Category measures on Baire spaces

Publicacions Ma emá iques, Vol 34 (1990), 299-305 . A bs ac CATEGORY MEASURES ON BAIRE SPACES J .M . AYERBE TOLEDANO The pu pose o his pape is o gi e a necessa y and su ñcien condi ion o de ine a ca ego y measu e on a Bai e opological space . In he las sec ion we gi e some examples o spaacs in hese condi ions . 1 . In oduc ion A ca ego y measu e in a opological space is a ini e, coun ably addi i e measu e m de ined on he class o he se s ha ing he p ope y o Bai e and such ha m(E) = 0 i and only i E is o i s ca ego y . The se s ha ing he p ope y o Bai e in a space X a e hose o he o m G+ P, whe e G is open, P is o i s ca ego y, and " + " deno es symme ic di e ence . These se s cons i u e a u- ield o subse s o X . In any opological space he open se s and he nowhe e dense se s gene a e a ield o subse s o X called he comple e basic ing o X . I consis s o all se s o he o m G - - N, whe e G is open and N is nowhe e dense . I G is equi ed o be egula open, ha is, such ha G=G - ' - ', hen his ep esen a ion o any elemen o he ing is unique . The egula open subse s o X cons i u e a Boolean algeb a R(X) in which he Boolean sum, p oduc and complemen a e de ined espec i ely by (G l U G2)-'-', Gl l G2 and G - ', and he Boolean o de ela ion by se inclusion . We shall call R(X) he egula algeb a o X . R(X) is always comple e . Mo eo e , any se ha ing he p ope y o Bai e can be ep esen ed in he o m A = G -I- P, whe e G is a egula open se and P is o i s ca ego y . This ep esen a ion is unique in any Bai e space . In any opological space a se is called clopen i i is open and closed . The clopen subse s o X cons i u e a Boolean algeb a B(X) in which he Boolean sum, p oduc and complemen a e de ined espec i ely by G U G2, G 1 1 G2 and G', and he Boolean o de ela ion by se inclusion . I is known ha gi en a ini e, comple e measu e space (X, S, m) we can de ine a opology on X wi h espec o which m will be a ca ego y measu e (see [7, chap e 22], [3], [8]) . In his pape we s udy he con e se p oblem, ha is, gi en a opological space, when can we de ine a ini e and coun ably addi i e measu e m on he 30 0  J .M . AYERBE class o se s ha ing he p ope y o Bai e wi h espec o which m will be a ca ego y measu e? In his di ec ion Ox oby has shown [8, h . 1] ha i S is he union o all open se s o i s ca ego y in a opological space X, hen X admi s a ca ego y measu e i and only i X -S - is non-emp y and R(X -S - ) admi s a ini e, s ic ly posi i e and coun ably addi i e measu e . In pa icula , a Bai e space admi s a ca ego y measu e i and only i i s egula algeb a admi s a ini e, s ic ly posi i e and coun ably addi i e xneasu e . Mo eo e , because X -S - is a Bai e space, i ollows ha in he s udy o his subjec , we may con ine ou a en ion o Bai e space wi hou essen ial loss o gene ali y . This heo em shows ha he p oblem o de ining a ca ego y measu e on a opological Bai e space can be educed o he p oblem o de ining a ini e, s ic ly posi i e and coun ably addi i e measu e on a Boolean algeb a . The gene al p oblem o he exis ence o measu es on Boolean algeb as has been abundan ly s udied (see [4], [5] and [6]) . In [5, Addendum] Ryll-Na dzewski gi es a necessa y and su icien condi ion . The concep o in e sec ion numbe has been de ined by Kelley o Boolean algeb as, and has been ansla ed o opological spaces in [1, De . 0 .3] . We shall use he e minology o his pape . 2 . A necessa y and su lcien condi ion De ini ion 2 .1 . A opological space (X, T) has he p ope y (***) i and only i he e exis s a decomposi ion o T* = T- {O} in o a sequence {T n : n < w} such ha : 1) k(T n ) > 0 o each n < w . 