The purpose of this paper is to give a necessary and sufñcient condition to define a category measure on a Baire topological space. In the last section we give some examples of spaacs in these conditions

Ayerbe Toledano, J. M. (José María)

Revistes Catalanes amb Accés Obert

Publicacions Matemátiques, Vol 34 (1990), 299-305 .
Abstract
CATEGORY MEASURES ON BAIRE SPACES
J .M . AYERBE TOLEDANO
The purpose of this paper is to give a necessary and sufñcient condition
to define a category measure on a Baire topological space . In the last
section we give some examples of spaacs in these conditions .
1. Introduction
A category measure in a topological space is a finite, countably additive
measure m defined on the class of the sets having the property of Baire and
such that m(E) = 0 if and only if E is of first category.
The sets having the property of Baire in a space X are those of the form
G + P, where G is open, P is of first category, and " + " denotes symmetric
difference . These sets constitute a u-field of subsets of X .
In any topological space the open sets and the nowhere dense sets generate a
field of subsets of X called the complete basic ring of X . It consists of all sets
of the form G -f- N, where G is open and N is nowhere dense . If G is required
to be regular open, that is, such that G = G -'-', then this representation of
any element of the ring is unique . The regular open subsets of X constitute
a Boolean algebra R(X) in which the Boolean sum, product and complement
are defined respectively by (Gl U G2)-'-', Gl fl G2 and G-', and the Boolean
order relation by set inclusion . We shall call R(X) the regular algebra of X.
R(X) is always complete . Moreover, any set having the property of Baire can
be represented in the form A = G -I- P, where G is a regular open set and P
is of first category. This representation is unique in any Baire space . In any
topological space a set is called clopen if it is open and closed . The clopen
subsets of X constitute a Boolean algebra B(X) in which the Boolean sum,
product and complement are defined respectively by Gr U G2, G1 f1 G2 and G',
and the Boolean order relation by set inclusion .
It is known that given a finite, complete measure space (X, S, m) we can
define a topology on X with respect to which m will be a category measure
(see [7, chapter 22], [3], [8]) .
In this paper we study the converse problem, that is, given a topological
space, when can we define a finite and countably additive measure m on the
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J.M . AYERBE
class of sets having the property of Baire with respect to which m will be a
category measure?
In this direction Oxtoby has shown [8, th . 1] that if S is the union of all
open sets of first category in a topological space X, then X admits a category
measure if and only if X - S- is non-empty and R(X - S-) admits a finite,
strictly positive and countably additive measure . In particular, a Baire space
admits a category measure if and only if its regular algebra admits a finite,
strictly positive and countably additive xneasure . Moreover, because X - S- is
a Baire space, it follows that in the study of this subject, we may confine our
attention to Baire space without essential loss of generality.
This theorem shows that the problem of defining a category measure on a
topological Baire space can be reduced to the problem of defining a finite,
strictly positive and countably additive measure on a Boolean algebra. The
general problem of the existence of measures on Boolean algebras has been
abundantly studied (see [4], [5] and [6]) . In [5, Addendum] Ryll-Nardzewski
gives a necessary and sufficient condition . The concept of intersection number
has been defined by Kelley for Boolean algebras, and has been translated to
topological spaces in [1, Def. 0 .3] . We shall use the terminology of this paper .
2 . A necessary and sufflcient condition
Definition 2.1 . A topological space (X, T) has the property (***) if and only
if there exists a decomposition of T* = T - {O} into a sequence {Tn : n < w}
such that:
1) k(Tn ) > 0 for each n < w .
2) If {Am : m < w} is a increasing sequence of open sets such that
U'-i Am E Tn , there is a natural number m with Am E Tn .
3) If V, W are open sets, V - W (%'.e ., V + W is a first category set) and
V E Tn , then W E Tn .
Theorem 2.2 . Let X be a topological Baire space. Then X has ¡he property
(***) if and only if X admits a category measure.
Proof. Suppose that X has the property (***) . Then there exists a decom-
position of T* in the conditions of Definition 2.1 .
Let Tn = {Vi(n) : i E I(n)} for each n < w.
For each i E J(n) let H; nl = Vi(n) ' . Then H;nl is a regular open set such
that Vün) C HIn) and are equivalents .
Let T _ {H(n) : ti E Ion>}. Then:
a) R(X)* = U0 1 Tn
b) K(Tn) > K(Tn ) for each n < w.
It follows immediately from Vi (n) C HI n) for each n < w, i E I(n) .
CATEGORY MEASURES
	
