GEORGIAN MATHEMATICAL JOURNAL : Vol period 6 comma No period 3 comma 1 9 9 9 comma\noindent 2 0 1GEORGIAN hyphen 2 1 2 MATHEMATICAL JOURNAL : Vol . 6 , No . 3 , 1 9 9 9 , 2 0 1 − 2 1 2 ON UNCOUNTABLE UNIONS .. AND .. INTERSECTIONS .. OF \ centerlineMEASURABLE{ON UNCOUNTABLE .. SETS UNIONS \quad AND \quad INTERSECTIONS \quad OF } M period BALCERZAK AND A period KHARAZISHVILI \ centerlineAbstract period{MEASURABLE We consider\ severalquad SETS natural} situations where the union or intersection of an uncountable family of measurable open parenthesis in various senses closing paren- \ centerline {M . BALCERZAKGEORGIAN MATHEMATICAL AND A . KHARAZISHVILI JOURNAL : Vol . 6 ,} No . 3 , 1 9 9 9 , 2 0 1 - 2 1 2 thesis ON UNCOUNTABLE UNIONS AND INTERSECTIONS OF sets with a quotedblleft good quotedblrightMEASURABLE additional structure is SETS again measurable or may fail \ centerlineto be measurable{ Abstract period We . We primarily consider deal with several Lebesgue natural measurable situations sets where the union or } M . BALCERZAK AND A . KHARAZISHVILI and sets with the Baire property period In particular comma uncountable unions Abstract . We consider several natural situations where the union or \ centerlineof sets homeomorphic{ intersection to a closed of Euclidean an uncountable simplex are family considered of measurable ( in various senses ) } intersection of an uncountable family of measurable ( in various senses ) in detail comma and it is shown that the Lebesgue measure and the Baire sets with a “ good ” additional structure is again measurable or may fail \ centerlineproperty differ{ sets essentially with in a this ‘‘ aspect good period ’’ additional Another difference structure between is again measurable or may fail } to be measurable . We primarily deal with Lebesgue measurable sets measure and category is illustrated in the case of some uncountable and sets with the Baire property . In particular , uncountable unions \ centerlineintersections{ to of sets be of measurable full measure . open We primarilyparenthesis comeager deal with sets Lebesgue comma respectively measurable closing sets } of sets homeomorphic to a closed Euclidean simplex are considered parenthesis period We in detail , and it is shown that the Lebesgue measure and the Baire \ centerlinealso discuss{ aand topological sets withform of the the Vitali Baire covering property theorem . In comma particular in con hyphen , uncountable unions } property differ essentially in this aspect . Another difference between nection with the Baire property of uncountable unions of certain sets period measure and category is illustrated in the case of some uncountable \ centerline0 period .. Introduction{ of sets homeomorphic to a closed Euclidean simplex are considered } intersections of sets of full measure ( comeager sets , respectively ) . We A mathematician working in probability theory comma the theory of random also discuss a topological form of the Vitali covering theorem , in con - \ centerlineprocesses or{ inin various detail fields , of and modern it is analysis shown is quite that frequently the Lebesgue obliged measure and the Baire } nection with the Baire property of uncountable unions of certain sets . to show that the union union of open brace X sub a : a in A closing brace or the intersection \ centerline { property differ essentially0 . inIntroduction this aspect . Another difference between } intersection of sub open braceA mathematician X sub a : a in working A closing in probability brace of theory , the theory of random pro- a given uncountable family open brace X sub a : a in A closing brace of measurable sets is measurable \ centerline {measurecesses and or in category various fields is of illustrated modern analysis in is quite the frequently case of obliged some touncountable show } provided that the setsthat X the sub union a haveS{ someX : additionala ∈ A} or the.. quotedblleft intersection goodT X quotedblright: a ∈ A} of .. structure period .. Here a { a a given uncountable family {X : a ∈ A} of measurable sets is measurable \ centerlinethe measurability{ intersections is meant in the of sense sets of the of respective fulla measure sigma hyphen ( comeager algebra S sets of , respectively ) . We } provided that the sets X have some additional “ good ” structure . Here subsets of a given nonempty basic set E containinga each set X sub a open parenthesis a in A closing the measurability is meant in the sense of the respective σ− algebra S of subsets parenthesis\ centerline period{ also discuss a topological form of the Vitali covering theorem , in con − } of a given nonempty basic set E containing each set X (a ∈ A). For instance , For instance comma let E coincide with the real line R and let S be the sigmaa hyphen algebra let E coincide with the real line R and let S be the σ− algebra of all Lebesgue \ centerlineof all Lebesgue{ nection measurable with subsets the of Baire R period property .. Let us considerof uncountable the family T unions of of certain sets . } measurable subsets of R. Let us consider the family T of all S− measurable all S hyphen measurable sets X subset equal R such that each point x from X is a density sets X ⊆ R such that each point x from X is a density point of X. It is well \ centerlinepoint of X{ period0 . \ ..quad It isIntroduction well known open} parenthesis see comma e period g period comma .. known ( see , e . g . , [ 1 ] , Chapter 22 ) that T forms a topology which is brackleft 1 brackright comma Chapter 22 closing parenthesis that T forms a usually called the density topo logy on the real line . Thus the union of each A mathematiciantopology which is usually working called in the probability density topo logy theory on the , real the line theory period Thus of random subfamily of T belongs to T , and hence is S− measurable . In particular , we processesthe union of or each in subfamily various of T fields belongs of to Tmodern comma analysis and hence is is S hyphen quite measurable frequently period obliged see that uncountable unions of S− measurable sets belonging toIn show particular that comma the we union see that $ uncountable\bigcup unions\{ ofX S hyphen{ a } measurable: a sets\ in belongingA \} $ 1 99 1 Mathematics Subject Classification . 28 A 5 , 28 A 1 2 , 54 E 52 . or1 the 99 1 intersection Mathematics Subject $ \ Classificationbigcap {\{} period 28X A 5{ commaa } 28: A a1 2 comma\ in 54A E 52\} period$ o f Key words and phrases . Lebesgue measure , Jordan measure , Baire property , Vitali Key words and phrases period Lebesgue measure comma Jordan measure comma Baire property covering , Jordan curve , Suslin condition , simplex , meager set . comma\noindent Vitali a given uncountable family $ \{ X { a } : a \ in A \} $ 20 1 ofcovering measurable comma sets Jordan is curve measurable comma Suslin condition comma simplex comma meager set period 1 72 - 947 X /99/0500 − 0201Dollar12.50/0circlecopyrt − c1997 P l enum P ubl provided20 1 that the sets $ X { a }$ have some additional \quad ‘ ‘ good ’ ’ \quad s t r u c t u r e . \quad Here ish ing Corporation the1 72 measurability hyphen 947 X slash is 9 9meant slash 0 in 5 0 the 0 hyphen sense 0 2 0of 1 Dollar the respective 1 2 period 5 0 slash $ \sigma 0 circlecopyrt-c− $ 1algebra 9 9 7 P l enum $ S P $ ubl of ish ing Corporation subsets of a given nonempty basic set $ E $ containing each set $ X { a } ( a \ in A ) . $ For instance , let $E$ coincide with the real line $R$ and let $ S $ be the $ \sigma − $ algebra of all Lebesgue measurable subsets of $ R . $ \quad Let us consider the family $ T $ o f a l l $ S − $ measurable sets $ X \subseteq R $ such that each point $x$ from $X$ is a density point of $X .$ \quad It is well known ( see , e . g . , \quad [ 1 ] , Chapter 22 ) that $ T $ forms a topology which is usually called the density topo logy on the real line . Thus the union of each subfamily of $ T $ belongs to $ T , $ and hence is $ S − $ measurable . In particular , we see that uncountable unions of $ S − $ measurable sets belonging

\ centerline {1 99 1 Mathematics Subject Classification . 28 A 5 , 28 A 1 2 , 54 E 52 . }

Key words and phrases . Lebesgue measure , Jordan measure , Baire property , Vitali covering , Jordan curve , Suslin condition , simplex , meager set .

\ centerline {20 1 }

\ hspace ∗{\ f i l l }1 72 − 947X$/ 9 9 / 0 5 0 0 − 0 2 0 1 Dollar 1 2 . 5 0 / 0 circlecopyrt −c 1 9 9 7 $ P l enum P ubl ish ing Corporation 202 .. M period BALCERZAK AND A period KHARAZISHVILI \noindentto T are S202 hyphen\quad measurableM . BALCERZAK comma too AND period A The . KHARAZISHVILI same situation holds for the von Neumann hyphen \noindentMaharam topologyto $ T which $ are is a generalization $ S − $ of measurablethe density topology , too and . The same situation holds for the von Neumann − Maharamcan be introduced topology for which an arbitrary is a measure generalization space open parenthesis of the density E comma topology S comma mu and closing parenthesiscan be introduced where E is a for an arbitrary measure space $ ( E , S , \mu )$nonempty where basic $E$ set comma isa S is a sigma hyphen algebra of subsets of E and mu is a nonzero nonempty basic set202 $M . , BALCERZAK S$ is AND a A $ . KHARAZISHVILI\sigma − $ algebra of subsets of sigma hyphen finiteto completeT are S− measuremeasurable defined , too on . The S open same parenthesis situation holds see forbrackleft the von 1 Neumann brackright comma$ E $ Chapter and $22\ closingmu $ parenthesis is a nonzero period .. Note that a $ \sigma − $- Maharam finite topologycomplete which measure is a generalization defined on of the $ density S ( topology $ see and [1 can ] , Chapter 22 ) . \quad Note that a number of interestingbe introducedexamples and for questions an arbitrary concerning measure the space measurability (E, S, µ) where E is a nonempty numberof uncountable of interesting unions of measurable examples sets and are discussed questions in brackleft concerning 2 brackright the measurabilityperiod of uncountable unionsbasic set of, S measurableis a σ− algebra sets of subsets are discussed of E and µ inis a [ nonzero 2 ] . σ− finite The main aim of thiscomplete paper is measure to illustrate defined that on theS( phenomenon see [ 1 ] ,produc Chapter hyphen 22 ) . Note that a ing measurable unionsnumber open of parenthesis interesting or examples intersections and closingquestions parenthesis concerning for the uncountable measurability families of ofThe quotedblleft main aim good of quotedblright this paper is to illustrate that the phenomenon produc − ing measurable unionsuncountable ( or unions intersections of measurable sets ) for are discussed uncountable in [ 2 ] families. of ‘‘ good ’’ measurable sets dependsThe on main very aim delicate of this conditions paper is period to illustrate .. Another that goal the phenomenon is to produc - measurableexpress some sets similarities depends and differences on very between delicate measure conditions and category . \ inquad Another goal is to express some similaritiesing measurable and unions differences ( or intersections between ) for uncountable measure and families category of “ good in that aspect period” .. measurable In Section 1 sets comma depends we briefly on very consider delicate the conditions most classical . Another situation goal is to thatdealing aspect with an . arbitrary\quad In family Section open brace 1 , we S sub briefly t : t in consider T closing brace the most of closed classical n hyphen situation dealing with anexpress arbitrary some similarities family and $ \{ differencesS { betweent } : measure t and\ in categoryT in\} that$ simplexes in the aspect . In Section 1 , we briefly consider the most classical situation dealing o fn c hyphenl o s e d dimensional $ n − $ Euclidean simplexes space R in to the the power of n period .. Applying some well hyphen with an arbitrary family {St : t ∈ T } of closed n− simplexes in the known properties n \noindent $ n n−− $dimensional dimensional Euclidean Euclidean space R . spaceApplying $ some R ˆ{ welln -} known. $ properties\quad Applying some well − known properties of the standard Lebesgue and Jordan measures in R to the power of n openn parenthesis see comma of the standardof Lebesgue the standard and Lebesgue Jordan and measures Jordan measures in $ in RR ˆ{ (n see} , e($ . g . , see,e.g., [ 3 ] ) \quad [ 3 ] ) , we e period g period comma, we .. brackleft 3 brackright closing parenthesis comma we show that the set unionshow of that open the brace set S sub{S t: :t t∈ inT T} closingis Lebesgue brace measurable is Lebesgue andmeasurable has the and Baire has the\noindent Baire show that the set $ \bigcupt \{ S { t } : t \ in T \} $ is Lebesgue measurableproperty and . In Section has the 2 , weBaire are concerned with arbitrary families of sets that property period Inare Section homeomorphic 2 comma we to are a closed concernedn− simplex with arbitrary in Rn( suchfamilies sets of are sets called that topo log propertyare homeomorphic . In Section to a closed 2 n , hyphen we are simplex concerned in R to thewith power arbitrary of n open familiesparenthesis suchof sets sets that are homeomorphic- to a closed $ n − $ simplex in $Rˆ{ n } ( $ such sets are called topo log − are called topo log hypheni cal n− s implexes ) . It turns out that , in this situation , the measure and i cal n hyphen s implexesthe category closing cases parenthesis differ essentially period ... Namely It turns , an out arbitrary that comma family in of this topological situation comma\noindent the measurei c a l and $ the n − $ s implexes ) . \quad It turns out that , in this situation , the measure and the category cases differn− simplexes essentially has the union . with Namely the Baire , an property arbitrary . On the family other hand of , topological there category cases differexists essentially a family period of topological Namely comman− simplexes an arbitrary in Rn family(n ≥ 2) of with topological the union non - $n hyphenn − simplexes$ simplexes has the unionhas the with union the Baire with property the period Baire On property the other hand . On comma the other there hand , there exists a familymeasurable of topological in the Lebesgue $ n sense− $ . simplexes Therefore , from in this $Rˆ point{ ofn view} ( , the n exists a family of topologicalBaire property n hyphen behaves simplexes better than in R Lebesgue to the power measure of n open. Section parenthesis 3 witnesses n greater that equal\geq 2 closing2 )parenthesis $ with with the the union union non hyphen− measurable in thein some Lebesgue situations sense it can . happen\quad converselyTherefore , that , is from , measure this behaves point better of view , the measurable in the Lebesguethan the Baire sense property period .. . Therefore Namely comma , in connection from this with point a theorem of view ofcomma Goldstern the BaireBaire property property behaves behaves better better than Lebesgue than measureLebesgue period measure Section . 3 witnesses Section 3 witnesses that in some situations[ 4 ] concerning it certain can happen uncountable conversely intersections , ofthat coanalytic is , sets measure with the behaves full that in some situationsprobability it can happen measure conversely , we show comma that an that analogous is comma result measure fails to behaves be true in the betterbetter than than the the Baire Baire property property period .. .Namely\quad commaNamely in connection , in connection with a theorem with of a theorem of Goldstern [ 4 ]category concerning case . certain In Section uncountable 4 , we introduce intersections the notion of a topological of coanalytic Vitali sets Goldstern brackleftspace 4 brackright . It turns concerning out that certain these uncountablespaces characterize intersections the situation of coanalytic where sets, for a withwith the full full probability probability measure commameasure we show, we that show an thatanalogous an resultanalogous fails to result fails to be true in the categorywide class of case sets ( . called\quad admissibleIn Section ) , the union 4 , we of an introduce arbitrary subfamily the notion is of a be true in the categorycountably case period approximable .. In Section and consequently 4 comma we hasintroduce the Baire the property notion of . a topologicaltopological Vitali Vitali space periodspace .. . It\ turnsquad outIt that turns these out spaces that characterize these spacesthe characterize the situation where ,Our for notation a wide and class terminology of sets concerning ( called set admissible theory , general )topology , the union , of situation where commameasure for a and wide category class of setsare fairly open parenthesisstandard . In called particular admissible , we closing denote parenthesis : commaan arbitrary the union of subfamily is countably approximable and consequently has the Baire property . ω – the set of all natural numbers , i . e ., ω = {0, 1, 2, ..., n, ...}; an arbitrary subfamily is countablyµ – a measure approximable on a given and basic consequently space ( as has a rule the , we assume that µ is Baire property periodnonzero , σ− finite and complete ) ; OurOur notation notation and and terminology terminology concerning concerning set theory set comma theory general , topology general comma topology , measure and category are fairlydom(µ) standard – the family . of In all µ particular− measurable , sets we ; denote : measure and category are fairly standardν – period the classical In particular Jordan comma measure we in denoteRn; : omega endash the set of all natural numbersn comma i period e period comma omega = open brace \ centerline { $ \omega $ −− the set of all natural numbers ,n i . e $ . , 0 comma 1 comma 2 comma period periodλn – period the classical comma Lebesgue n comma measure period period in R period closing brace semicolon\omega = \{ 0,1,2,...,n,.... \} mu; endash $ } a measure onLet aE givenbe an basic arbitrary space opentopological parenthesis space as . a If ruleX comma⊆ E, then we classume (X), int that (X mu) is and bd (X) stand for the closure , the interior and the boundary of X, re - \ hspacenonzero∗{\ commaf i l l } sigma$ \mu hyphen$ −− finitea and measure complete on closing a given parenthesis basic semicolon space ( as a rule , we assume that $ \dommu $ open i s parenthesis mu closing parenthesis endash the family of all mu hyphen measurable sets semicolon \noindentnu sub n endashnonzero the classical $ , Jordan\sigma measure− $ in R finite to the power and of complete n semicolon ) ; lambda sub n endash the classical Lebesgue measure in Case 1 n Case 2 period \ centerlineLet E be an{ arbitrary$ dom topological ( \mu space period) $ −− .. If Xthe subset family equal E of comma all then $ \mu cl open− parenthesis$ measurable sets ; } X closing parenthesis comma int open parenthesis X closing parenthesis \ centerlineand bd open{ parenthesis$ \nu { Xn closing}$ −− parenthesisthe classical stand for Jordanthe closure measure comma the in interior $ R ˆ and{ n the} boundary; $ } of X comma re hyphen \ centerline { $ \lambda { n }$ −− the classical Lebesgue measure in $\ l e f t .R\ begin { a l i g n e d } & n \\ &. \end{ a l i g n e d }\ right . $ }