2) I {A m : m < w} is a inc easing sequence o open se s such ha U' -i A m E T n , he e is a na u al numbe m wi h A m E T n . 3) I V, W a e open se s, V - W (%' .e ., V+ W is a i s ca ego y se ) and V E T n , hen W E T n . Theo em2 .2 . Le X be a opological Bai e space . Then X has ¡he p ope y (***) i and only i X admi s a ca ego y measu e . P oo . Suppose ha X has he p ope y (***) . Then he e exis s a decom- posi ion o T* in he condi ions o De ini ion 2 .1 . Le T n = {Vi (n) : i E I(n)} o each n< w . Fo each i E J(n) le H ; nl = Vi (n)  ' . Then H ; nl is a egula open se such ha Vün) C H In) and a e equi alen s . Le T  _ {H(n) : i E Ion>} . Then : a) R(X )* = U0 1 Tn b) K(Tn) > K(T n ) o each n <w . I ollows immedia ely om V i (n) C H I n) o each n < w, i E I(n) . CATEGORY MEASURES  30 1 c) Le {An, : m < w} be an inc easing sequence in R(X) such ha U'=1 Am E T,, . We shall show ha he e is a na u al numbe m such ha A  ,ET  . Indeed : Since Um=1 A  , E T,,, i ollows ha (Um=1 A m ) - ' - ' E T ., and he e o e he e is V E T  such ha V - ' - ' = (U°° -1 A  ,) - ' - ' . Thus V E T n and V - U°° =1 A  ,, and hence U°° =1 Am EIn . I ollows ha he e is a na u al numbe m such ha A mE T n . Thus A - ' - ' =Am E T,, . The e o e we ob ain ha he Boolean algeb a R(X) sa is ies he condi ion o Ryll-Na dzewski o admi a ini e, s ic ly posi i e and coun ably addi i e measu e . Thus (X, T) admi s a ca ego y measu e by [8, Th . 11 . Con e sely, suppose ha X admi s a ca ego y measu e . Then R(X) admi s a ini e, s ic ly posi i e and coun ably addi i e measu e . The e o e he e exis s a decomposi ion o R(X)* in o a sequence {T .* : n < w} in he condi ions o Ryll-Na dzewski's heo em . We conside now he ollowing decomposi ion o T* : Le H E T* and G E R(X) such ha G - H . Suppose ha G E Tá . Then H E T n . I is easy o see ha e e y open se belongs o a unique In . Thus he ob ained sequence {T n : n < w} sa is ies he ollowing p ope ies : a) T* = Un=1 Tn b) By cons uc ion, i U, V E T*, U- V and U E In, hen V E In . c) K(In) > 0 o each n < w . Indeed : We w i e o n ixed T,*, = {Gl : l E L} and le In = {[Gil : l E L}, whe e o each l E L[Gi] = {V i l : i E P}, V i l E T*, V i ' - Gl o each i EI i . We conside he decomposi ion o R(X)* by he sequence {T .** : n < w}, whe e Tñ* = {[Gil* : i E L}, [Gil* = {Gi : i E I`} . Ob iously k(Tñ*) > 0 and ITni = ¡Tn*j . Mo eo e , gi en a ini e se o indices J = {j1, . .-,in} co esponding in T n o he open se s Vi, , . . . 1 Vi  and in In* o he egula open se s Gj  ... G ; and such ha Vj ;  G ; ; o each i = 1, . . . , n, i is easy o p o e ha n iE y Vj ; 7É 0 i and only i (~ =E J' Gji :~ 0, J'CJ  J'CJ because i he e se s a e non emp y, hen hey a e equi alen second ca ego y se s . I ollows ha calT  (J) = calT . - .(J) o each J ini e, and hence k(T n ) _ k(Tn*) > 0 . d) Le {Am : m < w} be an inc easing sequence o open se s such ha U'=1 Am E T n . Fo each na u al numbe m we conside he egula open 00 se s A  '-' (equi alen o Am ) .  Thus U°°=1 Am -' - U n=1 Am, and since U °° A -,-, - V °° A -1- i ollows ha 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 cons uc ion he e exis s a egula open se V such ha V- V,°,°=1AM-' and V E In . I is easy o check ha V = V,°,°=1A--', 302  J .M . AYERBE and he e o e V,°,°-1A--'m - E Tn* . Thus V°,°- 1 Am -' E T,*, and {Am -' : m < w} is an inc easing sequence o egula open se s . I ollows ha he e exis s a na u al numbe m such ha AM -~ E T,*,*, and since Ann -A m ' - ', we ob ain ha A  , E T n . Thus he p oo is inished . Rema k 2 .3 . P ope y (***) implies p ope y (**) (see [1, De . 