301
c) Let {An, : m < w} be an increasing sequence in R(X) such that
U'=1 Am E T,, . We shall show that there is a natural number m such that
A ,ET .
Indeed : Since Um=1 A, E T,,, it follows that (Um=1 Am ) - '- ' E T., and
therefore there is V E T such that V- ' - ' = (U°°-1 A,) - '- ' . Thus V E Tn
and V - U°°=1 A,, and hence U°°=1 Am E In.
It follows that there is a natural number m such that A m E Tn . Thus
A- ' - ' = Am E T,, .
Therefore we obtain that the Boolean algebra R(X) satisfies the condition
of Ryll-Nardzewski to admit a finite, strictly positive and countably additive
measure . Thus (X, T) admits a category measure by [8, Th . 11 .
Conversely, suppose that X admits a category measure . Then R(X) admits a
finite, strictly positive and countably additive measure . Therefore there exists
a decomposition of R(X)* into a sequence {T.* : n < w} in the conditions of
Ryll-Nardzewski's theorem .
We consider now the following decomposition of T* :
Let H E T* and G E R(X) such that G - H . Suppose that G E Tá . Then
H E Tn . It is easy to see that every open set belongs to a unique In .
Thus the obtained sequence {Tn : n < w} satisfies the following properties :
a) T* = Un=1 Tn
b) By construction, if U, V E T*, U - V and U E In, then V E In .
c) K(In) > 0 for each n < w.
Indeed : We write for n fixed T,*, = {Gl : l E L} and let In = {[Gil : l E L},
where for each l E L[Gi] = {Vil : i E P}, Vi l E T*, Vi ' - Gl for each i E I i .
We consider the decomposition of R(X)* by the sequence {T.** : n < w},
where Tñ* = {[Gil* : i E L}, [Gil* = {Gi : i E I`} .
Obviously k(Tñ*) > 0 and ITni = ¡Tn*j . Moreover, given a finite set of
indices J = {j1,.-,in} corresponding in Tn to the open sets Vi, , . . . 1 Vi and
in In* to the regular open sets Gj . . . G ; and such that Vj; G;; for each
i = 1, . . . , n, it is easy to prove that n iEy Vj ; 7É 0 if and only if (~ =E J' Gji :~ 0,
J'CJ J'CJ
because if there sets are non empty, then they are equivalent second category
sets .
It follows that calT (J) = calT.-.(J) for each J finite, and hence k(Tn) _
k(Tn*) > 0 .
d) Let {Am : m < w} be an increasing sequence of open sets such that
U'=1 Am E Tn . For each natural number m we consider the regular open
00sets A '-' (equivalent to Am ) . Thus U°°=1 Am -' - Urn=1 Am, and since
U °° A-,-, - V °° A -1- it follows that U°° A- ~ - ~ V°°_ A-1-1m=1 m m=1 m m=1 m m=1 m
U m=1 Am E Tn . Hence VM°=1Am -' E Tn .
Since Vm=1A--' E In, by construction there exists a regular open set V such
that V - V,°,°=1AM-' and V E In . It is easy to check that V = V,°,°=1A--',
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J.M . AYERBE
and therefore V,°,°-1A--'m- E Tn*.
Thus V°,°- 1 Am-' E T,*, and {Am-' : m < w} is an increasing sequence of
regular open sets . It follows that there exists a natural number m such that
AM
-~ E T,*,*, and since Ann - Am' -', we obtain that A  , E Tn. Thus the proof
is finished .
Remark 2.3 . Property (***) implies property (**) (see [1, Def . 0 .4]) . The
converse is not true . Indeed : Let R be the set of real numbers with the Eu-
clidean topology T. This topology is second countable, and therefore there is
a countable base 8 = {Bn : n < w} of T . For each n < w let Tn = {V E T
Bn C V} . Then for each n < w, k(T) = 1 . Thus R has property (**) .
If 18 had property (***), then the topological space (R,T) would admit a
category measure by theorem 2.2 . This is not true because in R there are sets
of first category that are not nowhere dense (for example Q), contradicting [8,
Th . 3], Thus (R,T) has not property (***) .
Remark 2 .4 . Since property (**) implies property (*) (see[l, Def. 0 .2]),
and property (*) implies CCC (countable chain condition), it follows that the
following diagram holds
The converse of there implications is not true in general . Nevertheless, we
have proved (see [2, note 7]) that if every open set contains a minimal open set
then CCC implies that X admits a category measure, and therefore
(*) ==> CCC
(**) (*) -4~ CCC
Remark 2.5 . A concrete example of a topological space with the property
(***) is the following :
Let R be the set of real numbers, .M the o-algebra of Lebesgue measurable
sets, N the class of Lebesgue nullsets, and
Td={0(A)-N :AEM,NEN},
where 0 is the Lebesgue lower density (see [7, page 16]) .
It is proved in [7, page 88-90] that Td is a topology of Baire that admits
a category measure . Thus, it follows from theorem 2.2 that (R, Td) has the
property (***) . This topology has been abundantly studied in [9] .
3. Examples
Example 3.1 . Le¡ (X, T) be a Baire topological space ¡ha¡ satisfaes:
a) X is second countable.
b) X is countably compaci.
CATEGORY MEASURES
	
