N o] X such that X / N is metrizable and g is essentially defined on X / N . The mapping ~ is the ordinary l~rojection o] X onto X / N .
This theorem is a slight generalization of the theorem on page 60 of Montgomery and Zippin [12].
K, A. ROSS~ K. STROMBEHG~ Baire sets and Baire measures
Proo/.
Let g be a Baire function on X into Y. I t is easy to see t h a t it suffices to find N and establish t h a tx y - 1 EN implies
g(x)=g(y).
L e t B be a countable basis for Y. Each set g-l(B), BE B, is a Baire set and hence, by 2.4, there is a compact normal subgroup NB such t h a t
g-l(B)=g-I(B)N~
a n dX[Ns
is metrizable. Let N = N s=~ hrs; clearly N is a compact normal subgroup of X.Since each
X / N s
is metrizable, each N B is the countable intersection of open sets.Therefore N is also the countable intersection of open sets and it follows t h a t
X / N
is metrizable.Suppose now t h a t
xy ~1 E N
andg(x) = ~ E Y.
Then (=) = N~r for appropriat~Bj E ]g, and hence
)
g-l(o~) = iT-1(Bj) = g-i(Bj)Ns,~ [7 g-I(B~)N~ g-l(Bj) N=g-l(~)N~g-l(oO.
t~1 t=1 t=1 1
Thus
g-l(~)=g-l(:c)N,
both x and y belong to g-l(~), andg(x)=~=g(y).
This com- pletes the proof.The assumption in Theorem 2.5 t h a t X be a-compact is necessary, at least for Abelian groups, as Theorem 2.8 below will show.
We now prove Theorem 1.6 as promised in Section 1.
Proo] o/ Theorem
1.6.Step
I. Suppose t h a t E is a Baire set in a compactly gener- ated, locally compact group X. By 2.4, there is a compact normal subgroup N of X such t h a tE = E N
andX / N
is metrizable. I t follows t h a tE - ~ E - N .
If ~ denotes the projection of X ontoX[N,
then v ( E : ) !s closed inX / N
and there is an ] inC(X/N)
such t h a tZ(/)=v(E-).
T h e n / e v eC(X)
andZ(/ov)=E-
so t h a t E - is a Baire set.Step II.
Suppose t h a t E is a Baire set in an arbitrary locally compact group X.Then X contains an open and closed, compactly generated subgroup H. Clearly every coset of H is a closed Baire set. Each
E fl xH
is a Baire set inxH
b y 1.4 and so, by Step I, eachE- ~ xH
is a zero-set inxH:
Consequently, E - is the zero-set of some continuous function on X:Step
I I I . Suppose t h a t U is an open subset of a compactly generated, locally compact group X. Let 2 denote the H a a r measure on X. There exists a sequence {"~n}~=l Of compact subsets of U such t h a t 2 ( U - [ J ~ ) = 0 . Choose functions/~E C+(X)
such t h a t fn(_~.)= 1 and f~ ( X - U ) = 0 ; define F . =/;1(1). Then each F~is a Baire set and 2 ( U - ( J .%1F~)=0" Since [ J , ~ l Fn is a Baire set, its closure U - is also a Baire set by Step I.
Step
IV. Suppose t h a t U is an open subset of an arbitrary locally compact group X.An argument similar to Step I I and using Step I I I shows t h a t U - is a Baire set.
Step
V. Suppose t h a t E is a Baire set in a product X of separable metric spacesX~, aEA.
As noted in 2.3, every Baire function is determined by eountably m a n y indices. I n particular, this applies to the characteristic function of E. Hence for some countable subset C of A,E=Eo • a
where E o is a subset of I I a e c X a.Hence
E - = E o
• IIaeA-c Xa. Since E~ is a zero-set in the metric spaceIIaec Xa,
it follows that E - is a zero-set in X.Step
VI. Suppose that U is an open subset of X---IIaEA Xa, where each X a is sepa- rable metric. B y Theorem 3 of [13], U - is determined b y countably m a n y indices.Arguing as in Step V, we infer t h a t U - is a zero-set in X.
ARKIV FSR MATEMATIK. B d 6 n r 8 The next two lemmas are needed to prove Theorem 2.8. E x a m p l e 3.9 shows t h a t the natural analogue of L e m m a 2.7 for compact spaces is not true.
L e m m a 2.6. Let ~o be the set o/ordinals less than the first uncountable ordinal ~ . I / A is an uncountable Abelian group, then there is a non-decreasing transfinite sequence {H~:ccE ~0} o/proper subgroups o / A such that U ~ H~ = A.
Proo[. Let J be a subgroup of A having cardinality tr 1. F r o m Theorem 20.1 of Fuchs [3], it follows t h a t there is an isomorphism [ of J into a divisible group D such t h a t card (D) =tO 1. B y 16.1 [3], [ can be extended to a h o m o m o r p h i s m / ' of A into D.