\ hspace ∗{\ f i l l } Let $ E $ be an arbitrary topological space . \quad I f $ X \subseteq E ,$ thencl $( X ) ,$ int $( X )$

\noindent and bd $ ( X ) $ stand for the closure , the interior and the boundary of $ X , $ re − ON UNIONS AND INTERSECTIONS OF MEASURABLE SETS 203 spectively . We denote : NWD(E) – the ideal of all nowhere dense subsets of E; K(E) – the σ− ideal of all meager ( i . e . , first category ) subsets of E; Br(E) – the σ− algebra of all subsets of E with the Baire property . Recall that a set X ⊆ E has the Baire property iff X can be represented as the symmetric difference X = U M P where U is open and P is meager in E. Moreover , if X ⊆ E can be expressed in the form X = U M P where U is open and P is nowhere dense in E, then X is called an open s e t modulo a nowhere dense s e t ( cf . [ 5 ] , §8 , V ) or , briefly , an open s e t modulo NWD(E). We shall say that a subset X of a topological space E is admissible if

X ⊆ cl(int(X)). Clearly , every open set is admissible . If X is a regular closed set ( i . e ., X = cl ( int (X))), then X is admissible , as well . In addition , if X is admissible , then from the relations int (X) ⊆ X ⊆ cl ( int (X)), cl ( int (X))\ int (X) ∈ NWD(E) it follows that X is open modulo a nowhere dense set ( in general , the con - verse is false ) . In our further considerations , the notion of a Vitali covering plays an essen- tial role . So , we introduce this notion for a general topological space and for an arbitrary subset of that space . We say that a family V of subsets of a topo- logical space E forms a Vitali covering of a given set X ⊆ E if , for each point x ∈ X and for each neighbourhood U of x, there exists a set V ∈ V such that x ∈ V ⊆ U. Consequently , if V is a Vitali covering of X, then the family of sets notdef {V ∩ arrowdblright − colon notdef − elementnotdef − V universal − V braceright − F f − existential − notdefnotdef − rm − existentialT − notdef − negationslash−sa − braceleftn − universal − notdefnotdef−tf − universaluniversal− o r notdef − notdefnotdef − tnotdef − enotdefinfinity − s b T − notdef − sp − notdefunion−ace−notdef notdef − X o f . A sa r le , t h e V talico verings c nsidered b lo e e a su m e − dt wa cnss − it a dmssible s ts . W ea so r callth at a t pologicalsp ace E s t i − s o fies t e a s l − i n c ndi i − toen( he c untable c ain c ndiio n ) i e ch d s − joint f mly ho nem p − t y o en s tsin Es − i c untable . U n − ions o f n Si m l − p ex e − s ya n n s i m p − l e x w em a − e n a c osed n ndegenerate s m p − l ex i t e E hyphen − u ideB an s ace Rn − period a r al α > 0 i fi ed , w es y t at a b unded s t ⊆ Rn i αendash − r egular if λn(X) ≥ αλ(V (Xparenright−parenright w h − e re V (X) d notes a c osed ll w th t e m nim a − l di am t − e er , f o r w h − i ch w eh ve t ein clusion X ⊆ V (X) i l ear l − y, in ht isd fini i − t o n a b llca n b e r − e placed b ya c be ) . e h N t − o t − h atev ery sim p − l ex egular s tfo r s m e e s − i anαendash − r α > 0 ON UNIONS AND INTERSECTIONS OF MEASURABLE SETS .. 203 \ hspacespectively∗{\ periodf i l l }ON We UNIONS denote : AND INTERSECTIONS OF MEASURABLE SETS \quad 203 NWD open parenthesis E closing parenthesis endash the ideal of all nowhere dense subsets of E semicolon\noindent spectively . We denote : K open parenthesis E closing parenthesis endash the sigma hyphen ideal of all meager open paren- thesis\ centerline i period e{ period$ NWD comma ( first E category ) $ closing−− the parenthesis ideal of subsets all of nowhere E semicolon dense subsets of $ EBr open ; $ parenthesis} E closing parenthesis endash the sigma hyphen algebra of all subsets of E with h e − orem 11. L t {St : ∈ T } b a na b t − i r a ry f − a m y − l o n− s i m p − l exesin the Baire property period T S \ centerline { $ Kn − (period ET )en $ h −−−t ethe s t $ X\sigma= braceleft− $− S idealt : ∈ T of} i all L meagerbesgue ( i . e . , first category ) subsets of Recall that a set Xm subseta − e equalsurable E has a d the Baire p ssesses property e iff B Xa can− i be representedoperty . $ Eas the ; symmetric $ } difference X = U vartriangle P where U is open andrepr P is meager in E period Moreover comma if X subset equal E can be expressed in the form X = U vartriangle P where\ centerline U is { $ Br ( E ) $ −− the $ \sigma − $ algebra of all subsets of $ Eopen $ and with P is the nowhere Baire dense property in E comma . } then X is called an open s e t modulo a nowhere dense s e t open parenthesis cf period .. brackleft 5 brackright comma S 8 comma V closing parenthesisRecall that or comma a set briefly $ X comma\subseteq an open s e t moduloE $ hasNWD the open Baire parenthesis property E closing iff parenthesis $ X $ periodcan be represented asWe the shall symmetric say that a subset difference X of a topological $ X space= U E is admissible\ vartriangle if P $ where $U$ isX open subset and equal $ cl P open $parenthesis is meager int in open parenthesis X closing parenthesis closing parenthesis period$E . $ Moreover , if $X \subseteq E $ can be expressed in the form $ XClearly = comma U \ everyvartriangle open set is admissible P$ where period If $U$ X is a regular is closed set open parenthesis i periodopen e and period $P$ comma X is = nowhere dense in $E , $ then $X$ is called an open s e t modulo a nowherecl open parenthesis dense s int e opent ( parenthesis cf . \quad X closing[ 5 ] parenthesis , \S 8 closing , V ) parenthesis or , briefly closing ,parenthesis an open s e t modulo comma$ NWD then ( X is E admissible ) . comma $ as well period .. In addition comma if X is admissible comma Wethen shall from saythe relations that a subset $ X $ of a topological space $ E $ is admissible if int open parenthesis X closing parenthesis subset equal X subset equal cl open parenthesis int open parenthesis\ [X \subseteq X closing parenthesiscl ( closing int parenthesis ( X comma ) .. cl ) open . parenthesis\ ] int open parenthesis X closing parenthesis closing parenthesis backslash int open parenthesis X closing parenthesis in NWD open parenthesis E closing parenthesis \noindentit follows thatClearly X is open , every modulo open a nowhere set isdense admissible set open parenthesis . If $ in X general $ is comma a regular the con closed set ( i . e hyphen$ . , X = $ verse is false closing parenthesis period \noindentIn our furthercl(int considerations $( comma X the ) notion ) of ) a Vitali ,$ covering then plays $X$ an is admissible ,aswell. \quad In addition , if $X$essential is role admissible period .. So , comma we introduce this notion for a general topological space thenand for from an arbitrary the relations subset of that space period We say that a family V of subsets of a topological space E forms a .. Vitali covering of a given set X subset equal E if comma \ centerlinefor each point{ i n x t in X $ and ( for X each ) neighbourhood\subseteq U ofX x comma\subseteq there exists$ ac set l ( i n t $ ( XV )in V such) that , $ x in\quad V subsetcl(int equal U period $( .. X Consequently ) ) comma\setminus if V is a$ Vitali i n t covering $ ( of X)X comma \ in NWD ( E ) $ } then the family of sets open brace V cap Row 1 notdef Row 2 V . notdef-V universal-braceright-F f- existential-notdef\noindent it notdef-r follows m-existential that $ X T-notdef-negationslash-s $ is open modulo a-braceleft a nowhere n-universal-notdef dense set ( notdef- in general , the con − t f-universal universal-o r notdef-notdef notdef-t notdef-e notdef infinity-s b T-notdef-s p-notdef sub union-a\noindent c to theverse power is of e-notdef false ) notdef-X . o period A .. sa r le comma t h e .. V talico verings c nsidered b lo e sub wa e a su m e-d Case 1 f CaseIn our 2 c ns further s-i t considerations , the notion of a Vitali covering plays an essentiala dmssible s role ts period . \ Wquad .. eaSo so r , callth we introduce at a t pologicalsp this ace notion E s t i-s for o fies a t general e topological space ands l-i for n .. c an ndi arbitrary i-t sub o e n subsetopen parenthesis of that he space.. c untable . We c ain say c ndiio that n closinga family parenthesis $ V $i .. e of subsets chof d s-j a sub topological oint to the power space of a $ .. E f .. $ mly forms h sub o a \quad Vitali covering of a given set $ Xnem\ p-tsubseteq y o en s tsinE .. E$ s-i i .. f c , untable period forU .. each n-i sub point ons .. o $ f n x Si m\ l-pin ex ..X$ e-s and for each neighbourhood $U$ of $ x , $ya n there n s i m exists p-l e x w a em set a-e n a c osed n ndegenerate s m p-l ex i .. t e .. E hyphen-u $ide V to the\ in powerV$ of B an such s ace that R n-period $x .. a\ rin al alphaV greater\subseteq 0 .. i .. fi edU comma . $ .. w\quad .. es yConsequently t , if at$V$ a b unded is s a t Vitali covering of $X , $ thensubset the equal family R n i .. alphaof sets endash-r $ \{ egularV if lambda\cap n openarrowdblright parenthesis X closing−colon parenthesis\ begin { array greater}{ c} notdef \\ Vequal\end alpha{ array lambda} notdef open− parenthesiselement V notdef open parenthesis−V u n i X v e to r s thea l − powerbrac eright of parenright-parenright−F f−existential w −notdef h-enotdef re V− openr parenthesis m−existential X closing parenthesis T−notdef− dnegationslash notes a c osed −s a−b r a c e l e f t n−u n i v e r s a l −notdef notdefll w th−t t e .. f− muniversal nim a-l di am t-euniversal er comma−o f o $ r w r h-i $ ch notdef w eh ve− tnotdef ein clusion notdef X subset−t equal notdef V −e opennotdef parenthesis infinity X closing−s parenthesis$ b $ T−notdef−s p−notdef { union−a c ˆ{ e−notdef }} notdefl ear− l-yX comma$ o in h t isdRow 1 i Row 2 e isd i-t o n a b llca n b e r-e placed b ya c be closing parenthesis period N t-o sub e t-h atev ery \noindentsim p-l exRow.A 1\ hquad Row 2sa s-i r a n l e alpha , t endash-r h e \quad ex s tfoV r talico s m e alpha verings greater c 0 nsidered b lo $ eh e-o{ subwa rem}$ 1 e 1 perioda su m .. L $ ..\ tl e open f t . brace e−d S t t\ :begin in T closing{ a l i g n brace e d } b& .. a naf \\ b t-i r a ry f-a .. m y-l& .. o n c hyphen ns s i ms− p-li exesin t \end{ a l i g n e d }\ right . $ n-period to the power of T sub T .. en h-t e .. s t X = union of braceleft-S sub t : in T closing brace\ centerline i .. L besgue{a ..dmssible m a-e surable s ts a d .W .. p\ ssessesquad ea so r callth at a t pologicalsp ace $ Ee .. $ B sa-i t sub $ repr i−s operty $ o period f i e s t e } \noindent s $ l−i $ n \quad c ndi $ i−t { o } e{ n } ( $ he \quad c untable c ain c ndiio n ) i \quad e ch d $ s−j { o i n t }ˆ{ a }$ \quad f \quad mly $ h { o }$ nem $ p−t$ yoens tsin \quad $ E s−i $ \quad c untable .

\ centerline {U \quad $ n−i { ons }$ \quad o f $ n $ Si m $ l−p $ ex \quad $ e−s $ }

\noindent yan $n$ sim $p−l $ e x w em $ a−e $ n a c osed n ndegenerate s m $ p−l $ ex i \quad t e \quad E $ hyphen−u $ $ i d e ˆ{ B }$ an s ace $ R n−period $ \quad a r a l $ \alpha > 0 $ \quad i \quad $ f i $ ed , \quad w \quad es y t at a b unded s t $ \subseteq R n $ i \quad $ \alpha endash−r $ egular if $ \lambda n ( X ) \geq \alpha \lambda ( V ( X ˆ{ parenright −p a r e n r i g h t }$ w $ h−e$ re $V ( X )$ dnotesacosed

\noindent l l w th t e \quad m nim $ a−l $ di am $ t−e$ er,forw $h−i $ ch w eh ve t ein clusion $X \subseteq V ( X ) $

\noindent l ear $ l−y , $ in $h{ t }\ l e f t . i s d \ begin { array }{ c} i \\ e \end{ array } f i ni \ right . i−t$ onabllcanbe $r−e$ placedbyacbe ) .N $t−o { e } t−h $ atev ery sim $ p−l \ l e f t . ex\ begin { array }{ cc } h \\ s−i & a n \alpha endash−r \end{ array } e g u l a r \ right . $ s t f o r s m e $ \alpha > 0 $

\ centerline {h $ e−o { rem } 1 1 . $ \quad L \quad t $ \{ S t : \ in T \} $ b \quad a na b $ t−i $ r a ry $ f−a $ \quad m $ y−l $ \quad o $ n − $ s i m $ p−l $ e x e s i n }