303
c) Each open set of ¡he topology is a closed set.
Then (X, T) has property (***) and, therefore ii admits a category measure .
Proof. Let B = {Gn : n < w} a countable base of T, and let Tn = {V E T
Gn C_ V} for each n < w. Then the sequence {Tn : n < w} satisfies property
(***) .
A less trivial example is the following :
Example 3.2 . Let (X, T) be a topological space that satisfies:
a) X is second countable.
b) X is Hausdorff compact.
c) X is 0-dimensional.
d) The closure of every open set in X is open .
e) If {A,n : m < w} is an increasing sequence of open sets, then there is a
natural number m such that A,, - U,n=1 A,,, .
Then (X, T) satisfies property (***) and, therefore it admits a category mea-
sure .
Proof: Let B(X) and R(X) the clopen algebra and the regular algebra of X,
respectively .
By c) and d) R(X) = B(X) (see [8, Th . 10 and Corollary1) .
Moreover, from a), b) and c) it follows that B(X) is countable .
LetB(X)={Gn :n<w}andT,,={VET :V-Gn}foreachn<w .
Since X is a Baire topological space, for each open set V there is a unique
clopen set Gn such that V - Gn . Thus Tn fl T,,, = 0 for each pair of natural
numbers n, m, n qÉ m. Moreover the sequence {Tn : n < w} satisfies property
(***) .
We shall show some examples of topological spaces in this conditions, utilizing
terminology and properties of [3] .
Proposition 3.1 . Let (X,T) be a Boolean a-space second countable . Let
B(X) be ¡he family of clopen sets of X. Suppose that B(X) satisfies ¡he finite
chain condition. Then (X, T) satisfies property (***) and, therefore it admits a
category measure.
Proof. We shall show that (X, T) is in the conditions of example 3 .2 . Since
X is a Boolean a-space second countable, X satisfies a), b) and c) . By a) and
c) it follows that any open set is a Baire open set . Thus we obtain d) . To check
e), let {A,, : m < w} be an increasing sequence of open sets . We can suppose
that the sequence is strictly increasing without loss of generality.
Then {Am : m < w} is an increasing sequence of clopen sets (again we can
suppose that the sequence is strictly increasing), and therefore finite (following
304
	