L e t K be the kernel o f / ' . Then card ( A / K ) = tr 1. Let {x~K:~r E ~0} be a well-ordering of A / K . Define ~/~ to be the countable subgroup of A / K generated b y {x~K :fl <~ o~}.
Then {~/~:~E~0} is a non-decreasing sequence of proper subgroups of A / K and O ~ t ~ = A / K . Finally, define H ~ = { x E A : x K E ~ t ~ } ; the sequence { H ~ : a E ~ o } has the desired properties.
L e m m a 2.7. Let G be a locally compact Abelian group with identity e, and assume that G is not metrizable. Then there is a set A c G such that card (A ) =tr e (~A, and every Go set containing e intersects A in a nonvoid set.
Proo/. There is a c o m p a c t l y generated open subgroup H of G. B y virtue of P o n t r y a - gin's structure theorem for locally compact, compactly generated Abelian groups (see 9.8 [7]), H m a y be chosen to be topologically isomorphic with R m • G 0, where m is a non-negative integer, R is the real line, and G O is a compact group. E v i d e n t l y G O cannot be metrizable, a n d it suffices to find a set A 0 in G O as in the lemma. {Note t h a t G O is itself a G~ set in (7.) We therefore assume t h a t G is compact.
B y 23.17 and 24.48 of [7], the character group (~ of G is discrete and uncountable.
B y L e m m a 2.6, there is a non-decreasing transfinite sequence {~/~: a E ~o} of proper subgroups of G such t h a t O ~ ~/~ = G. F o r a E ~o, ~/~ =~ ~ a n d hence ~/~ does n o t sepa- rate points of G b y 23.20 [7]. F o r a E ~o, select x~E G, x~ =~ e, such t h a t ~(x~)= 1 for all ~ E ~/~. Let A be the set {x~: a E ~o}.
Sets of the form {x E G : ] ~ ( x ) - 11 < s for v 2 E :~} constitute a base for open sets a t e where e > 0 a n d :~ is a finite subset of ~ (see 23.15 a n d 24.3 of [7]). E v e r y Ga set containing e contains a countable intersection of sets of this form. Hence it contains a set n~EcyJ-l(1) for some countable subset C of (~, a n d it suffices to show t h a t A intersects n ~ c ~-1(1). To do this, choose aE ~o such t h a t C c ~/~. Then x~ belongs to A a n d to n ~Gc~-l(1). This p r o p e r t y of A also shows t h a t A m u s t be uncountable a n d hence t h a t card (A)=~1.
Theorem 2.8. Let G be a locally compact Abelian group, which is not metrizable or a-compact. There is a continuous /unction / o/ G into the circle group T such that / is not essentially de/ined on any metrizable G/N, where N is compact and normal.
Proo/. Let H be an open compactly generated subgroup of G. T h e n H is n o t metri- zable a n d G/H is n o t countable. L e t A be a subset of H as given in L e m m a 2.7;
card (A) = bt 1. F o r each a E A, there is a continuous character Za of H for which za(a) ~ 1.
Let {xa:aEA} be a subset of G consisting of elements from distinct cosets of H.
Define / on G as follows:
K. A. ROSS, K. S T R O M B E R G ,
Baire sets and Balre
m e a s u r e s/(x)=):~(xx~ I)
ifxEx~H,
= 1 if
xr x~H.
a e A
Clearly ] is continuous on G.
Suppose now t h a t N is a c o m p a c t n o r m a l s u b g r o u p of G a n d t h a t
G/N
is m e t r i - zable. T h e n N is a Ge set containing e; hence t h e r e exists an e l e m e n t a in A ~ N.P l a i n l y
ax~
a n d x~ belong to the same coset of N. Since/(ax~)=)~(aX~Xa 1)
4:1 a n d/(x~):Z,(x~x-~)=l, /
is not c o n s t a n t on cosets of N a n d hence is n o t e s s e n t i a l l y d e f i n e d onG/N.
3. Examples
Example
3.1. L e t N a n d R d e n o t e the set of integers a n d t h e real line, r e s p e c t i v e l y . L e t X be t h e space f i R - ( f i N - N ) discussed in Exercise 6P of G i l l m a n a n d J e r i s o n [4]. T h e n X is locally compact, b u t not normal. L e tE : N c X .
E a c h set (n),nEE,
is a c o m p a c t zero-set a n d so E is a a - c o m p a c t Baire set. H o w e v e r , E is closed a n dE is n o t a zero-set b y 6P.5 [4].