\noindent $ n−period ˆ{ T } { T }$ \quad en $ h−t $ e \quad s t $ X = \bigcup b r a c e l e f t −S { t } : \ in T \} $ i \quad L besgue \quad m $ a−e $ surable a d \quad p s s e s s e s e \quad B $ a−i { repr }$ operty . 204 M . BALCERZAK AND A . KHARAZISHVILI Proof . Obviously , the set X can be expressed in the form [[ X = St

m ∈ ω, m > 0t ∈ Tm where

Tm = {t ∈ T : Stis1/m − regular}. It suffices to show that , for a fixed integer m > 0, the set [ Xm = {St : t ∈ Tm}

c is Lebesgue measurable . For any t ∈ Tm, x ∈ St and c ∈]0, 1[, let St (x) denote the image of St under the homothetic transformation

y → x + c(y − x)(y ∈ Rn). Observe that the family of n− simplexes

c Fm = {St (x): t ∈ TmAmpersandx ∈ StAmpersandc ∈]0, 1[} forms a Vitali covering of Xm. Additionally , Fm consists of simplexes which are 1/m – regular sets . Thus , by the generalized Vitali theorem ( see [ 6 ] , Chapter 4 , Theorem 3 . 1 ) , there exists a countable disj oint family ∗ Fm ⊆ Fm S ∗ S ∗ such that λn(Xm \ Fm) = 0. Evidently , Fm is Lebesgue measurable . S ∗ Also , we have Fm ⊆ Xm by the definition of Fm. Hence Xm is Lebesgue measurable . Our argument in the category case also uses measure - theoretical tools . n Namely , we apply the Jordan measure νn in R and the respective inner n measure (νn)∗. By the classical criterion , a bounded set Y ⊆ R is Jor- dan measurable iff νn( bd (Y )) = 0 where bd (Y ) stands for the bound- ary of Y. Since bd (Y ) is compact , the equality νn( bd (Y )) = 0 im- plies that bd (Y ) is nowhere dense . Consequently , every Jordan mea- surable set Y can be expressed as the union of the open set int (Y ) and parenright−T−period the nowhere dense set Y ∩ ⇒ Y notdef notdefnotdef−I F p notdef−rt−existential−notdefnotdef−i−cexistential−u l T −notdef−negationslash− abraceleft−rcomma−universal−notdefwnotdef universal−es − notdefnotdef−notdef− eT − notdeft − notdefunion − a t notdefY notdef − ps − notdef sesses t e B a − ire p operty . v ew o th e aa overe m a − r k , i t su ffi e − cstooe pressou rse t X a a c untable nion o Jo rdan m e − a surablese ts . F r a na bitrary n − i teger m > 0 w ed note

m={x ∈ Rn :| x ||≤ mbraceright−comma m={St : ∈ T AmpersandS ⊆ Bmcomma−braceright h − e re x || s ands f r t e u ual E u − cld ean n rm o x ∈ Rperiod − n W eo viously ve 204 .. M period BALCERZAK AND A period KHARAZISHVILI \noindentProof period204 .. Obviously\quad M comma . BALCERZAK the set X AND can beA expressed . KHARAZISHVILI in the form ProofLine 1 . X\ =quad unionObviously of union of S , sub the t Line set 2 m $ in X omega $ can comma be m expressed greater 0 t in T the sub mform where \ [ \Tbegin sub m{ a = l iopen g n e d brace} X t in = T : S\ subbigcup t i s 1 slash\bigcup m minus regularS { t closing}\\ brace period mIt suffices\ in to show\omega that comma, m for a> fixed0 integer t m greater\ in 0T comma{ m the}\end set { a l i g n e d }\ ] X sub m = union of open brace S sub t : t in T sub m closing brace is Lebesgue measurable period .. For any t in T sub m comma x in[ S sub t and c in brackright 0 = Gm. comma\noindent 1 brackleftwhere comma let S sub t to the power of c open parenthesis x closing parenthesis denote the image of S sub t under the homotheticelement − transformationomega, mzero − greater \ [Ty right{ m arrow} x= plus\{ c opent parenthesis\ in yT:S minus x closing{ parenthesist } i open s parenthesis 1 / y m in R− to ther e g u l a r \} . \ ] o − w,le t u sc nsider t e n − i ner J rdan d ns i − t y o a na bitrary s t Z ⊆ Rn power of n closing parenthesisa p int periodx ∈ Rn − comma g hven b yt e o − f rm u − l a e Observe that the family of n hyphen simplexes F sub m = open brace S sub t to the power of c open parenthesis x closing parenthesis : t in T sub \noindent It suffices to show that , for a fixed integer $m > 0 , $ m Ampersand x in S sub(Z, tx) Ampersand = inf{((νn) c∗ in(Z brackright∩arrowdblright 0 comma−parenright 1 brackleft−parenrightnu closing brace−notdef−slashT −nparenleft − notdefQnotdefparenright−Funiversal−colonQ−existential−notdef element−notdefexistential−Q−T −notdef−negationslashparenleft − braceleftparenright − notdef − braceright, ∀T −notdef notdef theforms s e t a Vitali covering of X sub m period Additionally comma F sub m consists of simplexes which are .. 1 slash m endash regular sets period .. Thus comma .. by the generalized Vitali theorem .. open\ [X parenthesis{ m } see= .. brackleft\bigcup 6 brackright\{ S comma{ t } : t \ in T { m }\}\ ] Chapter 4 comma Theorem 3 period 1 closing parenthesis comma there exists a countable disj oint family F sub m to the power of * subset equal F sub m \noindentsuch that lambdais Lebesgue sub n open measurable parenthesis . X\ subquad m backslashFor any union $ t of F\ subin m toT the{ powerm } of, * closingx \ in parenthesisS { =t 0} period$ and .... Evidently $ c \ in comma] union 0 of F , sub 1 m to [ the power ,$let$Sˆ of * is Lebesgue{ c } { t } measurable( x ) period $ denoteAlso comma the weimage have ofunion $ of S F sub{ t m} to$ the under power ofthe * subset homothetic equal X sub transformation m by the definition of F sub m period Hence X sub m is Lebesgue \ [measurable y \rightarrow period x + c ( y − x ) ( y \ in R ˆ{ n } ).Our argument\ ] in the category case also uses measure hyphen theoretical tools period Namely comma we apply the Jordan measure nu sub n in R to the power of n and the respective inner \noindentmeasure openObserve parenthesis that nu the sub n family closing parenthesis of $ n sub− *$ period simplexes By the classical criterion comma a bounded set Y subset equal R to the power of n is Jordan \ [Fmeasurable{ m } iff= nu sub\{ n openS ˆ parenthesis{ c } { bdt } open( parenthesis x ) Y closing: t parenthesis\ in T closing{ m paren-} Ampersand thesisx \ =in 0 whereS bd{ opent } parenthesisAmpersand Y closing c parenthesis\ in ] stands 0 for , the 1 boundary [ of\}\ Y period] Since bd open parenthesis Y closing parenthesis is compact comma the equality nu sub n open parenthesis bd open parenthesis Y closing parenthesis closing parenthesis = 0 implies that bd open parenthesis\noindent Yforms closingparenthesis a Vitali covering of $ X { m } . $ Additionally $ , F is{ nowherem }$ dense consists period of .. Consequently simplexes comma which .. every Jordan measurable set Y can be areexpressed\quad as the$ 1 union / of the m $ open−− setregular int open parenthesis sets . \quad Y closingThus parenthesis , \quad andby the the nowhere generalized Vitali theorem \quad ( see \quad [ 6 ] , denseChapter set 4 , Theorem 3 . 1 ) , there exists a countable disj oint family $ F ˆ{ ∗ } { m } \subseteqY cap doubleF stroke{ m right}$ arrow Y notdef to the power of parenright-T-period notdef sub notdef- I F p notdef-r to the power of t-existential-notdef notdef-i-c existential-u l T-notdef-negationslash-a braceleft-r\noindent subsuch comma-universal-notdef that $ \lambda w notdef{ n universal-e} (X s-notdef{ m }\ notdef-notdef-esetminus T-notdef\bigcup t-notdef Funion-a ˆ{ ∗ t }notdef{ m Y} notdef-p) = s-notdef 0 sesses . $ t e\ h .. f B i l a-i l subEvidently re p operty $ period , \bigcup F ˆ{ ∗ } { m }$ isv Lebesgue ew o th e a measurable sub a overe .. . m a-r k comma i t su ffi e-c sub sto o sub e pressou rse t X a .. a c untable \noindentn ion o JoAlso rdan m , e-a we surablese have ts$ period\bigcup F r a naF bitrary ˆ{ ∗ }n-i{ tegerm }\ m greatersubseteq 0 .. w edX note{ m }$ bym the sub definition= open brace x of in R $ n :F bar{ xm bar} bar. less $ or Hence equal m $ braceright-comma X { m }$ is m sub Lebesgue = open brace Smeasurable t : in T Ampersand . S subset equal B m sub comma-braceright h-e re bar-bar x bar bar s .. ands .. f r t e .. u ual E u-c sub ld ean n rm o .. x in R period-n .. W ..\ hspace eo viously∗{\ f i l l }Our argument in the category case also uses measure − theoretical tools . ve \noindentLine 1 = unionNamely of G , m we sub apply period Linethe 2 Jordan element-omega measure comma $ \ mnu zero-greater{ n }$ in $ R ˆ{ n }$ ando-w the sub respective comma l e t u innersc nsider t e n-i ner J rdan d ns i-t y o .. a na bitrary s t Z subset equal R n measurea p int .. x $ in ( R n-comma\nu { ..n g h} ven) b{ yt ∗e .. } o-f. rm $ u-l By a e the classical criterion , a bounded set $ Yopen\ parenthesissubseteq Z commaR ˆ{ xn closing}$ parenthesis i s Jordan = in f open brace open parenthesis open parenthesis numeasurable n closing parenthesis iff $ sub\nu * open{ n parenthesis} ($ Z bd cap $( arrowdblright-parenright-parenright Y ) ) = 0$ nu-notdef-wherebd slash$ ( T-n Y parenleft-notdef ) $ stands Q for notdef the sub boundary parenright-F of universal-colon $Y . $ Q-existential-notdef element- notdefSince existential-Q-T-notdef-negationslash bd $ ( Y ) $ is compact parenleft-braceleft , the equality parenright-notdef-braceright $ \nu { n } comma( $ forallbd T-notdef$( Y notdef ) ) = 0$ impliesthatbd $( Y )$ is nowhere dense . \quad Consequently , \quad every Jordan measurable set $ Y $ can be expressed as the union of the open set int $ ( Y ) $ and the nowhere dense set $ Y \cap \Rightarrow Y notdef ˆ{ parenright −T−period } notdef { notdef−I } F $ p $ notdef−r ˆ{ t−existential −notdef notdef−i−c } existential −u $ l $ T−notdef−negationslash −a b r a c e l e f t −r { comma−u n i v e r s a l −notdef w notdef } u n i v e r s a l −e s−notdef notdef−notdef−e T−notdef t−notdef union−a$ t $notdef Y notdef−p s−notdef $ sesses t e \quad B $ a−i { re }$ p operty .

\noindent v ew o th e $ a { a }$ overe \quad m $ a−r$ k,itsu $ffi e−c { sto } o { e }$ pressou rse t $X$ a \quad a c untable $n{ ion }$ oJordanm $e−a $ surablese ts . F r a na bitrary $ n−i $ t e g e r $ m > 0 $ \quad w ed note

\ [ m { = }\{ x \ in R n : \mid x \mid \mid \ leq m brac erigh t −comma m { = }\{ S t : \ in T Ampersand S \subseteq B m { comma−b r a c e r i g h t }\ ]

\noindent $ h−e $ re $ \bracevert x \mid \mid $ s \quad ands \quad f r t e \quad u ual E $ u−c { ld }$ ean n rm o \quad $ x \ in R period −n $ \quad W \quad eo v i o u s l y ve

\ [ \ begin { a l i g n e d } = \bigcup G m { . }\\ element−omega , m zero−g r e a t e r \end{ a l i g n e d }\ ]

\noindent $ o−w { , l e }$ tusc nsider te $n−i $ ner J rdan d ns $ i−t $ y o \quad a na bitrary s t $Z \subseteq R n $ a p i n t \quad $ x \ in R n−comma $ \quad g $h{ ven }$ b yt e \quad $ o−f $ rm $ u−l $ a e

\ [ ( Z , x ) = in f \{ (( \nu n ) { ∗ } (Z \cap arrowdblright −parenright −parenright nu−notdef−s l a s h T−n p a r e n l e f t −notdef Q notdef { parenright −F } u n i v e r s a l −colon Q−existential −notdef element−notdef existential −Q−T−notdef−negationslash parenleft −braceleft parenright −notdef−b r a c e r i g h t , \ f o r a l l T−notdef notdef \ ] ON UNIONS AND INTERSECTIONS OF MEASURABLE SETS .. 205 \ hspacewhere∗{\ Q openf i l parenthesisl }ON UNIONS x closing AND parenthesis INTERSECTIONS stands for OF the MEASURABLE family of all closedSETS cubes\quad with205 centre x and with \noindentdiameters lesswhere or equal $Q 1 period ( .. x According ) $ to stands the well for hyphen the known family result of open all parenthesis closed cubes see with centre brackleft$ x $ 3 and brackright with comma Chapter 3 closing parenthesis comma if diametersthere exists epsilon $ \ leq greater1 0 such . $ that\quad commaAccording for each point to z the of a bounded well − setknown Z subset result equal( R see [ 3 ] , Chapter 3 ) , if tothere the power exists of n comma $ \ varepsilon we > 0 $ such that , for each point $ z $ of a bounded set $ Z \subseteqON UNIONS ANDR ˆ{ INTERSECTIONSn } , $ we OF MEASURABLE SETS 205 have d sub * openwhere parenthesisQ(x) Z stands comma for z the closing family parenthesis of all closed greater cubes equal with epsilon centre commax and then with Z ishave Jordan $ measurable d { ∗ period} ( Observe Z , that z ) \geq \ varepsilon , $ then $ Z $ is Jordan measurablediameters . Observe≤ 1. According that to the well - known result ( see [ 3 ] , Chapter 3 ) Line 1 G sub m = union, if there of G exists subε m > to0 the such power that ,of for k Line each 2 point k inz omegaof a bounded comma k set greaterZ ⊆ R 0n, we where have d (Z, z) ≥ ε, then Z is Jordan measurable . Observe that \ [ \Gbegin sub m{ a to l i theg n e power d } G of{∗ km = open} = brace\bigcup S sub t in GG sub ˆ{ mk :} open{ m parenthesis}\\ forall x in S sub k \ in \omega , k > 0 \end{ a l i g n e d }\ ] t closing parenthesis open parenthesis d sub * open parenthesis[ S subk t comma x closing parenthesis Gm = G greater equal 1 slash k closing parenthesis closing brace period m Then the set union of G sub m to the power of k is boundedk ∈ ω, k and > 0 d sub * open parenthesis union \noindent where of G sub m to the powerwhere of comma to the power of k x closing parenthesis greater equal 1 slash k for each point x in \ [ G ˆ{ k } { m } = \{ S { t }\ in G { m } :( \ f o r a l l x union of G sub m to the power of periodk to the power of k Consequently comma union of G sub m G = {St ∈ Gm :(∀x ∈ St)(d∗(St, x) ≥ 1/k)}. to\ in the powerS { oft k} is Jordan) ( measurable d { ∗periodm } (S We thus see{ t that} the, set x ) \geq 1 / k ) \} . \ ] S k S k X is expressible asThen a countable the set unionGm ofis Jordan bounded measurable and d∗( setsGm, x period) ≥ 1/k ..for Hence each it point x ∈ S k S k has the Baire propertyGm period. Consequently , Gm is Jordan measurable . We thus see that the set X We want to finish thisis expressible section with as severala countable simple union remarks of Jordan concerning measurable the sets . Hence it has \noindenttheorem jThen ust proved thethe period setBaire .. property $ First\bigcup of . all let usG note ˆ{ k that} { them union}$ of is a familybounded of and $ d { ∗ } ( n hyphen\bigcup simplexesG ˆ{ mayWek have} want{ am torather ˆ{ finish, bad}} this descriptivex section ) with structure\geq several period1 simple ../ remarks Indeed k$ comma concerning for suppose each the point $ xthat\ nin greater$ equaltheorem 2 and consider j ust proved some . hyperplane First of Capital all let Gammaus note thatin the the space union Row of 1 a n family Row 2 period . Y be of n− simplexes may have a rather bad descriptive structure . Indeed , \noindent $ \bigcup G ˆ{ k } { m ˆ{ . }}$ Consequently $ , \bigcupn any subset of Capitalsuppose Gamma that periodn ≥ 2 It and is easy consider to construct some hyperplane a family of Γ n in hyphen the space simplexesR inLet R Y to Gthe ˆ power{ k } of{ nm comma}$ such is Jordan measurable . We thus see that the set . n $that X $the intersection is expressiblebe of any its unionsubset as with a of countable Γ the. It hyperplane is easy union to construct Capital of Gamma Jordan a family coincides measurable of n− simplexes with the sets in R . \, quad Hence i t hasset Ythe period Baire .. Consequently propertysuch that thecomma. intersection if Y is not of lambda its union sub with n minus the hyperplane 1 hyphen measurable Γ coincides or with does not possess the the set Y. Consequently , if Y is not λn−1−measurable or does not possess the WeBaire want property to finish in CapitalBaire this property Gamma section in comma Γ, withthen .. several then the above the above simple - mentioned hyphen remarks unionmentioned concerning is not union an analytic is not the an ( theorem j ust proved . \quad Firstn of all let us note that the union of a family of analytic coanalytic ) subset of R In a similar way one can construct a family of n− $open n parenthesis− $ simplexes coanalytic mayclosing have parenthesis a rather. subset bad of Row descriptive 1 n Row 2 period structure . a similar . way\quad one Indeed , suppose canthat construct $ n a family\geq of 2 $ and consider some hyperplane $ \nGamma $ in the space $\ l e f t .R\ begin { arraysimplexes}{ c} whosen \\ union. \ isend not{ aarray projective} Let \ subsetright of.R Y $ be n hyphen simplexes whose union is not a projective subset of Case 1 n Case. 2 period anyIt immediately subset of follows $ \GammaIt from immediately Theorem. $ 1follows period It is from 1 easythat Theorem the to union construct 1 . of 1 an that arbitrary the a union family of an of arbitrary $ n − $ simplexes in $Rˆ{ n } , $ suchn family of convex bodiesfamily in of R convexto the power bodies of in nR is lambdais λn− submeasurable n hyphen ( measurable because any open convex parenthesis body becausethat anythe convex intersection bodyin Rn can of be itsrepresented union as with the union the of hyperplane a family of n− $simplexes\Gamma )$ .coincides Analo with the s ein t R $ to Y the power . $ of-\ gouslyquad n can , beConsequently the represented union of an as arbitrary ,the if union family$Y$ of a of family convex is not of bodies n hyphen $ \ inlambdaR simplexesn has the{ closing Bairen − parenthesis1 − measurable period .. Analoproperty}$ hyphen or. doesThe latter not fact possess remains the true for any family of convex bodies in a Bairegously property comma the union intopological $ of\ anGamma arbitrary vector space, family $ ( see\ ofquad , convex e . gthen . bodies , Theorem the in R above 2 to . the 1 below− powermentioned containing of n has the a union more Baire is not an analytic (property coanalytic period ) .. Thesubsetgeneral latter result of fact ) $ remains .\ l e f t .R true\ begin for any{ array family}{ ofc convex} n \\ bodies. \ inend{ array } In \ right . $ a similar way one can construct a family of a topological vector space open parenthesis2 . Unions see of comma Topological e period g periodn− S implexescomma Theorem 2 period 1 below$ n containing− $ simplexes a moreBy a topological whose unionn− s is implex notwe a projectivemean a topological subset space of homeomorphic $\ l e f t .R\ begin { a l i g n e d } & n \\general result closingto parenthesisan n− simplex period . Our purpose in this section is twofold . First , we are going &.2 period\end .. Unions{ a l i g ofto n e Topologicalshow d }\ right that the n. $ hyphenunion of S an implexes arbitrary family of topological n− simplexes in Rn By a topological nhas hyphen the Baires implex property we mean . a Next topological , we shall space prove homeomorphic that , for n ≥ to2, an analogous Itan immediately n hyphen simplex followsstatement period Ourfrom fails purpose to Theorem be true in this if 1 the section . possession 1 that is twofold the of the period union Baire First property of comma an arbitrary is replacedwe are going by tofamily of convexLebesgue bodies measurability in $ R ˆ{ .n The}$ first i s of these$ \lambda two results{ willn } be − derived$ measurable from ( because any convex body show that the unionthe of followingan arbitrary general family theorem of topological : n hyphen simplexes in R to the power of n \noindenthas the Bairein property $ R ˆ{ periodn }$ .. can Next be comma represented we shall prove as thethat commaunion for of n a greater family equal of 2 comma$ n an− analogous$ simplexes ) . \quad Analo − gouslystatement , thefails to union be true of if thean possession arbitrary of the family Baire propertyof convex is replaced bodies in $ R ˆ{ n }$ hasby the Lebesgue Baire measurability period .. The first of these two results will be derived propertyfrom the following . \quad generalThe theorem latter : fact remains true for any family of convex bodies in a topological vector space ( see , e . g . , Theorem 2 . 1 below containing a more