J.M . AYERBE
an argument as in [3, §14, lemma 2]) . Thus there is a natural number mo such
that A-, = Um=i Am'
00 00But A,,, ^' Am o and AM. = Un=, Amo UM=1 Am'
Um=1 Am and we obtain e) .
Hence Amo
Thus (X, T) is in the conditions of example 3.2 and therefore satisfies (***) .
Some examples of topological spaces in the conditions of proposition 3.1
follow immediately from the theory of Boolean algebras . Thus we have
Proposition 3.2 . Le¡ A be a countable Boolean a-algebra with ¡he finite
chain condition. Then ¡he Stone's space X of A satisfies property (***) and,
therefore it admits a category measure.
Proof. The Stone's space X of A is a Hausdorff compact and 0-dimensional
space, and its clopen algebra B(X) is isomorphic to A by the classical Stone's
representation theorem . It follows that B(X) is a u-algebra, and therefore X
is a v-space (see [3, §22, Th. 12]) . Since A is countable we obtain that X is
second countable and, therefore, B(X) is countable . Thus B(X) is complete
and hence the closure of every open set in X is open (see [8, Th . 10 and
Corollary]) . Finally, since B(X) satisfies the finite chain condition, we obtain
the condition e) following an argument as in the proof of the proposition 3 .1 .
In this paper we have always used the Ryll-Nardzewski's condition in order
to a Boolean algebra admits a finite, strictly positive and countably additive
measure . An analogue condition has been shown by Kelley in [5, Th . 4 and 9] .
We shall use Chis condition in the following
Example 3.3 . Le¡ (X, T) a topological space that satisfies:
a) X is second countable.
b) X is a Hausdorff compact.
c) X is 0-dimensional.
d) The closure of every open set in X is open.
e) Every set of first category in X is nowhere dense .
Then (X,T) admits a category measure and, therefore it satisfies property
Proof. An argument as in the example 3.2 shows that R(X) = B(X) and
that B(X) is countable .
From [5, Th . 8] it follows that B(X) is weakly countably distributive and,
therefore B(X) admits a finite, strictly positive and countably additive mea-
sure . Thus (X, T) admits á category measure .
Some examples of topological spaces in the conditions of example 3.3 are
given in the following
Proposition 3.3 . Let A be a weakly countably distributive and countable
Boolean Q-algebra. Then ¡he Stone's space X of A is in the conditions of
example 3.3 and therefore it admts a category measure and ii satisfies property
Proof. We argue as in proposition 3 .2 .
CATEGORY MEASURES
	
305
References
1 . S . ARGYROS, On compact spaces without strictly positive measure, Pacific
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2 . J .M . AYERBE, Algunos espacios topologicos que admiten una medida cat-
egoria, Collectanea Math . XXXV, Fasc. 3 (1984), 221-231 .
3 . P .R . HALMOS, "Lectures on Boolean algebras," Springer-Verlag, 1974 .
4 . A . HORN, A . TARSIQ, Measures in Boolean algebras, Trans . Amer. Math .
Soc . 64 (1948), 467-497 .
5 . J .L . KELLEY, Measures on Boolean algebras, Pacific J. Math 9 (1959),
1165-1177 .
6 . D . MAHARAM, An algebraic characterization of measure algebras, Annals
of Math 48 (1947), 154-167 .
7 . J.C . OXTOBY, "Measure and Category," Springer-Verlag, 1971 .
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Math 205 (1961), 156-170 .
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Departamento de Análisis Matemático
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Rebut el 25 d'Octubre de 1989


Category measures on Baire spaces

Ayerbe Toledano, J. M.

