Category measures on Baire spaces

Publicacions Matemátiques, Vol 34 (1990), 299-305 . A bstract 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 (G l 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, G 1 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 30 0  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 {T n : n < w} such that : 1) k(T n ) > 0 for each n < w . 2) If {A m : m < w} is a increasing sequence of open sets such that U' -i A m E T n , there is a natural number m with A m E T n . 3) If V, W are open sets, V - W (%' .e ., V+ W is a first category set) and V E T n , then W E T n . Theorem2 .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 decomposition of T* in the conditions of Definition 2 .1 . Let T n = {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 H In) and are equivalents . Let T  _ {H(n) : ti E Ion>} . Then : a) R(X )* = U0 1 Tn b) K(Tn) > K(T n ) for each n <w . It follows immediately from V i (n) C H I n) for each n < w, i E I(n) . CATEGORY MEASURES  30 1 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 A m ) - ' - ' E T ., and therefore there is V E T  such that V - ' - ' = (U°° -1 A  ,) - ' - ' . Thus V E T n and V - U°° =1 A  ,, and hence U°° =1 Am EIn . It follows that there is a natural number m such that A mE T n . 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 T n . It is easy to see that every open set belongs to a unique In . Thus the obtained sequence {T n : 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] = {V i l : i E P}, V i l E T*, V i ' - Gl for each i EI 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 T n 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 iE y 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(T n ) _ k(Tn*) > 0 . d) Let {Am : m < w} be an increasing sequence of open sets such that U'=1 Am E T n . For each natural number m we consider the regular open 00 sets 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 -1-1 m=1 m  m=1 m  m=1 m  m=1 m U m=1 A m E T n . Hence VM°=1Am -' E T n . Since Vm=1 A- -' 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--', 302  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 -A m ' - ', we obtain that A  , E T n . 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 Euclidean topology T . This topology is second countable, and therefore there is a countable base 8 = {B n : n < w} of T . For each n < w let T n = {V E T B n 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 . CATEGORYMEASURES  30 3 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 T n = {V E T G n C_ V} for each n < w . Then the sequence {T n : 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 measure . 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)={G n :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 G n such that V-G n . Thus T n fl T,,, = 0for 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 {A m : 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 m o such that A-, = Um=i Am' 00  00 But A  ,,, ^' Am o and AM . = Un=, Amo  UM=1 Am' Um = 1 Am and we obtain e) . Hence A mo 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 measure . 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 . 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Departamento de Análisis Matemático Facultad de Matemáticas Apartado de Correos 1160 41080-Sevilla SPAIN Rebut el 25 d'Octubre de 1989