\noindent general result ) .

\ centerline {2 . \quad Unions of Topological $ n − $ S implexes }

By a topological $ n − $ s implex we mean a topological space homeomorphic to an $ n − $ simplex . Our purpose in this section is twofold . First , we are going to show that the union of an arbitrary family of topological $ n − $ simplexes in $ R ˆ{ n }$ has the Baire property . \quad Next , we shall prove that , for $ n \geq 2 , $ an analogous statement fails to be true if the possession of the Baire property is replaced by Lebesgue measurability . \quad The first of these two results will be derived from the following general theorem : 206 .. M period BALCERZAK AND A period KHARAZISHVILI \noindentTheorem 2206 period\quad 1 periodM . .. BALCERZAK Let E be an arbitrary AND A . topological KHARAZISHVILI space and le t F be a family consisting .... of admissible subsets .... of E period .... Then union of F is .... open modulo \ hspaceNWD∗{\ openf i parenthesis l l }Theorem E closing 2 . 1 parenthesis . \quad periodLet ..$ EIn $particular be an comma arbitrary union of topological F has th e space and le t Baire$ F $ property be a period Proo f-period .. We put V = union of open brace int open parenthesis F closing parenthesis : F in F\noindent closing bracefamily period Since consisting F consists\ ofh fadmissible i l l of admissible sets comma subsets \ h f i l l o f $ E . $ \ h f i l l Then $ \bigcup206 M . BALCERZAKF $ i s \ ANDh f i A l l . KHARAZISHVILIopen modulo we have comma for eachTheorem F in F comma 2 . 1 . Let E be an arbitrary topological space and le t F F subset equal cl open parenthesis int open parenthesis F closing parenthesis closing parenthesis \noindent $ NWDbe a ( E ) . $ \quad In particular $ , \bigcup F $ subset equal cl open parenthesis V closing parenthesis period S has th e Baire propertyfamily consisting . of admissible subsets of E. Then F is open modulo Thus comma we obtain union of F subset equal clS open parenthesis V closing parenthesis and comma Proo $ f−periodNWD $ \(Equad). InWe particular put $ V, F =has th\bigcup e Baire property\{ $ . iProo n tf $− (period F therefore comma S ):F \ in WeF put \}V = {.int $ (F Since): F ∈ F} $. FSince $F consistsconsists of of admissible admissible sets , sets , V = union of openwe brace have int , for open each parenthesisF ∈ F, F closing parenthesis : F in F closing brace subset equal union of F subsetF equal⊆ cl ( cl int open (F )) parenthesis⊆ cl (V ). Thus V closing , we parenthesis obtain S F period ⊆ cl (V ) and , therefore , \noindentHence unionwe of have F = V , cup for Z foreach some $ set F Z subset\ in equalF cl ,open $ parenthesis V closing parenthesis [ [ backslash V comma which gives the assertionV = period{int(F ): F ∈ F} ⊆ F ⊆ cl(V ). \noindentCorollary 2$ period F 1\ periodsubseteq .. For$ each cl(int integer n greater $( equal F 1 and ) for ) ea ch\subseteq family of topo$ logical c l $ ( V ) . $ n hyphen simplexesHence openS braceF = SV sub∪ Z tfor : t insome T closing set Z ⊆ bracecl (V in) \ RV, towhich the power gives ofthe n assertion comma th . e s Thus , we obtain $ \bigcup F \subseteq $ cl $( V )$ and,therefore , e t union of open brace SCorollary sub t : t in 2 T . closing 1 . braceFor each has the integer Bairen property≥ 1 and period for ea ch family of topo Obviously comma Corollarylogical n− 2 periodsimplexes 1 yields{S : anothert ∈ T } openin R parenthesisn, th e s e t purelyS{S topological: t ∈ T } has closing the \ [ V = \bigcup \{ i n t (t F ) : F \ in F t \}\subseteq parenthesis proof of theBaire property . \bigcup F \subseteq c l ( V ) . \ ] category part of TheoremObviously 1 period , Corollary 1 period 2 . 1 yields another ( purely topological ) proof of the The next statementcategory shows that part the of Theoremmeasure case 1 . 1 for . unions of topological simplexes is completelyThe different next period statement shows that the measure case for unions of topological \noindent Hence $ \bigcup F = V \cup Z$ for some set $Z \subseteq $ Theorem 2 period 2simplexes period .. is For completely each integer different n greater . equal 2 comma there exists a family open c l $ ( V ) \setminus V , $ which gives the assertion . brace Z sub t : t in T closingTheorem brace 2 . 2 . For each integer n ≥ 2, there exists a family {Zt : t ∈ of topological n hyphenT } of s topological implexes inn R− tos implexes the power in ofR nn .. suchsuch that that the s s e e t t unionS{Z of: opent ∈ braceT } Corollary 2 . 1 . \quad For each integer $ n \geq 1 $ andt for ea ch family of topo logical Z sub t : t in T closingis brace not measurable.. is not in the Lebesgue s ense . $ n − $ simplexes $ \{ S { t } : t \ in T \} $ in $ R ˆ{ n } measurable in the LebesgueProof . It s ense is enough period to consider the case n = 2 since , if a family {Z : , $ th e s e t $ \bigcup \{ S { t } : t \ in T \} $ hast the Baire property . Proof period .... Itt is∈ enoughT } satisfies to consider the assertion the case of nour = theorem 2 since comma for n = .... 2, then if a family the family open brace Z sub t : {Z × [0, 1]n−2 : t ∈ T } satisfies the assertion of the theorem for an arbitrary Obviously , Corollaryt 2 . 1 yields another ( purely topological ) proof of the t in T closing braceinteger satisfiesn >the2. assertionSo , we of restrict our theorem our further for n =consideration 2 comma then to the the case familyn = category part of Theorem 1 . 1 . open brace Z sub t2 times . brackleft In the 0 sequel comma , by 1 brackright a Jordan curve to the we power mean of a n homeomorphic minus 2 : t in image T closing of brace satisfies the assertionthe unit of the circle theorem . for an It isarbitrary well known that there exists a Jordan curve L in \ hspace ∗{\ f i l l }The2 next statement shows that the measure case for unions of topological integer n greater 2R periodpossessing .. So comma a positive we restrict two - dimensional our furtherLebesgue consideration measure to the , i case . e ., n λ =2(L) > 2 period .... In the0 sequel . The comma construction by a Jordan of L curvecan be we done mean directly a homeomorphic . Another image idea of is to derive \noindent simplexes is completely different . the unit circle periodfrom .... the It is Denjoy well known - Riesz that theorem there (exists see , a e Jordan . g . , [ curve 5 ] , § L6 in 1 , V , Theorem 5 ) R to the power of 2that possessing each a positive two hyphen dimensional Lebesgue measure comma i period Theorem 2 . 2 . \quad For each integer $ n \geq 2 , $ there exists a family e period comma lambdacompact sub 2 open zero - parenthesis dimensional L set closingC in parenthesisR2 is contained greater in a Jordan curve . Taking $ \{ Z { t } : t \ in T \} $ 0 period .. The constructionas C a Cantor of L can- type be set done in directlyR2 with periodλ (C) ..> Another0, we get idea the isdesired to derive curve L. of topological $ n − $ s implexes in $Rˆ2 { n }$ \quad such that the s e t from the Denjoy hyphenNow , by Riesz the Jordan theorem Curve open Theorem parenthesis,R2 see\ L has comma exactly e period two components g period comma : one $ \bigcup \{ Z { t } : t \ in T \} $ \quad i s not brackleft 5 brackright commabounded S and 6 1 comma one unbounded V comma . Theorem The bounded 5 closing component parenthesis will that be denoted each by measurable in the Lebesgue s ense . compact zero hyphenU. dimensionalWe shall use set C the in Sch R too ¨ thenflies power theorem of 2 is ( contained see [ 5 ] , in§6 a 1 Jordan , I I , curve Theorem period 1 Taking 1 ) according to which , for any points x ∈ L and y ∈ U, there is a simple arc \noindent Proof . \ h f i l l It is enough to consider the case $ n = 2 $ as C a Cantor hyphenl, with type end set in- points R to thex and powery, ofsuch 2 with that lambdal \{x} sub ⊆ U. 2 openIn parenthesis fact , a bit C sharper closing s i n c e , \ h f i l l i f a family $ \{ Z { t } : $ parenthesis greater 0 commaversion we is neededget the where desired the curve simple L period arc l is a quasi - polygonal curve , i . e . , Now comma by thethe Jordan set l Curve\{x} consists Theorem of comma countably R to many the power linear of segments 2 backslash which L has converge exactly ( two in \noindent $ t \ in T \} $ satisfies the assertion of our theorem for components : the Hausdorff metric ) to {x}. ( Cf . , e . g . , [ 7 ] , Appendix to Chapter IX $n = 2 ,$ thenthefamily one bounded and one.) unbounded period The bounded component will be denoted by U period .. We shall use the Sch o-dieresis nflies theorem open parenthesis see brackleft 5 brackright\noindent comma$ \{ S 6 1Z comma{ t I I}\ commatimes Theorem[ 1 1 0 closing , parenthesis 1 ] ˆ{ n − 2 } : t according\ in T to which\} comma$ satisfies for any points the x assertion in L and y in of U comma the theorem there is a for simple an arc arbitrary i nl t comma e g e r with $ n end> hyphen2 points . $ x and\quad y commaSo , such we that restrict l backslash our open further brace x consideration closing brace to the case subset$ n equal = $ U period .. In fact comma a bit sharper version is needed where the simple arc l is a quasi hyphen polygonal curve comma i period e period comma\noindent the 2 . \ h f i l l In the sequel , by a Jordan curve we mean a homeomorphic image of set l backslash open brace x closing brace consists of countably many linear segments which converge open\noindent parenthesisthe in unit circle . \ h f i l l It is well known that there exists a Jordan curve $ Lthe $ Hausdorff in metric closing parenthesis to open brace x closing brace period open parenthesis Cf period comma e period g period comma brackleft 7 brackright comma Appendix to Chapter IX period closing\noindent parenthesis$ R ˆ{ 2 }$ possessing a positive two − dimensional Lebesgue measure , i . e $ . , \lambda { 2 } (L) > $ 0 . \quad The construction of $ L $ can be done directly . \quad Another idea is to derive from the Denjoy − Riesz theorem ( see , e . g . , [ 5 ] , \S 6 1 , V , Theorem 5 ) that each

\noindent compact zero − dimensional set $C$ in $Rˆ{ 2 }$ is contained in a Jordan curve . Taking

\noindent as $C$ a Cantor − type set in $Rˆ{ 2 }$ with $ \lambda { 2 } (C) > 0 , $ weget the desired curve $L . $

\noindent Now , by the Jordan Curve Theorem $ , R ˆ{ 2 }\setminus L $ has exactly two components : one bounded and one unbounded . The bounded component will be denoted by $ U . $ \quad We shall use the Sch $ \ddot{o} $ nflies theorem ( see [ 5 ] , \S 6 1 , I I , Theorem 1 1 ) according to which , for any points $ x \ in L $ and $ y \ in U , $ there is a simple arc $l ,$ withend − points $x$ and $y ,$ such that $ l \setminus \{ x \}\subseteq U . $ \quad In fact , a bit sharper version is needed where the simple arc $ l $ is a quasi − polygonal curve , i . e . , the s e t $ l \setminus \{ x \} $ consists of countably many linear segments which converge ( in the Hausdorff metric ) to $ \{ x \} . ( $ Cf . , e . g . , [ 7 ] , Appendix to ChapterIX . ) ON UNIONS AND INTERSECTIONS OF MEASURABLE SETS 207 Now , let us pick a Lebesgue nonmeasurable set L∗ ⊆ L. ( The existence of L∗ is well known - see , e . g . , [ 1 ] , Chapter 5 . ) We shall define ∗ a family {Zx,y : x ∈ L , y ∈ U} of sets lying in the plane and satisfying the following

conditions :

(1)Zx,y is homeomorphic to a closed nondegenerate triangle ;

(2)Zx,y \{x} ⊆ U;

(3)x and y are boundary points of Zx,y. So , fix x ∈ L∗ and y ∈ U and pick a point y0 such that the segment [y, y0] is contained in U. By the above - mentioned sharp version of the Scho ¨ nflies theorem , we choose quasi - polygonal curves Px,y( joining x and y) and Px,y0 0 ( j oining x and y ) such that Px,y \{x} ⊆ U, Px,y0 \ {x} ⊆ U. First , by a simple { T − notdef modification , we can ensure that P intersection − ycomma − arrowdblrightynotdefbrackright − Tnotdef − equal F − braceright, notdef notdefy − prime T − notdef − negationslash − intersectionyarrowdblright − comma − universal − notdefprime − notdef − notdef]T triangle − equal − notdefnotdef − element x,y y existential−notdef−P ∃ notdef s−t . N e x t − comma us i g the fac tha tthe seg men of P x a n of P x0c on v e − r ge he p o − it − n {x}, we mo i − f y P comma−xya n P, 0( s t p ,by te ) to ns ue tha t F ∩xarrowdblright − commaprime − notdef = notdef{notdef − x .notdef − existentialf − notdef i existential−tnegationslash−notdef−T {d − notdef − existentialnotdef− o n e − universal, notdef − T − th−notdefe−unions − notdef t notdefL, y = e P xy∪[y,yprime − brackright ∪ x − P , 0 or m s a n cu r − v e W e d fin Z, y st h ec osu e of th b oun ed comp on n − e t \ L, . Fi n a y − comma we e p t t = (x)y Zt = Zx,y T = asteriskmath− L × U. T h − en mil y {Zt : t ∈ T }s − ai − tfi es the a ser ion of th th eor em . In d − ee d , each t(t ∈ T )i − s ho me om r − o ph i − c to sa lo e nond gen r − ea − t e trian le ( this tr n g ve r − s i n of th eJ ord n th o − e r em fort e p ane whc h d go es n − ot oran Eu c l − i ean space of h g − i he rdim e si o − n). Sinc e the s t S {Zt : t ∈ U = L∗ si Lb es gueno nm e − a su a − r ble , the s e t S {Zt : t ∈ T } s Le es gue e − a su a − r ble , too . 3 . Uncountable Intersections of Thick Sets Here we consider the following problem concerning uncountable inter - sections of measurable sets . Let S be a σ− algebra of subsets of a given nonempty set E and let I be a σ− ideal of subsets of E such that I ⊆ S. In such a case , the triple (E, S, I) is called a measurable space with a σ− ideal . Additionally , let the pair (S, I) fulfil the countable chain condition ( in short , ccc ) , which means that each disj oint subfamily of S\I is count - able . One can ask the question what should be assumed , for an uncountable family {Xt : t ∈ T } of subsets of E, such that (∀t ∈ T )(E \ Xt ∈ I), to T get the relation E \ {Xt : t ∈ T } ∈ I. The main difficulty is to have T {Xt : t ∈ T } ∈ S. It seems natural to introduce the set