Diposit Digital de Documents de la UAB

Publicacions Matemátiques, Vol 34 (1990), 299-305 .AbstractCATEGORY MEASURES ON BAIRE SPACESJ .M . AYERBE TOLEDANOThe purpose of this paper is to give a necessary and sufñcient conditionto define a category measure on a Baire topological space . In the lastsection we give some examples of spaacs in these conditions .1. IntroductionA category measure in a topological space is a finite, countably additivemeasure m defined on the class of the sets having the property of Baire andsuch that m(E) = 0 if and only if E is of first category.The sets having the property of Baire in a space X are those of the formG + P, where G is open, P is of first category, and " + " denotes symmetricdifference . These sets constitute a u-field of subsets of X .In any topological space the open sets and the nowhere dense sets generate afield of subsets of X called the complete basic ring of X . It consists of all setsof the form G -f- N, where G is open and N is nowhere dense . If G is requiredto be regular open, that is, such that G = G -'-', then this representation ofany element of the ring is unique . The regular open subsets of X constitutea Boolean algebra R(X) in which the Boolean sum, product and complementare defined respectively by (Gl U G2)-'-', Gl fl G2 and G-', and the Booleanorder relation by set inclusion . We shall call R(X) the regular algebra of X.R(X) is always complete . Moreover, any set having the property of Baire canbe represented in the form A = G -I- P, where G is a regular open set and Pis of first category. This representation is unique in any Baire space . In anytopological space a set is called clopen if it is open and closed . The clopensubsets of X constitute a Boolean algebra B(X) in which the Boolean sum,product and complement are defined respectively by Gr U G2, G1 f1 G2 and G',and the Boolean order relation by set inclusion .It is known that given a finite, complete measure space (X, S, m) we candefine a topology on X with respect to which m will be a category measure(see [7, chapter 22], [3], [8]) .In this paper we study the converse problem, that is, given a topologicalspace, when can we define a finite and countably additive measure m on the300	J.M . AYERBEclass of sets having the property of Baire with respect to which m will be acategory measure?In this direction Oxtoby has shown [8, th . 1] that if S is the union of allopen sets of first category in a topological space X, then X admits a categorymeasure if and only if X - S- is non-empty and R(X - S-) admits a finite,strictly positive and countably additive measure . In particular, a Baire spaceadmits a category measure if and only if its regular algebra admits a finite,strictly positive and countably additive xneasure . Moreover, because X - S- isa Baire space, it follows that in the study of this subject, we may confine ourattention to Baire space without essential loss of generality.This theorem shows that the problem of defining a category measure on atopological Baire space can be reduced to the problem of defining a finite,strictly positive and countably additive measure on a Boolean algebra. Thegeneral problem of the existence of measures on Boolean algebras has beenabundantly studied (see [4], [5] and [6]) . In [5, Addendum] Ryll-Nardzewskigives a necessary and sufficient condition . The concept of intersection numberhas been defined by Kelley for Boolean algebras, and has been translated totopological spaces in [1, Def. 0 .3] . We shall use the terminology of this paper .2 . A necessary and sufflcient conditionDefinition 2.1 . A topological space (X, T) has the property (***) if and onlyif there exists a decomposition of T* = T - {O} into a sequence {Tn : n < w}such that:1) k(Tn ) > 0 for each n < w .2) If {Am : m < w} is a increasing sequence of open sets such thatU'-i Am E Tn , there is a natural number m with Am E Tn .3) If V, W are open sets, V - W (%'.e ., V + W is a first category set) andV E Tn , then W E Tn .Theorem 2.2 . Let X be a topological Baire space. Then X has ¡he property(***) if and only if X admits a category measure.Proof. Suppose that X has the property (***) . Then there exists a decom-position of T* in the conditions of Definition 2.1 .Let Tn = {Vi(n) : i E I(n)} for each n < w.For each i E J(n) let H; nl = Vi(n) ' . Then H;nl is a regular open set suchthat Vün) C HIn) and are equivalents .Let T _ {H(n) : ti E Ion>}. Then:a) R(X)* = U0 1 Tnb) K(Tn) > K(Tn ) for each n < w.It follows immediately from Vi (n) C HI n) for each n < w, i E I(n) .CATEGORY MEASURES	301c) Let {An, : m < w} be an increasing sequence in R(X) such thatU'=1 Am E T,, . We shall show that there is a natural number m such thatA ,ET .Indeed : Since Um=1 A, E T,,, it follows that (Um=1 Am ) - '- ' E T., andtherefore there is V E T such that V- ' - ' = (U°°-1 A,) - '- ' . Thus V E Tnand V - U°°=1 A,, and hence U°°=1 Am E In.It follows that there is a natural number m such that A m E Tn . ThusA- ' - ' = Am E T,, .Therefore we obtain that the Boolean algebra R(X) satisfies the conditionof Ryll-Nardzewski to admit a finite, strictly positive and countably additivemeasure . Thus (X, T) admits a category measure by [8, Th . 