0 .4]) . The con e se is no ue . Indeed : Le R be he se o eal numbe s wi h he Eu- clidean opology T . This opology is second coun able, and he e o e he e is a coun able base 8 = {B n : n < w} o T . Fo each n < w le T n = {V E T B n C V} . Then o each n < w, k(T) = 1 . Thus R has p ope y (**) . I 18 had p ope y (***), hen he opological space (R,T) would admi a ca ego y measu e by heo em 2 .2 . This is no ue because in R he e a e se s o i s ca ego y ha a e no nowhe e dense ( o example Q), con adic ing [8, Th . 3], Thus (R,T) has no p ope y (***) . Rema k 2 .4 . Since p ope y (**) implies p ope y (*) (see[l, De . 0 .2]), and p ope y (*) implies CCC (coun able chain condi ion), i ollows ha he ollowing diag am holds The con e se o he e implica ions is no ue in gene al . Ne e heless, we ha e p o ed (see [2, no e 7]) ha i e e y open se con ains a minimal open se hen CCC implies ha X admi s a ca ego y measu e, and he e o e (*) ==> CCC (**)  (*) -4~ CCC Rema k 2 .5 . A conc e e example o a opological space wi h he p ope y (***) is he ollowing : Le R be he se o eal numbe s, .M he o-algeb a o Lebesgue measu able se s, N he class o Lebesgue nullse s, and Td={0(A)-N :AEM,NEN}, whe e 0 is he Lebesgue lowe densi y (see [7, page 16]) . I is p o ed in [7, page 88-90] ha Td is a opology o Bai e ha admi s a ca ego y measu e . Thus, i ollows om heo em 2 .2 ha (R, Td) has he p ope y (***) . This opology has been abundan ly s udied in [9] . 3 . Examples Example 3 .1 . Le¡ (X, T) be a Bai e opological space ¡ha¡ sa is aes : a) X is second coun able . b) X is coun ably compaci . CATEGORYMEASURES  30 3 c) Each open se o ¡he opology is a closed se . Then (X, T) has p ope y (***) and, he e o e ii admi s a ca ego y measu e . P oo . Le B= {Gn : n < w} a coun able base o T, and le T n = {V E T G n C_ V} o each n < w . Then he sequence {T n : n < w} sa is ies p ope y (***) . A less i ial example is he ollowing : Example 3 .2 . Le (X, T) be a opological space ha sa is ies : a) X is second coun able . b) X is Hausdo compac . c) X is 0-dimensional . d) The closu e o e e y open se in X is open . e) I {A,n : m < w} is an inc easing sequence o open se s, hen he e is a na u al numbe m such ha A,, - U,n=1 A,,, . Then (X, T) sa is ies p ope y (***) and, he e o e i admi s a ca ego y mea- su e . P oo : Le B(X) and R(X) he clopen algeb a and he egula algeb a o X, espec i ely . By c) and d) R(X) = B(X) (see [8, Th . 10 and Co olla y1) . Mo eo e , om a), b) and c) i ollows ha B(X) is coun able . Le B(X)={G n :n<w}andT,,={VET :V-Gn} o eachn<w . Since X is a Bai e opological space, o each open se V he e is a unique clopen se G n such ha V-G n . Thus T n l T,,, = 0 o each pai o na u al numbe s n, m, n qÉ m . Mo eo e he sequence {Tn : n < w} sa is ies p ope y (***) . We shall show some examples o opological spaces in his condi ions, u ilizing e minology and p ope ies o [3] . P oposi ion 3 .1 . Le (X,T) be a Boolean a-space second coun able . Le B(X) be ¡he amily o clopen se s o X . Suppose ha B(X) sa is ies ¡he ini e chain condi ion . Then (X, T) sa is ies p ope y (***) and, he e o e i admi s a ca ego y measu e . P oo . We shall show ha (X, T) is in he condi ions o example 3 .2 . Since X is a Boolean a-space second coun able, X sa is ies a), b) and c) . By a) and c) i ollows ha any open se is a Bai e open se . Thus we ob ain d) . To check e), le {A,, : m < w} be an inc easing sequence o open se s . We can suppose ha he sequence is s ic ly inc easing wi hou loss o gene ali y . Then {A m : m < w} is an inc easing sequence o clopen se s (again we can suppose ha he sequence is s ic ly inc easing), and he e o e ini e ( ollowing 304  J .M . AYERBE an a gumen as in [3, §14, lemma 2]) . Thus he e is a na u al numbe m o such ha A-, = Um=i Am' 00  00 Bu A  ,,, ^' Am o and AM . = Un=, Amo  UM=1 Am' Um = 1 Am and we ob ain e) . Hence A mo Thus (X, T) is in he condi ions o example 3 .2 and he e o e sa is ies (***) . Some examples o opological spaces in he condi ions o p oposi ion 3 .1 ollow immedia ely om he heo y o Boolean algeb as . Thus we ha e P oposi ion 3 .2 . Le¡ A be a coun able Boolean a-algeb a wi h ¡he ini e chain condi ion . Then ¡he S one's space X o A sa is ies p ope y (***) and, he e o e i admi s a ca ego y measu e . P oo . The S one's space X o A is a Hausdo compac and 0-dimensional space, and i s clopen algeb a B(X) is isomo phic o A by he classical S one's ep esen a ion heo em . I ollows ha B(X) is a u-algeb a, and he e o e X is a -space (see [3, §22, Th . 12]) . Since A is coun able we ob ain ha X is second coun able and, he e o e, B(X) is coun able . Thus B(X) is comple e and hence he closu e o e e y open se in X is open (see [8, Th . 10 and Co olla y]) . Finally, since B(X) sa is ies he ini e chain condi ion, we ob ain he condi ion e) ollowing an a gumen as in he p oo o he p oposi ion 3 .1 . In his pape we ha e always used he Ryll-Na dzewski's condi ion in o de o a Boolean algeb a admi s a ini e, s ic ly posi i e and coun ably addi i e measu e . An analogue condi ion has been shown by Kelley in [5, Th . 4 and 9] . We shall use Chis condi ion in he ollowing Example 3 .3 . Le¡ (X, T) a opological space ha sa is ies : a) X is second coun able . b) X is a Hausdo compac . c) X is 0-dimensional . d) The closu e o e e y open se in X is open . e) E e y se o i s ca ego y in X is nowhe e dense . Then (X,T) admi s a ca ego y measu e and, he e o e i sa is ies p ope y P oo . An a gumen as in he example 3 .2 shows ha R(X) = B(X) and ha B(X) is coun able . F om [5, Th . 8] i ollows ha B(X) is weakly coun ably dis ibu i e and, he e o e B(X) admi s a ini e, s ic ly posi i e and coun ably addi i e mea- su e . Thus (X, T) admi s á ca ego y measu e . Some examples o opological spaces in he condi ions o example 3 .3 a e gi en in he ollowing P oposi ion 3 .3 . Le A be a weakly coun ably dis ibu i e and coun able Boolean Q-algeb a . Then ¡he S one's space X o A is in he condi ions o example 3 .3 and he e o e i adm s a ca ego y measu e and ii sa is ies p ope y P oo . We a gue as in p oposi ion 3 .2 . CATEGORYMEASURES  30 5 Re e ences 1 .  S . ARGYROS, On compac spaces wi hou s ic ly posi i e measu e, Paci ic J . Ma h . 105 (1983), 257-272 . 2 .  J .M . AYERBE, Algunos espacios opologicos que admi en una medida ca - ego ia, Collec anea Ma h . XXXV, Fasc . 3 (1984), 221-231 . 3 .  P .R . HALMOS, "Lec u es on Boolean algeb as," Sp inge -Ve lag, 1974 . 4 .  A . HORN,A . TARSIQ, Measu es in Boolean algeb as, T ans . Ame . Ma h . Soc . 6 4 (1948), 467-497 . 5 .  J .L . KELLEY, Measu es on Boolean algeb as, Paci ic J . Ma h 9 (1959), 1165-1177 . 6 .  D . MAHARAM, An algeb aic cha ac e iza ion o measu e algeb as, Annals o Ma h 48 (1947), 154-167 . 7 .  J.C . OXTOBY, "Measu e and Ca ego y," Sp inge -Ve lag, 1971 . 8 .  J .C . OXTOBY, Spaces ha admi a ca ego y measu e, J . Reine Angew . Ma h 205 (1961), 156-170 . 9 .  F .D . TALL, The densi y opology, Paci ic J . Ma h . 62 (1976), 275-284 . Depa amen o de Análisis Ma emá ico Facul ad de Ma emá icas Apa ado de Co eos 1160 41080-Se illa SPAIN Rebu el 25 d'Oc ub e de 1989