W = {(t, x) ∈ T × E : x ∈ Xt}, ON UNIONS AND INTERSECTIONS OF MEASURABLE SETS .. 207 \ hspaceNow comma∗{\ f i letl l } usON pick UNIONS a Lebesgue AND INTERSECTIONS nonmeasurable set OF L to MEASURABLE the power of SETS* subset\quad equal207 L period open parenthesis The existence Nowof ,L tolet the us power pick of a * is Lebesgue well known nonmeasurable hyphen see comma set e period $ L gˆ{ period ∗ } comma \subseteq .. brackleftL 1 brackright. ( $ comma The existence Chapter 5 period closing parenthesis .. We shall define a family o fopen $ brace L ˆ{ Z ∗ sub }$ x comma is well y : x known in L to− thesee power , of e * . comma g . , y\ inquad U closing[ 1 brace ] , of Chapter sets lying 5 in . ) \quad We shall define a family the$ plane\{ andZ satisfying{ x the , following y } : x \ in L ˆ{ ∗ } , y \ in U \} $ of sets lying inand the thus plane the sets andXt satisfyingare equal to the the vertical following sections Wt = {x ∈ E :(t, x) ∈ conditions : W } of W. So , our problem can now be formulated as follows . Let W be a open parenthesis 1 closing parenthesis Z sub x comma y is homeomorphic to a closed nondegenerate \ begin { a l i g n ∗} set of some “ good ” structure in the product set T × E, with thick sections triangle semicolon W , t ∈ T, ( i . e ., E \W ∈ I). How is it possible to obtain the relation conditionsopen parenthesis : 2 closingt parenthesis Z sub x commat y backslash open brace x closing brace subset equal\end{ Ua semicolonl i g n ∗} open parenthesis 3 closing parenthesis x and y are\ boundary points of Z sub x comma y period E \ {Wt : t ∈ T } ∈ I? \ centerlineSo comma fix{ $ x in ( L to 1 the )power Z of{ * andx y ,in U y and}$ pick is a point homeomorphic y to the power to of a prime closed such nondegenerate triangle ; } that the segment brackleft y comma y to the power of prime brackright \ [is ( contained 2 ) in U Z sub{ periodx .. , By ythe}\ abovesetminus hyphen mentioned\{ sharpx \}\ version ofsubseteq the Sch dieresis-oU nflies; \ ] theorem comma we choose quasi hyphen polygonal curves P sub x comma y open parenthesis joining x and y closing parenthesis and P sub x comma y prime \ centerlineopen parenthesis{ $( j oining 3 x and ) y to x$ the power and of $y$ prime closing areboundary parenthesis such points that P of sub x$Z comma{ x y, backslash y } open. $ brace} x closing brace subset equal U comma P sub x comma y prime backslash open brace x closing brace subset equal U period First comma by a \ hspacesimple∗{\ modificationf i l l }So comma , f i x we can $ x ensure\ in that PL sub ˆ{ x comma ∗ }$ y and intersection-y $ y comma-arrowdblright\ in U $ and pick a point y$ notdef y ˆ{\ brackright-Tprime }$ Row such 1 open that brace thesegment Row 2 y . comma $ notdef [ sub y existential-notdef-P , yˆ{\prime notdef} y-prime] $ sub exists T-notdef-negationslash-intersection to the power of y arrowdblright-comma-universal-notdef prime-notdef-notdef\noindent is contained brackright T in triangle-equal-notdef $ U { . }$ \ Casequad 1 T-notdefBy the Case above 2 notdef− mentioned sharp version of the Sch $ \periodddot{ No} e x$ t-comma n f l i e sus i g the fac .. tha tthe seg men to the power of s-t of .. P x .. a n of .. P x primetheorem sub c on , we v e-r choose ge quasi − polygonal curves $ P { x , y } ( $ j o i n i n g $x$he p .. and o-i t-n $y open brace )$ x closing and brace$P comma{ x we , .. mo y i-f}\ y ..prime P comma-x$ y sub a n .. P comma prime sub open parenthesis s t .. p comma sub by te closing parenthesis to .. ns ue tha t \noindentcap sub x arrowdblright-comma( j oining $x$ prime-notdef and $yˆ = notdef{\prime sub open} brace) $ notdef-x such to that the power $P of F{ pe-x riod, notdef-existential y }\setminus f-notdef\{ i existential-tx \}\ sub negationslash-notdef-Tsubseteq U,P open brace{ x d-notdef-existential , y }\prime notdef-o\setminus n e-universal\{ commax \}\ notdef-T-tsubseteq sub h-notdefU e-union .$ s-notdef First t ,bya notdef L comma y = P x y subsimple cup brackleft modification y sub comma , we y prime-brackright can ensure that cup x-P $ to\ l the e f t power.P of{ ex comma , prime y } orintersection m s −y commaa n− cuarrowdblright r-v e .. W e d .. fin y .. Z comma notdef y .. st brackright h ec osu e of−T .. th notdef .. b oun− edequal .. comp\ begin on n-e{ array t }{ c}\{\\ y \backslashend{ array L comma}F−braceright period Fi n a .., y-comma notdef .. we{ eexistential p t t = open parenthesis−notdef− xP closing} notdef parenthesis y−prime {\ exists } Ty− Znotdef sub t =− Znegationslash sub x comma sub−intersection y T = asteriskmath-L ˆ{ y times} arrowdblright U period T h-e sub−comma n −u n i v e r s a l −notdef primemil− ynotdef open brace−notdef Z t : t in ] T closing T trianglebrace s-a i-t−equal fi es the−notdef a ser ion ofnotdef th th− eorelement em period\ begin In d-e{ a l i g n e d } & Tsub−notdef e .. d comma\\ each &t open notdef parenthesis\end t{ ina l T i g closing n e d }\ parenthesisright . $ i-s ho me om r-o ph i-c to .. s a .. lo e .. nond gen r-e a-t. e N .. e trian x le $ open t−comma parenthesis $ us this i g the fac \quad tha tthe seg $ men ˆ{ s−t }$ o f tr\quad n g ve r-s$ iP n of xth $ eJ ord\quad n th o-ea n r em o f fort\quad e p ane$ .. P whc x h d g\prime sub o es n-o{ c sub}$ t on v $ e−r $ ge oran .. Eu c l-i ean space of .. h g-i he rdim e si o-n closing parenthesis period Sinc e .. the s t unionhe p of\ openquad brace$ Zo− ti : t in t−n \{ x \} , $ we \quad mo $ i−f $ y \quad $ PU = comma L * si− Lbx es gueno y { nma e-a}$ su n a-r\quad ble comma$ P the s , e t union\prime of open{ ( brace}$ Z st : t t in\quad T closingp brace$ , s{ Leby es gue}$ te ) to \quad ns ue tha t e-a su a-r ble comma too period \noindent3 period .. Uncountable$ \cap { Intersectionsx } arrowdblright of Thick Sets−comma prime−notdef = notdef {\{} notdefHere− wex consider ˆ{ F } the. following notdef problem−existential concerning uncountable f−notdef inter $ hyphen i $ existential −t { negationslash −notdef−T } \{ sectionsd−notdef of measurable−existential sets period .... notdef Let S− beo $a sigma n$ hyphen e−universal algebra of subsets , of notdef a given−T−t { h−notdef e−unionnonempty} sets− Enotdef$ and let I be t a sigma $notdef hyphen ideal L of subsets, y of E= such P that x I subset y {\ equalcup S period} [ y In{ such, } a casey comma prime− theb r triple a c k r openi g h t parenthesis\cup Ex comma−P ˆ{ Se comma} , I closing\prime parenthesis$ or ism called s aa measurable n cu $ space r−v with $ e a sigma\quad hyphenW e d \quad f i n \quad $ Z , y $ \quad st h ec osu e of \quad th \quad b oun ed \quad comp on $ nideal−e $ period t .. Additionally comma let the pair open parenthesis S comma I closing parenthesis fulfil the countable chain condition \noindentopen parenthesis$ \setminus in short commaL ccc , closing . $ parenthesis Fi n a comma\quad which$ y− meanscomma that $ each\quad disjwe oint e p t subfamily$ t = of S ( backslash x ) I is count y hyphen Z { t } = Z { x } , { y } T = asteriskmath −L \timesable periodU One . can $ askT the $ h question−e { n what}$ should be assumed comma for an uncountable milfamily y open $ \{ braceZ X sub t t : :t in T t closing\ in braceT of subsets\} ofs− Ea comma i−t such fi that $ open es parenthesis the a ser ion of th th eorem . In forall$ d−e t in{ Te closing}$ parenthesis\quad d , open each parenthesis E backslash X sub t in I closing parenthesis comma to $ t ( t \ in T ) i−s $ ho me om $ r−o $ ph $ i−c $ to \quad $ s {geta the}$ relation\quad E backslashl o e \quad intersectionnond gen of open $ brace r−e X sub a−t t :$ t in e T\quad closingtrian brace in le I period ( this .. The main difficulty is to have \noindentintersectiont r of n open g ve brace $ X r sub−s$ t : t inof in T closing theJordnth brace in S period $oIt seems−e $ natural rem to fort introduce e p ane \quad whc h d the$ g set{ o }$ es $ n−o { t }$ oranW =\ openquad braceEu open c $ parenthesis l−i $ ean t comma space x closing of \quad parenthesish $ in g T−i times $ E herdime : x in X sub si t closing $o−n brace) . comma $ Sinc e \quad the s t $ \bigcup \{ Z t : t \ in $ $and U thus = the L sets X∗ sub$ t si are Lb equal es to guenonm the vertical $ sections e−a $ W su sub t $ = a− openr$ brace ble x in ,theset E : open parenthesis$ \bigcup t comma\{ xZ closing t parenthesis : t in \ in T \} $ s Le es gue $W e closing−a $ brace su of $ W a− periodr $ .. b Sol e comma , too our . problem can now be formulated as follows period Let W be a \ centerlineset of some{ quotedblleft3 . \quad goodUncountable quotedblright Intersections structure in the product of Thick set TSets times} E comma with thick sections \ hspaceW sub∗{\ t commaf i l l } tHere in T comma we consider open parenthesis the following i period e period problem comma concerning E backslash uncountable W sub t in I inter − closing parenthesis period .. How is it possible to obtain the relation \noindentE backslashsections intersection of of measurable open brace W subsets t : . t in\ h T f iclosing l l Let brace $ in S I $ ? be a $ \sigma − $ algebra of subsets of a given

\noindent nonempty set $E$ and let $ I $ be a $ \sigma − $ ideal of subsets of $E$ such that $ I \subseteq S . $ In such a case , the triple $ ( E , S , I ) $ is called a measurable space with a $ \sigma − $ i d e a l . \quad Additionally , let the pair $ ( S , I ) $ fulfil the countable chain condition ( in short , ccc ) , which means that each disj oint subfamily of $ S \setminus I $ i s count −

\noindent able . One can ask the question what should be assumed , for an uncountable family $ \{ X { t } : t \ in T \} $ of subsets of $E ,$ such that $ ( \ f o r a l l t \ in T)(E \setminus X { t } \ in I ) , $ to get the relation $ E \setminus \bigcap \{ X { t } : t \ in T \}\ in I . $ \quad The main difficulty is to have $ \bigcap \{ X { t } : t \ in T \}\ in S . $ It seems natural to introduce the set

\ [ W = \{ ( t , x ) \ in T \times E : x \ in X { t } \} , \ ]

\noindent and thus the sets $ X { t }$ are equal to the vertical sections $ W { t } = \{ x \ in E : ( t , x ) \ in $ $ W \} $ o f $ W . $ \quad So , our problem can now be formulated as follows . Let $ W $ be a set of some ‘‘ good ’’ structure in the product set $ T \times E , $ with thick sections $ W { t } , t \ in T , ($i.e$. , E \setminus W { t } \ in I ) . $ \quad How is it possible to obtain the relation