Example
3.2. L e t X d e n o t e t h e real line a n d retopologize X so t h a t t h e open sets are t h e sets of t h e form U tJ S, where U is open in t h e usual t o p o l o g y a n d S is a s u b s e t of t h e i r r a t i o n a l numbers. The space X is p a r a c o m p a c t (see Michael [10] or [11]), b u t n o t locally compact. L e t E be the set of r a t i o n a l s in X. E a c h set(r), rEE,
is a c o m p a c t zero-set a n d so E is a closed a - c o m p a c t Baire set. H o w e v e r , E is n o t a zero-set in X; otherwise E w o u l d be a Go s u b s e t of the real line with t h e usual topol- ogy, which is impossible b y t h e Baire c a t e g o r y theorem. The space X a n d t h i s p r o p e r t y of E are discussed b y K a t ~ t o v [8], page 74.Example
3.3. L e t X be a well-ordered set with the order t o p o l o g y . T h e n e v e r y closed Baire set E in X is a zero-set. N o t e t h a t X is a l w a y s normal, a n d X is p a r a - c o m p a c t if a n d o n l y if X has a c o u n t a b l e cofinal set.I f X has a c o u n t a b l e cofinal set, t h e n E is a zero-set b y 1.3. Suppose t h a t X has no c o u n t a b l e cofinal set. I f E is bounded, t h e n E is c o m p a c t a n d hence a zero-set b y 1.2. If E is u n b o u n d e d ; t h e n
X---E
is b o u n d e d and, for some fl in X, [fl, ~ [ c E.Clearly E N [1, fl] is t h e zero-set of some function g on [1, ill. Define ~ on X b y t h e rule:
g(x)=g(x)
for x<3,= 0 for x > ~ f l + l .
Since [ 1, 8] is open a n d closed, ~ is continuous; a n d it is obvious t h a t Z(~) = E.
Example
3.4. L e t X be t h e c o m p a c t space fiR, where R is still t h e real line. L e tU=
(J ~ _ l ] 4 n - 1 , 4 n + l [ c R . The set U is open in R a n d hence is open in X. W e h a v e U = LJ ~ - l F k where the Fk's are compact. E a c h Fk is a zero-set in X a n d there- fore U is a B a i r e set in X.F i n a l l y , we claim t h a t U - is n o t a Baire set. Otherwise
U- =Z(g)
for some g E C+(X).Clearly
g(x)
= 0 for x E R if a n d o n l y if x E [J ~-114n - 1,4n + 1]. F o r n = 1, 2 . . . choosexnER
such t h a t l < [ 4 n - - x , I < 2 a n d Ig(x,)[ < I / n ; letS=(xl, x 2 ...).
T h e n S is a zero-set for some b o u n d e d continuous f u n c t i o n on R a n d , b y T h e o r e m 6.4 [4], S N U- = ~ . Since S is n o t c o m p a c t , S is n o t closed in X a n d t h e r e is a n e l e m e n t y in S - S . Theng(y)=0
a n dyE Z(g),
a n d y e t y q U - . This c o n t r a d i c t i o n shows t h a tU : is n o t a Baire set.
ARKIV FSR MATEMATIK. B d 6 n r 8
Example
3.5. L e t X be a n y space t h a t is n o t a - c o m p a c t . T h e n t h e m e a s u r e /~given b y
~u(A) = 0 if A is a s u b s e t of a a, c o m p a c t set,
= ~ otherwise,
defines a n o n t r i v i a l infinite B a i r e m e a s u r e on X t h a t is n o t regular.
Example
3.6. L e t X consist of all o r d i n a l s less t h a n t h e first u n c o u n t a b l e o r d i n a l ~ . W i t h t h e o r d e r t o p o l o g y , X is n o r m a l a n d l o c a l l y c o m p a c t , b u t n o t p a r a c o m p a c t (see Ex. Y, p a g e 172, K e l l e y [9]). F o r B a i r e sets E c X , define j u ( E ) = 1 if E contains a n u n c o u n t a b l e closed set, a n d # ( E ) = 0 otherwise. T h e n / z is a finite B a i r e m e a s u r e t h a t is n o t regular; see Exercise 52.10 [5].Example
3.7. L e t X d e n o t e t h e S o r g e n f r e y line; i.e., t h e real line w i t h sets [a, b[as a n open basis. T h e n X is p a r a c o m p a c t as shown b y S o r g e n f r e y [15]. Since t h e sets [a, b[ are Lebesgue m e a s u r a b l e , i t follows t h a t t h e B a i r e sets of X are all Le- besgue m e a s u r a b l e . L e t / z be t h e Lebesgue m e a s u r e r e s t r i c t e d to [0, 1] a n d d e f i n e d for all Baire sets of X, Then/~ is n o t regular, since e v e r y c o m p a c t set, being countable, has m e a s u r e zero.