11 .Conversely, suppose that X admits a category measure . Then R(X) admits afinite, strictly positive and countably additive measure . Therefore there existsa decomposition of R(X)* into a sequence {T.* : n < w} in the conditions ofRyll-Nardzewski's theorem .We consider now the following decomposition of T* :Let H E T* and G E R(X) such that G - H . Suppose that G E Tá . ThenH E Tn . It is easy to see that every open set belongs to a unique In .Thus the obtained sequence {Tn : n < w} satisfies the following properties :a) T* = Un=1 Tnb) By construction, if U, V E T*, U - V and U E In, then V E In .c) K(In) > 0 for each n < w.Indeed : We write for n fixed T,*, = {Gl : l E L} and let In = {[Gil : l E L},where for each l E L[Gi] = {Vil : i E P}, Vi l E T*, Vi ' - Gl for each i E I i .We consider the decomposition of R(X)* by the sequence {T.** : n < w},where Tñ* = {[Gil* : i E L}, [Gil* = {Gi : i E I`} .Obviously k(Tñ*) > 0 and ITni = ¡Tn*j . Moreover, given a finite set ofindices J = {j1,.-,in} corresponding in Tn to the open sets Vi, , . . . 1 Vi andin In* to the regular open sets Gj . . . G ; and such that Vj; G;; for eachi = 1, . . . , n, it is easy to prove that n iEy Vj ; 7É 0 if and only if (~ =E J' Gji :~ 0,J'CJ J'CJbecause if there sets are non empty, then they are equivalent second categorysets .It follows that calT (J) = calT.-.(J) for each J finite, and hence k(Tn) _k(Tn*) > 0 .d) Let {Am : m < w} be an increasing sequence of open sets such thatU'=1 Am E Tn . For each natural number m we consider the regular open00sets A '-' (equivalent to Am ) . Thus U°°=1 Am -' - Urn=1 Am, and sinceU °° A-,-, - V °° A -1- it follows that U°° A- ~ - ~ V°°_ A-1-1m=1 m m=1 m m=1 m m=1 mU m=1 Am E Tn . Hence VM°=1Am -' E Tn .Since Vm=1A--' E In, by construction there exists a regular open set V suchthat V - V,°,°=1AM-' and V E In . It is easy to check that V = V,°,°=1A--',302	J.M . AYERBEand therefore V,°,°-1A--'m- E Tn*.Thus V°,°- 1 Am-' E T,*, and {Am-' : m < w} is an increasing sequence ofregular open sets . It follows that there exists a natural number m such thatAM-~ E T,*,*, and since Ann - Am' -', we obtain that A  , E Tn. Thus the proofis finished .Remark 2.3 . Property (***) implies property (**) (see [1, Def . 0 .4]) . Theconverse is not true . Indeed : Let R be the set of real numbers with the Eu-clidean topology T. This topology is second countable, and therefore there isa countable base 8 = {Bn : n < w} of T . For each n < w let Tn = {V E TBn C V} . Then for each n < w, k(T) = 1 . Thus R has property (**) .If 18 had property (***), then the topological space (R,T) would admit acategory measure by theorem 2.2 . This is not true because in R there are setsof first category that are not nowhere dense (for example Q), contradicting [8,Th . 3], Thus (R,T) has not property (***) .Remark 2 .4 . Since property (**) implies property (*) (see[l, Def. 0 .2]),and property (*) implies CCC (countable chain condition), it follows that thefollowing diagram holdsThe converse of there implications is not true in general . Nevertheless, wehave proved (see [2, note 7]) that if every open set contains a minimal open setthen CCC implies that X admits a category measure, and therefore(*) ==> CCC(**) (*) -4~ CCCRemark 2.5 . A concrete example of a topological space with the property(***) is the following :Let R be the set of real numbers, .M the o-algebra of Lebesgue measurablesets, N the class of Lebesgue nullsets, andTd={0(A)-N :AEM,NEN},where 0 is the Lebesgue lower density (see [7, page 16]) .It is proved in [7, page 88-90] that Td is a topology of Baire that admitsa category measure . Thus, it follows from theorem 2.2 that (R, Td) has theproperty (***) . This topology has been abundantly studied in [9] .3. ExamplesExample 3.1 . Le¡ (X, T) be a Baire topological space ¡ha¡ satisfaes:a) X is second countable.b) X is countably compaci.CATEGORY MEASURES	303c) Each open set of ¡he topology is a closed set.Then (X, T) has property (***) and, therefore ii admits a category measure .Proof. Let B = {Gn : n < w} a countable base of T, and let Tn = {V E TGn C_ V} for each n < w. Then the sequence {Tn : n < w} satisfies property(***) .A less trivial example is the following :Example 3.2 . Let (X, T) be a topological space that satisfies:a) X is second countable.b) X is Hausdorff compact.c) X is 0-dimensional.d) The closure of every open set in X is open .e) If {A,n : m < w} is an increasing sequence of open sets, then there is anatural number m such that A,, - U,n=1 A,,, .Then (X, T) satisfies property (***) and, therefore it admits a category mea-sure .Proof: Let B(X) and R(X) the clopen algebra and the regular algebra of X,respectively .By c) and d) R(X) = B(X) (see [8, Th . 