\ begin { a l i g n ∗} E \setminus \bigcap \{ W { t } : t \ in T \}\ in I ? \end{ a l i g n ∗} 208 .. M period BALCERZAK AND A period KHARAZISHVILI \noindentIn connection208 with\quad the generalM . BALCERZAK question posed AND above A . comma KHARAZISHVILI .. let us mention especially the result of Goldstern .. brackleft 4 brackright .. which deals with a particular .. open parenthesisIn connection but with the general question posed above , \quad let us mention especiallyimportant for the various result applications of Goldstern closing parenthesis\quad case[ 4 period ] \quad Namelywhich comma deals let E bewith a perfect a particular \quad ( but Polishimportant for various applications ) case . Namely , let $ E $ be a perfect Polish spacespace with with a aprobability probability Borel measure Borel mu measure period .. $ The\mu completion. $ of\quad mu is denotedThe completion of 208 M . BALCERZAK AND A . KHARAZISHVILI $ \bymu mu-macron$ i s denoted sub period .. Then S and I stand for the sigma hyphen algebra of macron-mu hyphen by $ \bar{\mu} {In. connection}$ \quad withThen the general $S$ question and posed $ I $ above stand , let for us the mention $ \sigma measurable sets and theespecially the result of Goldstern [ 4 ] which deals with a particular ( but − $sigma algebra hyphen ideal of of $ mu-macron\bar{\mu} hyphen − null$ measurablesets open parenthesis sets respectivelyand the closing parenthesis $ \sigma − $important i d e a lo for f various $ \bar applications{\mu} − ) case$ . null Namely sets , let (E respectivelybe a perfect Polish ) . \quad It is well known that period .. It is well knownspace that with open a probability parenthesis Borel S sub measure commaµ. I closingThe completionparenthesis of fulfilsµ is the denoted by $ (ccc period S { The, } roleI of T )$ is played fulfils by the zero the hyphen dimensional perfect Polish product ccc . The role ofµ¯. Then $ T $S and isI playedstand for by the theσ− algebra zero − ofµ ¯dimensional− measurable sets perfect and the Polishσ− product space omega to theideal power of ofµ ¯− omeganull sets where ( respectively omega is equipped ) . Itis with well the known discrete that topology (S I) fulfils period the .. Aspace natural order $ \omega ˆ{\omega }$ where $ \omega $ is equipped with, the discrete topology . \quad A natural order in this productccc space . The is role defined of T is played by $by t the zero\ leq - dimensionalt ˆ{\prime perfect Polish}\Leftrightarrow product in this product space is definedω by t less or equal t to the power of prime Leftrightarrow open paren- ( \ f o r a l l n space\ inω where\omegaω is equipped) ( with t the ( discrete n topology ) \ leq . A naturalt ˆ{\ orderprime } thesis forall n in omegain closing this product parenthesis space openis defined parenthesis by t ≤ t t0 ⇔ open(∀n parenthesis∈ ω)(t(n) ≤ nt closing0(n)) for parenthesis any less( or n equal ) t to )$ the power forany of prime open parenthesis n closing parenthesis closing parenthesis for any t and t to the power of prime from omega to the power of omega period \ begin { a l i g n ∗} M period Goldstern proved the following result opentand parenthesist0fromωω. see brackleft 4 brackright comma Lemmat and 6 closing t parenthesis ˆ{\prime period} from \omega ˆ{\omega } . \end{ a l i g n ∗} Theorem 3 period 1 periodM .. . Goldstern Assume that proved W subset the following equal omega result to ( seethe [ power 4 ] , Lemma of omega 6 ) times . E is a coanalytic s e t satisfying Theorem 3 . 1 . Assume that W ⊆ ωω × E is a coanalytic s e t \ centerline {M . Goldstern proved the following result ( see [ 4 ] , Lemma 6 ) . } the conditions : satisfying open parenthesis athe closing conditions parenthesis : open parenthesis forall t in omega to the power of omega closing\ hspace parenthesis∗{\ f i l l } openTheorem parenthesis 3 . 1 mu-macron . \quad openAssume parenthesis that W $W sub t\ closingsubseteq parenthesis\omega = 1 ˆ{\omega } closing\times parenthesisE $ issemicolon a coanalytic open parenthesis s e t b satisfying closing parenthesis open parenthesis forall t comma ω t to the power of prime in omega to the power of omega(a)(∀ closingt ∈ ω )(¯ parenthesisµ(Wt) = 1); open parenthesis t less \noindent the conditions : 0 ω 0 or equal t to the power of prime double stroke(b)(∀ rightt, t ∈ arrowω )(t W≤ t sub⇒ tW supseteqt ⊇ Wt0). W sub t prime closing parenthesis period Then mu-macron open parenthesis to theT power of intersection of open brace W sub T henµ¯( {W : t ∈ ωω}) = 1. t\ begin : t in omega{ a l i g ton ∗} the power of omega closing brace closing parenthesist = 1 period ( a ) ( \ f o r a l l t \ in \omega ˆ{\omega } )( \bar{\mu} We are going to show thatWe are the going category to show analogue that ofthe Goldstern category quoteright analogue of s Goldsterntheorem ’ s theorem (Wis false{ periodt } Now) comma = 1 we have ) that ; \\ E( = brackleft b ) 0 comma ( \ 1f o brackright r a l l t comma , the t sigma ˆ{\prime } \ in \omega ˆ{\isomega false . Now} ) , we (have that t E\ leq= [0, 1], tthe ˆ{\σ− primealgebra S}\consistsRightarrow of subsets ofW { t } hyphen algebra S consistsE with of subsets the Baire of property and the σ− ideal I consists of all meager subsets of \supseteqE with the BaireW { propertyt }\ andprime the sigma). hyphen\\ idealThen I consists\bar of{\ allmu meager} ( subsets ˆ{\bigcap } \{ W { t } :E. It t is well\ in known\omega that theˆ{\ pairomega (S, I) fulfils}\} the ccc) . = 1 . of E period It is well known that the pair open parenthesis S comma I closingω parenthesis fulfils the \end{ a l i g n ∗} Theorem 3 . 2 . There is an open s e t W ⊆ ω × [0, 1] such that ccc period ( a )(∀t ∈ ωω)(W is comeager in [ 0 , 1 ] ) , Theorem 3 period 2 period .. There is an open s e tt W subset equal omega to the power of omega We are going to show that the category analogue of Goldstern ’ s theorem times brackleft 0 comma 1 brackright such that(b)(∀t, t0 ∈ ωω)(t ≤ t0 ⇒ W ⊇ W 0), isfalse.Now,wehavethatopen parenthesis a closing parenthesis open $E parenthesis = [ forall 0 tin ,t omega 1t to ] the ,$ power the of omega $ \sigma − $ algebra $ S $ consists of subsetsT of ω closing parenthesis open parenthesis W( c sub ) the t is s e comeager t {Wt in: t brackleft∈ ω } is 0countable comma 1. brackright closing $ E $ with the Baire property and the $ \sigma − $ ideal $ I $ consists of all meager subsets parenthesis comma Proof . Let {pj}j ∈ ω be a one - to - one sequence of all rationals in [ 0 , 1 of $E .$ It is wellknownthat the pair $( S , I )$ fulfils the ccc . open parenthesis b] closing and let parenthesisB(p, r) stand open for parenthesis the open ball forall in [0 t,1] comma with centre t to thep and power radius of primer. Let in omega to the power ofthe omega set W closing⊆ ωω × parenthesis[0, 1] be given open by parenthesis t less or equal t to the power of prime\ centerline double stroke{Theorem right arrow 3 . W 2 sub . \quad t supseteqThere W sub is t an prime open closing s e parenthesis t $W comma\subseteq \omega ˆ{\omega }\times [ 0 , 1 ]$ suchthatk=j } open parenthesis c closing parenthesis the s e t intersection[ of openX brace W sub t : t in omega to (t, x) ∈ W ⇔ x ∈ B(p , 2(t1k)+k). the power of omega closing brace is countable period j ∞ \ centerlineProof period{( .. Let a open $ ) brace ( p sub\ f o j r closing a l l bracet sub\ inj j∈ inω omega\omega be aˆ{\ oneomega hyphen} to hyphen)( one Wsequence{ t of}$ all rationals is comeagerThen in brackleftW is in open [ 00 comma since , 1 it 1 ] can brackright ) be , written} and as let B open parenthesis p comma r closing parenthesis stand for the open ball in brackleft 0 sub comma 1\ [ brackright ( b with ) centre ( p\ andf o r radius a l l r periodt , Let tthe ˆ{\prime }\ in∞ \omega ˆ{\omega } )set ( W subset t \ equalleq omegat ˆ to{\ theprime power}\ of omegaRightarrow[[ times brackleft−W1 0{ commat }\ 1 brackrightsupseteq be givenW by{ t } W = fj,m[(−∞, 0)], \primeopen parenthesis), \ t] comma x closing parenthesis in W Leftrightarrow x in union of j in omega B open parenthesis p sub j comma sum from k = j to infinityj ∈ 2ω to the m = powerj + 1 of open parenthesis t 1 k ω closing parenthesis pluswhere k closing a function parenthesisfj,m period: ω × [0, 1] → R given by the formula \ centerlineThen W is open{( c since ) the it can s be e written t $ \ asbigcap \{ W { t } : t \ in \omega ˆ{\omega } \} Line$ is1 infinity countable Line 2 W . } = union of union of f sub j comma m to the power of minus 1 brackleft open parenthesis minus infinity comma 0 closing parenthesis brackright comma Line 3 j in omega m = j\ plusnoindent 1 Proof . \quad Let $ \{ p { j }\} { j }\ in \omega $ be a one − to − one sequence of all rationals in [ 0 , 1 ] and let $Bwhere a ( function p f , sub rj comma )$ m standfortheopenballin : omega to the power of omega times $[ brackleft 0 { 0 comma, } 1 1 brackright] $ with right centre arrow R $p$ given by andthe formula radius $ r . $ Let the s ef t sub $ j Wcomma\subseteq m open parenthesis\omega t commaˆ{\omega x closing}\ parenthesistimes = bar[ x minus 0 , p sub 1 j bar ]$ minus begivenby sum from k = j to m 2 to the power of open parenthesis t 1 k closing parenthesis plus k open parenthesis open\ [ ( parenthesis t , t comma xx ) closing\ in parenthesisW \Leftrightarrow in omega to the power ofx omega\ in times\ brackleftbigcup 0 comma{ j 1\ in brackright\omega closing} parenthesisB ( p { j } , \sum ˆ{ k = j } {\ infty } 2 ˆ{ ( { t } 1 { k ) } + k } ). \ ]

\noindent Then $W$ is open since it can be written as

\ [ \ begin { a l i g n e d }\ infty \\ W = \bigcup \bigcup f ˆ{ − 1 } { j , m } [( − \ infty , 0 ) ] , \\ j \ in \omega m = j + 1 \end{ a l i g n e d }\ ]

\noindent where a function $ f { j , m } : \omega ˆ{\omega }\times [ 0 , 1 ] \rightarrow R $ given by the formula

\ [ f { j , m } ( t , x ) = \mid x − p { j }\mid − \sum ˆ{ k = j } { m } 2 ˆ{ ( { t } 1 { k ) } + k } (( t , x ) \ in \omega ˆ{\omega }\times [ 0 , 1 ] ) \ ] k=j X (t1k)+k ω fj,m(t, x) =| x − pj | − 2 ((t, x) ∈ ω × [0, 1]) m ON UNIONS AND INTERSECTIONS OF MEASURABLE SETS .. 209 ONis UNIONS continuous AND period INTERSECTIONS Condition open OF parenthesis MEASURABLE a closing SETS parenthesis\quad 209 clearly follows from the fact thatis the continuous set . Condition ( a ) clearly follows from the fact that the set W sub t = union of j in omega B open parenthesis p sub j comma sum from k = j to infinity 2 to the\ [W power{ oft open} = parenthesis\bigcup t 1 k{ closingj parenthesis\ in \omega plus k} closingB parenthesis ( p { j } , \sum ˆ{ k =is jopen} {\ andinfty dense comma} 2 ˆ for{ each( { t int } omega1 to{ thek power ) } of+ omega k open} ) parenthesis\ ] thus it is comeager closing parenthesis period .. To check open parenthesis b closing parenthesis comma ON UNIONS AND INTERSECTIONS OF MEASURABLE SETS 209 is continu- consider any t commaous t . to Condition the power ( a of ) prime clearly in follows omega from to the the power fact that of omega the set with t less or equal t\noindent to the poweris of prime open period and dense Then , for each $ t \ in \omega ˆ{\omega } ( $ thus it is comeager ) . \quad To check ( b ) , sum from k = j to infinity 2 to the power of t to the powerk= ofj prime open parenthesis 1 k closing considerany $t , tˆ{\prime }\[in \Xomega ˆ{\omega }$ with $ t parenthesis plus k less or equal sum from k =W j to= infinityB(p 2, to the2(t1 powerk)+k) of open parenthesis t 1 k \ leq t ˆ{\prime } . $ Then t j closing parenthesis plus k comma j∈ω ∞ which implies that W sub t prime subset equal W sub t period Finally comma to obtain open is open and dense , for each t ∈ ωω ( thus it is comeager ) . To check ( b ) parenthesis\ [ \sum ˆ c{ closingk = parenthesis j } {\ commainfty let} us show2 ˆ{ thatt ˆ{\prime } ( 1 { k ) } + , consider any t, t0 ∈ ωω with t ≤ t0. Then k }\W toleq the power\sum of tˆ{ intersectionk = of j in} notdef{\ infty omega} notdef2 ˆ to{ the( power{ t } of =1 notdef{ k p j ) : notdef} + k } , \ ] notdef-intersectiontext in closing brace period notdefk=j k=j X 0 X side to the power of u r l a ny .. oi t e x in brackleft2t (1k 0)+ brackrightk ≤ backslash2(t1k)+k, p j sub : in closing brace W to the power of f e sha lfi y nd t in r sub omega∞ omega n s chthat∞ \noindentnegationslash-elementwhich implies t to the that power $W of W{ r tt period}\prime Fi ln sub u\subseteq mber to the powerW { oft stp} ik. s $ which implies that Wt0 ⊆ Wt. Finally , to obtain ( c ) , let us show that t-parenleftFinally 0, closing to obtain parenthesis ( c comma) , let to theus powershow of that t open parenthesis 0 closing parenthesis sub t open parenthesis sub 1 to the power of in omega comma to the power of 1 s u period c h comma t \ [ W ˆ{ t }\bigcap\ \ in notdef \omega notdef ˆ{ = } notdef p open parenthesis tha nW closingt parenthesis∈ notdefωnotdef ar t bar=notdefpj minus e sub: cnotdefnotdef p h 0 sub bar− seintersectiontext sub n to the power∈}.notdef of greaterj : 1 sub notdef s to the powernotdef of− slashintersectiontext sub o t h to the power{\ in of}\} t open parenthesis. notdef 0 closing\ ] parenthesis period .. Su p p o e to theu power of u s sub n owth at f side r l a ny oi t e x ∈ [0 ] \ pj: ∈ } W e sha lfi y nd t ∈ rωωn s u a .. closing parenthesis .. o .. a t \noindent $ s i d echthat ˆ{ u }$ r l a ny \quad o i t e $ x \ in [ 0 ] \setminus closing parenthesis6∈ barW r xt. minusFiln jstp barik greatert( = j 2 to0)∈ theω1 powersu. c h of ttha open parenthesist | −e p 0> k1 closing/t(0). p j { : }\ in t \} Wumber ˆ{ f s}t$−parenleft esha0), lfit(1 , ynd $t,t( n)ar\ in rc h {\|se nomegas oth parenthesis plus k for jSu = p 0 commaoeus owth period at u period a ) period o a comma t n period \omegaThen pickn t} open$ s parenthesis chthatp n n plus 1 closing parenthesis in omega so that Line 1 bar x minus p sub j bar greater sum) | x from− kj | =>= j toj2 nt(k 2)+ tokfor thej = power 0, ..., of n. open parenthesis t 1 k closing\noindent parenthesis$ \not plus\ kin plus{ 2 tot the}ˆ{ powerW }$ of t r open $ parenthesis t . Fi 1 1{\ plusln n closing} { u parenthesis mber } plusˆ{ stp } ni k plus{ 1s for j = t− 0parenleft sub commaThen pick period 0t(n + period ) 1) ∈ ω period , so}ˆ that{ commat ( n} comma0{ ) Line} { 2t bar x ( minus}ˆ{\ p nin plus} 1{ bar1 } greater\omega 2 to{ the, power}ˆ{ 1 of} t opens parenthesis u { . }$ 1 1 c plus $ n h closing{ , parenthesis t ( plus} ntha plus{ 1 periodn ) ar }$ k=j tWe $ \ havemid thus−{ definede } { thec sequence} p { t inh omega} 0 to{\ themid power{ se of}} omegaˆ{ > satisfying1 } { openn parenthesis} { s }ˆ{ * / }ˆ{ t X (t1k)+k t( ( 0 ) } { o t h }| x −.pj $ |>\quad2 Su+ $ 2 p11 { ) +p n}+ 1o for ej = ˆ{ 0,...,u n,}{ s } { n }$ closing parenthesis for each n in omega period If +n owthn right at arrow infinity in open parenthesis *n closing parenthesis comma we obtain u a \quad ) \quad o \quad a t | x − pn + 1 |> 2t(1 + n + 1. bar x minus p sub j bar greater equal sum from k = j to infinity 2 to the power1+n ) of open parenthesis t 1 k closing parenthesis plus k ω \ [) \mid x We− havej thus\mid defined> the sequence= jt ∈ 2ω ˆ{ satisfyingt ( (∗ k) for ) each +n ∈ kω. }If for each j in omegan period→ ∞ Hencein (∗), xwe element-negationslash obtain W sub t period for4 period j=0,. .. Vitali Spaces . . ,n. \ ] In this section comma we consider some strong version of thek=j measurability of X (t1k)+k uncountable unions of measurable sets period| Letx − openpj |≥ parenthesis2 E comma S comma I closing parenthesis\noindent beThenpick again a measurable $t ( n + 1 ) ∞ \ in \omega $ so that space with a sigmafor hyphen each idealj ∈ ω. periodHence ..xelement open parenthesis− negationslashW Recall that two. triples connected comma \ [ \ begin { a l i g n e d }\mid x − p { j }\mid > t\sum ˆ{ k = j } { n } respectively comma with 4 . Vitali Spaces 2 ˆ{ ( { t } 1 { k ) } + k } + 2 ˆ{ t ( } 1 { 1 { + { n }} measure and categoryIn are this classical section examples , we consider of a measurable some strong space version with a of the measurability of ) } + n + 1 f o r j = 0 { , } . . . , n , \\ uncountable unions of measurable sets . Let (E, S, I) be again a measurable \mid x − p n + 1 \mid > 2 ˆ{ t ( } 1 { 1 { + { n }} space with a σ− ideal . ( Recall that two triples connected , respectively , ) } + n + 1 . \end{ a l i g n e d }\ ] with measure and category are classical examples of a measurable space with a

\noindent We have thus defined the sequence $ t \ in \omega ˆ{\omega }$ satisfying $ ( ∗ )$ for each $n \ in \omega . $ I f $ n \rightarrow \ infty $ in $ ( ∗ ) ,$ weobtain

\ [ \mid x − p { j }\mid \geq \sum ˆ{ k = j } {\ infty } 2 ˆ{ ( { t } 1 { k ) } + k }\ ]

\noindent f o r each $ j \ in \omega . $ Hence $ x element−negationslash W { t } . $