Example
3.8. L e t X be t h e c o m p a c t space f i n +, where N § = {1, 2, 3 . . . . ). L e t g be t h e continuous f u n c t i o n on X d e f i n e d b yg(n)=l/n
forn E N +, g(x)=O
forx E X - N +.
Then t h e r e does n o t exist a m e t r i c space M , a c o n t i n u o u s open m a p p i n g v of X o n t o M , a n d a f u n c t i o n / on M such t h a t / o r = g .
A s s u m e t h a t such M , 3, a n d / exist. T h e n ~ m u s t be one-to-one on N+; e v i d e n t l y 9 (N +) ~=M. Select a n y x in M - v ( N + ) . T h e n x = l i m k
v(n~)
for some sequence (nk}k~=l of d i s t i n c t integers. L e t U = (n k E N + : k e v e n } - . Since X - U = (n E N + : n =~ nk for a n y e v e n k ) - , g is open a n d closed in X. P l a i n l yv(U)=(T(nk)
: k even} U (x}. Since X = l i m k T(n~+l), T(U) is n o t open. This c o n t r a d i c t s o u r a s s u m p t i o n .Example
3.9. L e t X be t h e w e l l - o r d e r e d set of o r d i n a l s less t h a n or e q u a l to t h e s m a l l e s t o r d i n a l 0 whose c o r r e s p o n d i n g c a r d i n a l is ~2" T h e n X is c o m p a c t a n d n o t first countable. I fA c X
has t h e p r o p e r t y t h a tO~A
a n d c a r d (A)=~r t h e n A is b o u n d e d a n d 0 has a n e i g h b o r h o o d missing A altogether.A C K N O W L E D G E M E N T
T h i s w o r k w a s s u p p o r t e d in p a r t b y t h e N a t i o n a l Science F o u n d a t i o n on g r a n t s N S F - G 2 3 7 9 9 a n d N S F C5-1501, respectively.
R E F E R E N C E S
1. BOURBAKI, 1~.,
Topologie gdndrale,
Actualit~s Sci. Ind. 1142. Paris: I-Iermann & Cie. 1951.2. CORSO~, I-I. H. & ISB~.LL, J. R., Some properties of strong uniformities, Quart. J. Math.
Oxford (2)
11,
17-33 (1960).3. FUCHS, L.,
Abelian Groups,
London: Pergamon Press 1960.4. GILL~AN, L. & JERXSO~, M.,
Rings of Continuous Functions,
Princeton, N.J.: D. Van No- strand Co. 1960.5. HALLOS, P. R.,
Measure Theory,
New York, N.Y.: D. Van Nostrand Co. 1950.K. A. ROSS, K. STROMBERG, B a i r e sets a n d B a i r e m e a s u r e s
6. HEWITT, E., L i n e a r f u n c t i o n a l s on s p a c e s of c o n t i n u o u s f u n c t i o n s , F u n d . M a t h . 37, 161-189 (1950).
7. I-IEWITT, n . & R o s s , K . A., Abstract Harmonic Analysis I, Berlin: S p r i n g e r - V e r l a g 1963.
8. KAT~TOV, M., M e a s u r e s in fully n o r m a l spaces, F u n d . M a t h . 38, 73-84 (1951).
9. KELLEY, J. L., General Topology, N e w Y o r k , N.Y.: D. V a n N o s t r a n d Co. 1955.
10. MICHAEL, E. A., T h e p r o d u c t of a n o r m a l s p a c e a n d m e t r i c s p a c e n e e d n o t be n o r m a l , Bull.
A m e r . M a t h . See. 69, 375-376 (1963).
11. - - T h e p r o d u c t of a n o r m a l s p a c e a n d m e t r i c s p a c e n e e d n o t be n o r m a l , N o t i c e s A m e r . M a t h . See. 10, 258-259 (1963).
12. MONTGOMERY, D. & ZIPPI•, L., Topological Transformation Groups, N e w Y o r k , N . Y . : I n t e r s c i e n c e P u b l i s h e r s , I n c . 1955.
13. R o s s , K . A. & STO~E, A. H . , P r o d u c t s of s e p a r a b l e spaces, A m e r . M a t h . M o n t h l y , 71, 398-403 (1964).
14. RUBIN, H . , A l m o s t Lindel6f m e a s u r e s o n p a r a c o m p a e t spaces, Notices A m e r . M a t h . See. 10, 472 (1963).
15. SORGENFREY, R . H . , O n t h e topological p r o d u c t of p a r a e o m p a c t spaces, Bull. A m e r . M a t h . Soc. 53, 631-632 (1947).
16. ULAM, S., Z u r M a s s t h e o r i e in d e r a l l g e m e i n e n M e n g e u l e h r e , F u n d . M a t h . 16, 140-150 (1930).
T r y c k t d e n 21 j u n i 1965
Uppssla 1965. Almqvist & WikseUs Boktryckeri AB