10 and Corollary1) .Moreover, from a), b) and c) it follows that B(X) is countable .LetB(X)={Gn :n<w}andT,,={VET :V-Gn}foreachn<w .Since X is a Baire topological space, for each open set V there is a uniqueclopen set Gn such that V - Gn . Thus Tn fl T,,, = 0 for each pair of naturalnumbers n, m, n qÉ m. Moreover the sequence {Tn : n < w} satisfies property(***) .We shall show some examples of topological spaces in this conditions, utilizingterminology and properties of [3] .Proposition 3.1 . Let (X,T) be a Boolean a-space second countable . LetB(X) be ¡he family of clopen sets of X. Suppose that B(X) satisfies ¡he finitechain condition. Then (X, T) satisfies property (***) and, therefore it admits acategory measure.Proof. We shall show that (X, T) is in the conditions of example 3 .2 . SinceX is a Boolean a-space second countable, X satisfies a), b) and c) . By a) andc) it follows that any open set is a Baire open set . Thus we obtain d) . To checke), let {A,, : m < w} be an increasing sequence of open sets . We can supposethat the sequence is strictly increasing without loss of generality.Then {Am : m < w} is an increasing sequence of clopen sets (again we cansuppose that the sequence is strictly increasing), and therefore finite (following304	J.M . AYERBEan argument as in [3, §14, lemma 2]) . Thus there is a natural number mo suchthat A-, = Um=i Am'00 00But A,,, ^' Am o and AM. = Un=, Amo UM=1 Am'Um=1 Am and we obtain e) .Hence AmoThus (X, T) is in the conditions of example 3.2 and therefore satisfies (***) .Some examples of topological spaces in the conditions of proposition 3.1follow immediately from the theory of Boolean algebras . Thus we haveProposition 3.2 . Le¡ A be a countable Boolean a-algebra with ¡he finitechain condition. Then ¡he Stone's space X of A satisfies property (***) and,therefore it admits a category measure.Proof. The Stone's space X of A is a Hausdorff compact and 0-dimensionalspace, and its clopen algebra B(X) is isomorphic to A by the classical Stone'srepresentation theorem . It follows that B(X) is a u-algebra, and therefore Xis a v-space (see [3, §22, Th. 12]) . Since A is countable we obtain that X issecond countable and, therefore, B(X) is countable . Thus B(X) is completeand hence the closure of every open set in X is open (see [8, Th . 10 andCorollary]) . Finally, since B(X) satisfies the finite chain condition, we obtainthe condition e) following an argument as in the proof of the proposition 3 .1 .In this paper we have always used the Ryll-Nardzewski's condition in orderto a Boolean algebra admits a finite, strictly positive and countably additivemeasure . An analogue condition has been shown by Kelley in [5, Th . 4 and 9] .We shall use Chis condition in the followingExample 3.3 . Le¡ (X, T) a topological space that satisfies:a) X is second countable.b) X is a Hausdorff compact.c) X is 0-dimensional.d) The closure of every open set in X is open.e) Every set of first category in X is nowhere dense .Then (X,T) admits a category measure and, therefore it satisfies propertyProof. An argument as in the example 3.2 shows that R(X) = B(X) andthat B(X) is countable .From [5, Th . 8] it follows that B(X) is weakly countably distributive and,therefore B(X) admits a finite, strictly positive and countably additive mea-sure . Thus (X, T) admits á category measure .Some examples of topological spaces in the conditions of example 3.3 aregiven in the followingProposition 3.3 . Let A be a weakly countably distributive and countableBoolean Q-algebra. Then ¡he Stone's space X of A is in the conditions ofexample 3.3 and therefore it admts a category measure and ii satisfies propertyProof. We argue as in proposition 3 .2 .CATEGORY MEASURES	305References1 . S . ARGYROS, On compact spaces without strictly positive measure, PacificJ. Math . 105 (1983), 257-272 .2 . J .M . AYERBE, Algunos espacios topologicos que admiten una medida cat-egoria, Collectanea Math . XXXV, Fasc. 3 (1984), 221-231 .3 . P .R . HALMOS, "Lectures on Boolean algebras," Springer-Verlag, 1974 .4 . A . HORN, A . TARSIQ, Measures in Boolean algebras, Trans . Amer. Math .Soc . 64 (1948), 467-497 .5 . J .L . KELLEY, Measures on Boolean algebras, Pacific J. Math 9 (1959),1165-1177 .6 . D . MAHARAM, An algebraic characterization of measure algebras, Annalsof Math 48 (1947), 154-167 .7 . J.C . OXTOBY, "Measure and Category," Springer-Verlag, 1971 .8 . J .C . OXTOBY, Spaces that admit a category measure, J. Reine Angew.Math 205 (1961), 156-170 .9 . F .D . TALL, The density topology, Pacific J. Math . 62 (1976), 275-284 .Departamento de Análisis MatemáticoFacultad de MatemáticasApartado de Correos 116041080-SevillaSPAINRebut el 25 d'Octubre de 1989

J. M. Ayerbe Toledano

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Category measures on Baire spaces

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