\ centerline {4 . \quad Vitali Spaces }

In this section , we consider some strong version of the measurability of uncountable unions of measurable sets . Let $ ( E , S , I ) $ be again a measurable space with a $ \sigma − $ i d e a l . \quad ( Recall that two triples connected , respectively , with measure and category are classical examples of a measurable space with a 2 10 .. M period BALCERZAK AND A period KHARAZISHVILI \noindentsigma hyphen2 10 ideal\quad periodM closing . BALCERZAK parenthesis AND .. WeA . shall KHARAZISHVILI say that a family F subset equal S is .. countably$ \sigma approximable− $ if i d e a l . ) \quad We shall say that a family $ F \subseteq S $there i s exists\quad a countablecountably subfamily approximable G of F such that if union of F backslash union of G in I period \noindentPlainly commathere if a exists family is a countably countable approximable subfamily comma $ then G $ its of union $ is F in $ S but such comma that in general comma the converse is not true period \ [ \bigcup F 2\ 10setminus M . BALCERZAK\bigcup AND AG . KHARAZISHVILI\ in I.σ−\ideal] . ) We shall say In connection withthat the problem a family concerningF ⊆ S is thecountably S hyphen approximable measurabilityif for unions of uncountable subfamiliesthere exists of S a comma countable it is subfamily natural toG askof F aboutsuch useful that criteria for the countable approximability of such subfamilies provided that the ele hyphen \noindent Plainly , if a family is countably[ [ approximable , then its union is in ments of a subfamily have some additional quotedblleftF\ goodG ∈ Iquotedblright. structure period In particular$ S $ but comma , in generalwe are going , the to consider conversePlainly this , if question is a family not for trueis acountably triple . open approximable parenthesis ,E then comma its union Br open is in parenthesisS but , in E closing parenthesis commageneral K open , the parenthesis converse is notE closing true . parenthesis closing parenthesis comma where InE connection is a topological with spaceIn the comma connection problem and forwith concerning a the family problem of admissible concerningthe $ subsets S the−S− of$ Emeasurability period measurability .. As for unions for unions ofwe uncountable know open parenthesis subfamiliesof uncountable see Introduction subfamilies of $ S closing of S ,, parenthesisit $ is naturalit is comma to natural ask about every to useful admissible ask criteria about set for is useful open the criteria for modulothe countable NWD open parenthesis approximabilitycountable E approximability closing parenthesis of such of such subfamilies subfamilies provided provided that that the ele the - ments ele of− mentsand comma of a consequently subfamilya subfamily comma have have somebelongs some additional to additional Br open parenthesis “ good‘‘ good ” structure E ’’ closing structure . parenthesis In particular . periodIn , we particular are , weWe are say goingthat a Vitali togoing consider covering to consider V thisof a this given question question subset X for for of a E triple a is admissible triple (E, Br(E $if),K ((E)) E, where , Br ( EV )consists , of K admissible (E is Ea sets topological period ) ) A space space ,$where , E and iscalled for a family a Vitali of space admissible if comma subsets for ofeachE. As we set X subset equalknow E and ( for see each Introduction admissible ) , Vitali every covering admissible V of set X is comma open modulo there existsNWD a(E) \noindentdisjoint countable$ E $ familyand is , consequently a W topological subset equal , belongs V space with to Br X ,( backslashE and). for union a family of W belonging of admissible to K open subsets of parenthesis$ E . $ E closing\quad parenthesisAsWe say periodthat a Vitali covering V of a given subset X of E is admissible if V weWe know say that ( see E almost Introductionconsists satisfies of admissible the Suslin ) , every sets condition . A admissible space if thereE is exists called set a a setVitali is open space moduloif , for each $ NWD set (Z E in K open) $ parenthesisX ⊆ E Eand closing for parenthesiseach admissible such Vitali that the covering subspaceV of EX, backslashthere exists Z of a E disjoint satisfies the Suslin condition periodcountable family W ⊆ V with X \ S W belonging to K(E). \noindentClearly commaand if , a consequently topologicalWe say that spaceE , satisfiesalmost belongs the satisfies Suslin to the $Br condition Suslin ( condition comma E then )if it there .almost $ exists a set satisfies the SuslinZ condition∈ K(E) but such it is that easy the to subspace find examplesE \ Z ofof spacesE satisfies which the Suslin condition . Wesaydisprove that the converse a VitaliClearly assertion covering , if a period topological $V$ space of satisfies a given the subsetSuslin condition $X$ , then of it $E$ almost isWe admissible are going to if givesatisfies a full characterization the Suslin condition of topological but it is spaces easy to for find which examples of spaces which $any V family $ consists of admissibledisprove of sets admissible the is countablyconverse sets assertion approximable . A . space period $ For E this $ purposeis called comma a Vitali space if , for each s ewe t need $ X several\subseteq auxiliaryWe facts areE going period $ andto give for a full each characterization admissible of topologicalVitali covering spaces for which $ V $ ofWe $X begin with , $ the thereany following family exists consequence of admissible a of sets the isclassical countably Banach approximable theorem . For this purpose , we disjointon the unions countable of openneed first familyseveral category auxiliary $ sets W period facts\subseteq . V $ with $ X \setminus \bigcupLemma 4W$ period 1 belongingto periodWe .. begin Every with .. $K topological the following ( space E consequence E ) .. can .$ .. of be the .. expressed classical Banach in .. the theorem form E = E sub 0 cup Eon sub the 1 ..unions where of E open sub 0first .. and category E sub sets 1 .. . are .. disjoint s e ts comma E sub 0 .. Weis .. say an ..that open Baire $ E $Lemma almost 4 .satisfies 1 . Every the topological Suslin condition space E can if there be expressed exists a set $subspace Z \ in of E andK Ein sub ( the 1 E is form a clos ) $E ed first such= categoryE that∪ E subspacethewhere subspace ofE E periodand $EE are\setminus disjoint s eZ $ of $ E $ satisfies the Suslin condition0 1 . 0 1 Proof period .... opents ,E parenthesis0 is Cf an period open .... brackleft Baire subspace 8 brackright of E commaand E Theorem1 is a clos 2 period ed first 4 periodClearly closing , parenthesis if a topologicalcategory According subspace to space the of BanachE. satisfies Category the Theorem Suslin condition , then it almost satisfiesopen parenthesis the see SuslinProof brackleft . condition( 1 Cf brackright . [ 8 but ] ,comma Theorem it isTheorem 2 easy . 4 . 1 ) to 6 According period find 1 examples closing to the parenthesis Banach of Categoryspaces comma which wedisprove have the equality the converse ETheorem = E to the assertion power of prime . cup E to the power of prime prime where E to the power of prime is the ( see [ 1 ] , Theorem 1 6 . 1 ) , we have the equality E = E0 ∪ E00 where E0 is Welargest are going open parenthesis to givethe with a full respect characterization to inclusion closing parenthesis of topological open meager spaces set in for E comma which andany E to family the power of of admissiblelargest prime prime ( with = sets respect E backslash is to countably inclusion E to the ) open power approximable meager of prime set inperiodE, . Forand E this00 = E purpose\ E0. , we need several auxiliary facts0 . 0 0 Let us put E sub 1Let = Eus to put theE1 power= E ∪ ofbd prime (E ),E cup0 = bdE open\ E1. parenthesisSince bd (E E) to∈ theNWD power(E), ofwe prime closing parenthesis commahave EE1 sub∈ K 0( =E) E and backslash , moreover E sub,E 11 periodis closed Since in E. bd openHence parenthesisE0 is open E to in theE Wepower begin of prime with closing theand parenthesis following , by the in maximality NWD consequence open of parenthesisE0, we of can the infer E closing classical that parenthesisE is a Banach Baire comma space theorem we. on the unions of open first category sets . 0 have E sub 1 in K openThe parenthesis set E0 inE the closing above parenthesis - mentioned and expression comma moreoverE = E0 ∪ commaE1 will Ebe sub called 1 is closed in E period .... Hencea Baire E kernel sub 0 isof open the inspace E E. Obviously , the Baire kernel of E is unique Lemmaand comma 4 . 1 by . the\quadmodulo maximalityEvery the σ of\−quad Eideal to theKtopological(E power). of prime space comma $ we E can $ infer\quad thatcan E sub\quad 0 is abe \quad expressed in \quad the form Baire$ E space = period E { 0 }\cup E { 1 }$ \quad where $ E { 0 }$ \quad and $ EThe{ set1 E}$ sub\ 0quad in theare above\quad hyphendisjoint mentioned expressionse ts E$ = , E sub E 0{ cup0 E}$ sub\ 1quad will bei s called\quad an \quad open Baire subspacea Baire kernel of of $E$ the space and E period $E ..{ Obviously1 }$ comma is a clos the Baire ed kernel first of category E is unique subspace of $ Emodulo . $ the sigma hyphen ideal K open parenthesis E closing parenthesis period \noindent Proof . \ h f i l l ( Cf . \ h f i l l [ 8 ] , Theorem 2 . 4 . ) According to the Banach Category Theorem

\noindent ( see [ 1 ] , Theorem 1 6 . 1 ) , we have the equality $E = E ˆ{\prime }\cup E ˆ{\prime \prime }$ where $ E ˆ{\prime }$ i s the

\noindent largest ( with respect to inclusion ) open meager set in $ E , $ and $ E ˆ{\prime \prime } = E \setminus E ˆ{\prime } . $

\noindent Let us put $ E { 1 } = E ˆ{\prime }\cup $ bd $ ( E ˆ{\prime } ),E { 0 } = E \setminus E { 1 } .$ Sincebd $( Eˆ{\prime } ) \ in NWD ( E ) , $ we

\noindent have $ E { 1 }\ in K ( E )$ and,moreover $, E { 1 }$ is closed in $E . $ \ h f i l l Hence $ E { 0 }$ is open in $E$

\noindent and , by the maximality of $ E ˆ{\prime } , $ we can infer that $ E { 0 }$ is a Baire space .

The s e t $ E { 0 }$ in the above − mentioned expression $ E = E { 0 } \cup E { 1 }$ will be called a Baire kernel of the space $E . $ \quad Obviously , the Baire kernel of $E$ is unique modulo the $ \sigma − $ ideal $K ( E ) .$ ON UNIONS AND INTERSECTIONS OF MEASURABLE SETS 2 1 1 Lemma 4 . 2 . For a topological space E, the following ass e rtions are

equivalent : (1)E almost satisfies the Suslin condition ; ( 2 ) the Baire kernel E0 of E satisfies the Suslin condition ; ( 3 ) for each disjoint family F of open s e ts in E, there exists a countable family G ⊆ F such that S F\ S G ∈ K(E). The proof of this lemma easily follows from Lemma 4 . 1 and will be omit - ted . Lemma 4 . 3 . Let Z be a subset of a given topological space E. Suppose als o that V is an admissible Vitali covering of Z. Then there exists a disjoint family W ⊆ V such that Z \ S W is a nowhere dense s e t . Proof . ( Cf . [ 9 ] , Proposition 1 . ) We apply Zorn ’ s lemma to choose W as a maximal ( with respect to inclusion ) disj oint subfamily of V. S To get the assertion , we suppose to the contrary that int ( cl (Z \ W)) = ∅. Hence we can pick a nonempty open set U ⊆ cl (Z \ S W). The latter relation implies that Z is dense in U and U∩ ⇒ notdefW T −parenright−notdef notdef−equal∅Fnotdef− f r notdefexistential−eT −notdef−negationslash−cbraceleft−hexistential− notdefW − notdefnotdef − element − notdefW − T − notdefunion − period − notdefS n − notdefbackslash−cenotdefa−notdef lse ts om W a e a dmssible , w ea so h ve U∩ ⇒ notdef − T notdef∅r − Fe − notdef − existentialac−notdef ∃W − negationslash−notdef−T {notdef−existentialnotdefperiod − elementnotdef− T − P − notdefc k notdefW i U∩ arrowdblright−periodS − notdefn − notdef − cnotdef V −F∀existential−notdef−isnotdef T − notdef − negationslash − existentialbraceleft − it−existential−notdef notdef − l − inotdef−notdef−c−element−negationslasho−T−notdefve−notdef r a∪n − notdef g notdef of Z, th e − r ex i − s ts V ∈∈ V u h e ∈ V ⊆ . T e n WV∪{V f rms adij i t su bfa m i − l y of V wh h − c di c − t s the Um axm aityof W. o Now , we can formulate and prove Theorem 4 . 1 . For a topological space E, the following three ass ertio ns are equivalent : (1)E almost satisfies the Suslin condition ; (2)E is a Vitali space ; ( 3 ) ea ch family F of admissible subsets of E is countably approximable . Proof . (1) ⇒ (2). Suppose that ( 1 ) is true . Let Z ⊆ E and let V be an admissible Vitali covering of Z. By Lemma 4 . 3 , there exists a disj oint family W ⊆ V such that Z \ S W ∈ NWD(E). From Lemma 4 . 2 it follows that there exists a countable family W∗ ⊆ W such that S{ int (X): X ∈ W}\ S{ int (X): X ∈ W∗} ∈ K(E). Hence we easily infer that Z \ S W∗ ∈ K(E). Consequently ,E is a Vitali space . (2) ⇒ (3). Suppose that ( 2 ) is true . Let F be a family of admissible subsets of E. We put Z = S{ int (X): X ∈ F}. Then S F\ Z ∈ NWD(E) ON UNIONS AND INTERSECTIONS OF MEASURABLE SETS .. 2 1 1 \ hspaceLemma∗{\ 4 periodf i l l }ON 2 period UNIONS .. For AND a .. INTERSECTIONS topological space .. OF E commaMEASURABLE .. the following SETS \quad .. ass e2 rtions 1 1 .. are \ hspaceequivalent∗{\ :f i l l }Lemma 4 . 2 . \quad For a \quad topological space \quad $ E , $open\quad parenthesisthe 1following closing parenthesis\quad Eass almost e r satisfies t i o n s the\quad Suslinare condition semicolon open parenthesis 2 closing parenthesis the Baire kernel E sub 0 of E satisfies the Suslin condition \ begin { a l i g n ∗} semicolon ( compare with the proof of Theorem 2 . 1 ) . Let equivalentopen parenthesis : 3 closing parenthesis for each disjoint family F of open s e ts in E comma there \end{ a l i g n ∗} V = {V : V is open in EAmpersand(∃X ∈ F)(V ⊆ int (X))}. exists a countable Clearly , V forms an admissible Vitali covering of Z. Since E is a Vitali family G subset equal F such that union of F backslash union of G in K open parenthesisS E closing \ centerline { $ (space 1 , there ) exists E$ a countablealmost satisfies disj oint family theW Suslin ⊆ V such condition that Z \ W ; ∈} parenthesis period K(E). For each W ∈ W, choose X ∈ F such that W ⊆ int (X ) and The proof of this lemma easily follows from Lemma 4 periodW 1 and will be omit hyphen W \ centerlineted period {( 2 ) the Baire kernel $ E { 0 }$ of $ E $ satisfies the Suslin condition ; } Lemma 4 period 3 period .. Let Z be a subset of a given topological space E period .. Suppose ( 3als ) o forthatV each is an disjoint admissible Vitali family covering $ F of $ Z period of open .. Then s e there ts exists in $E a disjoint , $ there exists a countable familyfamily W $ subset G equal\subseteq V such thatF$ Z backslash such that union of $ W\bigcup is a nowhereF dense\setminus s e t period \bigcup G Proof\ in periodK .. ( open E parenthesis ) . Cf $ period .. brackleft 9 brackright comma Proposition 1 period closing parenthesis .. We apply Zorn quoteright s lemma to choose W as \ hspacea maximal∗{\ f open i l l } parenthesisThe proof with of respect this lemma to inclusion easily closing follows parenthesis from disj Lemma oint subfamily 4 . 1 and of V will be omit − period .. To get the \noindentassertion commated . we suppose to the contrary that int open parenthesis cl open parenthesis Z backslash union of W closing parenthesis closing parenthesis = varnothing period Hence we Lemmacan pick 4 . a 3 nonempty . \quad openLet set $ UZ subset $ be equal a cl subset open parenthesis of a given Z backslash topological union of spaceW closing $ E parenthesis. $ \quad periodSuppose The latter relation implies alsthat o Z that is dense $V$ in U and is U capan doubleadmissible stroke right Vitali arrow covering notdef W to of the $ power Z of . T-parenright- $ \quad Then there exists a disjoint notdeffamily notdef-equal $ W varnothing\subseteq F notdef-fV$ r such notdef that existential-e $Z T-notdef-negationslash-c\setminus \bigcup braceleft-hW $ existential-notdefis a nowhere W-notdefdense s notdef-element-notdef e t . W-T-notdef union-period-notdef sub S n-notdef sub backslash-c e notdef a-notdef lse ts \noindentom W a e aProof dmssible . comma\quad w( ea Cf so h . ve\ Uquad cap double[ 9 ] stroke , Proposition right arrow notdef-T 1 . ) notdef\quad varnothingWe apply Zorn ’ s lemma to choose r-F$ W e-notdef-existential $ as sub a c-notdef exists W-negationslash-notdef-T open brace notdef-existential notdefa maximal period-element ( with notdef-T-P-notdef respect to inclusion sub c .. k notdef ) disj W oint subfamily of $ V . $ \quadU capTo arrowdblright-period get the S-notdef n-notdef-c sub notdef V-F sub forall existential-notdef-i s notdef toassertion the power of i, T-notdef-negationslash-existential we suppose to the contrary braceleft-i that subint t-existential-notdef ( cl $ ( Z notdef-l-i\setminus notdef- notdef-c-element-negationslash\bigcup W ) ) = sub o-T-notdef\ varnothing v e-notdef . $r a cup Hence n-notdef we g notdef of Z comma th e-r .. ex i-s ts V in in V u .. h \noindentin V subsetcan equal pick period a T nonempty e n .. W V open sub cup set open $ brace U V\subseteq f rms .. adij$ i t to c l the $ power ( of Z e su\ ..setminus bfa\bigcup m i-l y ofW V .. wh ) h-c . $ The latter relation implies di c-t s the U sub m axm aityof W period .. o \noindentNow commathat we can $Z$ formulate is and dense prove in $U$ and $U \cap \Rightarrow notdefTheorem W 4 period ˆ{ T− 1parenright period .. For−notdef a topological} notdef space E− commaequal the following\ varnothing three ass Fertio ns notdef−f $ rare $ notdefequivalent : existential −e T−notdef−negationslash −c b r a c e l e f t −h existential −notdef W−notdefopen parenthesis notdef 1 closing−element parenthesis−notdef E almost W−T satisfies−notdef the Suslin union condition−period semicolon−notdef { S } n−notdefopen parenthesis{ backslash 2 closing− parenthesisc e notdef E is a Vitali a− spacenotdef semicolon}$ l s e t s omopen $W$ parenthesis a e 3 a closing dmssible parenthesis ,wea ea ch so family hve F of $U admissible\cap subsets\Rightarrow of E is countablynotdef ap- −T proximablenotdef period\ varnothing r−F e−notdef−existential { a c−notdef }\ exists W−negationslashProof period open−notdef parenthesis−T 1 closing\{ notdef parenthesis−existential double stroke right notdef arrow open period parenthesis−element 2 closingnotdef parenthesis−T−P−notdef period{ Supposec }$ that\quad openk parenthesis $ notdef 1 closing W $ parenthesis is true period .. Let Z subset equal E and let V be an \noindentadmissible Vitali$ U covering\cap of Zarrowdblright period By Lemma−period 4 period 3 commaS−notdef there exists n−notdef a disj− ointc family{ notdef } V−FW ˆ subset{ i } equal{\ Vf o such r a l l that Zexistential backslash union−notdef of W in− NWDi s open notdef parenthesis} ET− closingnotdef parenthesis−negationslash −existential periodb r a c e .. l e From f t −i Lemma{ t− 4existential period 2 it follows−notdef that } notdef−l−i notdef−notdef−c−element−negationslash { o−T−notdef vthere e−notdef exists a countable}$ r $a family{\cup W to} then power−notdef$ of * subset g equal $notdef$ W such that of union $Z of open ,$ brace th int$ e open−r $ parenthesis\quad ex X closing $ i− parenthesiss $ t s : $ X V in W\ closingin brace\ in backslashV $ u \quad h union of open brace int open parenthesis X closing parenthesis : X in W to the power of * closing brace\noindent in K open$ parenthesis\ in V E closing\subseteq parenthesis. period $ THence e n \ wequad easily$ infer W that V Z{\ backslashcup union\{} Vof W $ to f the rms power\quad of * ina d K i j open i parenthesis$ t ˆ{ e E}$ closing su parenthesis\quad bfa period m $ i−l $ y o f $ V $ \quadConsequentlywh $ h comma−c $ E is a Vitali space period diopen $ parenthesis c−t $ s 2 the closing $ parenthesis U { m } double$ axmaityof stroke right arrow $W open . $parenthesis\quad 3o closing paren- thesis period Suppose that open parenthesis 2 closing parenthesis is true period .. Let F be a family of admissible\ centerline {Now , we can formulate and prove } subsets of E period We put Z = union of open brace int open parenthesis X closing parenthesis : X in\ hspace F closing∗{\ bracef i l l period}Theorem Then union 4 . 1 of. F backslash\quad For Z in a NWD topological open parenthesis space E closing $ E parenthesis , $ the following three ass ertio ns open parenthesis compare with the proof of Theorem 2 period 1 closing parenthesis period Let \noindentV = openare brace equivalent V : V is open :in E Ampersand open parenthesis exists X in F closing parenthesis open parenthesis V subset equal int open parenthesis X closing parenthesis closing parenthesis closing brace\ centerline period { $ ( 1 ) E$ almost satisfies the Suslin condition ; } Clearly comma V forms an admissible Vitali covering of Z period .. Since E is a Vitali \ centerlinespace comma{ there$( exists 2 a countable ) E$ disj isaVitalispace; oint family W subset equal} V such that Z backslash union of W in \ centerlineK open parenthesis{( 3 ) E ea closing chparenthesis family $ period F $ .... of For admissible each W in W subsets comma choose of X $ sub E $ W in is F countably approximable . } such that W subset equal int open parenthesis X sub W closing parenthesis and \noindent Proof $ . ( 1 ) \Rightarrow ( 2 ) .$ Supposethat(1) is true . \quad Let $ Z \subseteq E$ and let $V$ bean admissible Vitali covering of $ Z . $ By Lemma 4 . 3 , there exists a disj oint family $ W \subseteq V$ such that $Z \setminus \bigcup W \ in NWD ( E ) . $ \quad From Lemma 4 . 2 it follows that

\noindent there exists a countable family $Wˆ{ ∗ } \subseteq W $ such that $ \bigcup \{ $ i n t $ ( X ) : X \ in W \}\setminus $

\noindent $ \bigcup \{ $ i n t $ ( X ) : X \ in W ˆ{ ∗ } \} \ in K ( E ) . $ Henceweeasily infer that $Z \setminus \bigcup W ˆ{ ∗ } \ in K ( E ) . $ Consequently $ , E $ is a Vitali space .

$ ( 2 ) \Rightarrow ( 3 ) .$ Supposethat(2) is true . \quad Let $ F $ be a family of admissible subsets of $E .$ Weput $Z = \bigcup \{ $ i n t $ ( X ) :X \ in F \} . $ Then $ \bigcup F \setminus Z \ in NWD ( E ) $

\noindent ( compare with the proof of Theorem 2 . 1 ) . Let

\ centerline { $ V = \{ V : V$ is openin $E Ampersand ( \ exists X \ in F)(V \subseteq $ i n t $ ( X ) ) \} . $ }

\noindent Clearly $ , V $ forms an admissible Vitali covering of $ Z . $ \quad Since $E$ is a Vitali space , there exists a countable disj oint family $W \subseteq V $ such that $ Z \setminus \bigcup W \ in $

\noindent $ K ( E ) . $ \ h f i l l For each $ W \ in W , $ choose $ X { W }\ in F$ such that $W \subseteq $ i n t $ ( X { W } ) $ and 2 1 2 .. M period BALCERZAK AND A period KHARAZISHVILI \noindentput G = open2 1 brace 2 \quad X subM W . : BALCERZAK W in W closing AND brace A . period KHARAZISHVILI .... Then G is a countable subfamily of F and \noindentunion of Fput backslash $ G union = of G\{ in K openX { parenthesisW } :W E closing\ in parenthesisW period\} ... We $ have\ h f thusi l l Then proved$G$ that is the a family countable F is countably subfamily of $ F $ and approximable period \noindentopen parenthesis$ \bigcup 3 closing parenthesisF \setminus double stroke\bigcup right arrowG open\ in parenthesisK(E) 1 closing paren- 2 1 2 M . BALCERZAK AND A . KHARAZISHVILI thesis. $ period\quad SupposeWe have that thus open parenthesisproved that 3 closing the parenthesis family $ is Ftrue $ period is countably Let F be an arbitrary approximable . put G = {XW : W ∈ W}. Then G is a countable subfamily of F disj oint family and of open sets in E periodS F\ S ....G Then ∈ K(E F) consists. We have of admissible thus proved sets that period the .... family By assumptionF is countably open parenthesis\ hspace ∗{\ 3 closingf i l l } parenthesis$ ( 3 comma ) \Rightarrow ( 1 ) . $ Supposethat (3) is true . Let $ F $ be an arbitraryapproximable disj . oint family there exists a countable family(3) ⇒ G(1) subset. Suppose equal thatF such ( 3 that ) is true union . Let of FF backslashbe an arbitrary union of disj G oint in K open parenthesis E closing parenthesis period .... This comma \noindent of openfamily sets in $E . $ \ h f i l l Then $ F $ consists of admissible sets . \ h f i l l By assumption ( 3 ) , by Lemma 4 periodof 2 open comma sets immediately in E. Then yieldsF consists assertion of admissible open parenthesis sets . By 1 assumption closing parenthesis ( 3 ) , period there exists a countable family G ⊆ F such that S F\ S G ∈ K(E). This , \noindentFinally commathere we existswish to note a countable that the last family theorem can $ G be generalized\subseteq open parenthesisF $ such under that $ \bigcup F \bysetminus Lemma 4 . 2\ ,bigcup immediatelyG yields\ in assertionK ( 1 ( ) . E ) . $ \ h f i l l This , certain natural assumptionsFinally comma , wewish .. of courseto note closing that the parenthesis last theorem to the can situation be generalized of a measurable ( under space with a sigma hyphen ideal period .. This generalization can be obtained by using some \noindent by Lemmacertain 4 . natural 2 , immediately assumptions , yields of course assertion ) to the situation ( 1 ) of . a measurable abstract versions ofspace the Banach with a Categoryσ− ideal Theorem. This generalizationperiod One of suchcan be versions obtained is by using some contained comma for instance comma in the monograph by Morgan brackleft 1 0 brackright period \ hspace ∗{\ f i l l } Finallyabstract versions , we wish of the to Banach note Category that the Theorem last .theorem One of such can versions be generalized is ( under References contained , for instance , in the monograph by Morgan [ 1 0 ] . 1 period .. J period C period .. Oxtoby comma .. Measure and category period .. Springer hyphen References Verlag\noindent commacertain .. New .. York natural comma assumptions , \quad of course ) to the situation of a measurable space with a $ \sigma1 . J .− C .$ Oxtoby i d e a l , . \ Measurequad This and category generalization . Springer can - Verlag be obtained , by using some 1 980 period New York , 1 980 . abstract2 period .. versions J period Kupka of the and Banach K period Category Prikry comma Theorem The measurability . One of of such uncountable versions unions is contained , for instance2 . J . Kupka , in and the K monograph . Prikry , The by measurability Morgan [ of 1 uncountable 0 ] . unions . period Amer . Math . Monthly 9 1 ( 1 984 ) , No . 2 , 85 – 97 . Amer period .. Math period .. Monthly 9 1 open parenthesis 1 984 closing parenthesis comma No \ centerline { References3 . H .} Hadwiger , Vorlesungenu ¨ ber Inhalt , Oberfla ¨ che und Isoperimetrie period 2 comma 85 endash. Springer 97 period - Verlag , Berlin , 1 957 . 3 period H period Hadwiger4 . M .comma Goldstern Vorlesungen , An application u-dieresis of berShoenfield Inhalt comma’ s absoluteness Oberfl a-dieresis theorem cheto und1 . Isoperimetrie\quad J.C. period\quad Oxtoby , \quad Measure and category . \quad Springer − Verlag , \quad New \quad York , 1 980 . the theory of uniform distribution . Monatsh . Math . 1 16 ( 1 993 ) Springer hyphen Verlag, 237 comma – 243 . Berlin comma 1 957 period 4 period M period Goldstern5 . K comma . Kuratowski An application , Topology of Shoenfield , volumes quoteright 1 , 2 . Academic s absoluteness Press theorem , New to2 . \quad J . Kupka and K . Prikry , The measurability of uncountable unions . Amer . \quad MathYork . ,\quad1 966 ,Monthly 1 968 . 9 1 ( 1 984 ) , No . 2 , 85 −− 97 . the theory of uniform6 . distribution S . Saks , Theory period of .. the Monatsh integral period . G .. . Math E . Stechert period .. , New1 16 open York parenthesis , 1 937 . 1 993 closing parenthesis comma7 . J . 237Dieudonn endashe ´ 243Foundations period of modern analysis . Academic Press , 3 .5 Hperiod . Hadwiger .. K period , Kuratowski Vorlesungen comma $ Topology, \ddot{ commau} $ volumes ber Inhalt 1 comma , 2Oberfl period .. $ Academic\ddot{a} $ che und IsoperimetrieNew York . , 1 960 . Press comma New York comma8 . J . Cichon ´ A . Kharazishvili , and B . W e − cedilla glorz , Subsets Springer1 966 comma− Verlag 1 968 period , Berlin , 1, 957 . of the real line , part 1 . L o´ d z´ University Press , L o´ d z´,1995. 6 period S period Saks9 comma . M . OrlandoTheory of , Vitalithe integral covers period and category .. G period . EExposition period Stechert . Math comma . New4 . MYork . comma Goldstern 1 937 period , An application of Shoenfield ’ s absoluteness theorem to the theory of uniform1 1 ( 1 993 distribution ) , 1 63 – 1 67 . . \quad Monatsh . \quad Math . \quad 1 16 ( 1 993 ) , 237 −− 243 . 7 period J period Dieudonn1 0 . acute-e J . C .sub Morgan comma II Foundations, Point set theory of modern . Marcel analysis Dekker period , Inc .. Academic . , New Press comma New York and Basel , 1 990 . 5 .York\quad commaK .1 960 Kuratowski period , Topology , volumes 1 , 2 . \quad Academic Press , New York , 1 966 , 1 968 . ( Received 9 . 4 . 1 997 ) 8 period .. J period Cicho n-acute sub commaAuthors A period ’ addresses Kharazishvili : comma and B period W e-cedilla glorz comma Subsets of the real line comma \ centerline {6 . S . Saks , TheoryMarek Balcerzak of the integral Alexander . Kharazishvili\quad G . E . Stechert , New York , 1 937 . } part 1 period .. L acute-o dInstitute z-acute University of Mathematics Press comma Institute L acute-o of Applied d z-acute Mathematics sub comma 1 995 period Lo ´ dz ´ Technical University Tbilisi State University 7 .9 periodJ . Dieudonn M period Orlando $ \acute comma{e} Vitali{ , covers}$ and Foundations category period of .. modern Exposition analysis period .. . Math\quad Academic Press , New York , 1 960 . al . Politechniki 1 1 , 90 - 924 Lo ´ dz ´ 2, University St . , Tbilisi 380043 period .. 1 1 open parenthesis 1 993 closing parenthesisPoland comma Georgia 1 63 endash 1 67 period 8 .1 0\quad periodJ .. J . period Cicho C period $ \acute Morgan{n} II comma{ , }$ Point A set . theoryKharazishvili period .. Marcel , andB Dekker .W comma $ e−c e d i l l a $ Incglorz period , comma Subsets New of York the real line , partand Basel 1 . \ commaquad 1L 990 $ period\acute{o} $ d $ \acute{z} $ University Press , L $ \acute{o} $ dopen $ \acute parenthesis{z} Received{ , } 91 period 995 4 period . $ 1 997 closing parenthesis Authors quoteright addresses : 9 .Marek M . Balcerzak Orlando .. , Alexander Vitali Kharazishvili covers and category . \quad Exposition . \quad Math . \quad 1 1 ( 1 993 ) , 1Institute 63 −− 1 of 67Mathematics . .. Institute of Applied Mathematics L acute-o d acute-z Technical University .. Tbilisi State University 1 0al .period\quad PolitechnikiJ . C . 1 1Morgan comma 90 II hyphen , Point 924 Lset acute-o theory d z-acute . \quad 2 commaMarcel University Dekker St period , Inc . , New York commaand Basel Tbilisi 380043 , 1 990 . Poland .. Georgia \ centerline {( Received 9 . 4 . 1 997 ) }

\ centerline { Authors ’ addresses : }

\ centerline {Marek Balcerzak \quad Alexander Kharazishvili }

\ centerline { Institute of Mathematics \quad Institute of Applied Mathematics }

\ centerline {L $ \acute{o} $ d $ \acute{z} $ Technical University \quad Tbilisi State University }

\ centerline { al . Politechniki 1 1 , 90 − 924 L $ \acute{o} $ d $ \acute{z} 2 , $ University St . , Tbilisi 380043 }

\ centerline {Poland \quad Georgia }