Ideals with Smital properties
aa r X i v : . [ m a t h . GN ] F e b IDEALS WITH SMITAL PROPERTIES
MARCIN MICHALSKI, ROBERT RA LOWSKI, AND SZYMON ˙ZEBERSKI
Abstract. A σ -ideal I on a Polish group ( X, +) has Smital Property if for everydense set D and a Borel I -positive set B the algebraic sum D + B is a complement ofa set from I . We consider several variants of this property and study their connectionswith countable chain condition, maximality and how well they are preserved via Fubiniproducts. Introduction
We adopt the usual set-theoretical notation. We say that X is a Polish space if isseparable and completely metrisable topological space. Bor( X ) denotes the family ofBorel subsets of X . M ( X ) and N ( X ) are families of meager and null subsets of X respectively. Sometimes we will write briefly M and N if X is clear from the context.Let I be a σ -ideal on an Abelian Polish group ( X, +). If it is not stated otherwise, weassume that considered σ -ideals contain all singletons and have Borel bases, i.e. for each A ∈ I exists B ∈ Bor( X ) ∩ I such that A ⊆ B .For any sets A, B ⊆ X we denote the algebraic sum of these sets by A + B , i.e. A + B = { a + b : a ∈ A, b ∈ B } . We say that a set A is I -positive, if A / ∈ I . If A c ∈ I then we call it I -residual, alsodenoted by A ∈ I ⋆ . Definition 1.
We say that I has(i) a Steinhaus Property for any A, B ∈ Bor( X ) \I the set A − B has a nonemptyinterior;(ii) a Smital Property, briefly SP, if for every dense set D and every I -positive Borelset B a set B + D is I -residual;(iii) a Weaker Smital Property, briefly WSP if there exists a countable and dense set D such that for every I -positive Borel set B a set B+D is I -residual;(iv) a Very Weak Smital Property, briefly VWSP, if for every I -positive Borel set B there is a countable set D such that B + D is I -residual. Note that M and N have all of these properties.The following Proposition seems to be a folklore but we could not find the proof of thesecond part in the literature. Proposition 2.
SP is equivalent to Steinhaus Property.Proof.
Steinahus Property implies SP. Let A ∈ Bor \I and D be countable and dense.We may assume that D is a subgroup. Suppose that ( A + D ) / ∈ I ⋆ . Then ( A + D ) c / ∈ I .By Steinhaus Property ( A + D ) − ( A + D ) c contains an open neighborhood of 0. Acontradiction since 0 / ∈ ( A + D ) − ( A + D ) c . Mathematics Subject Classification.
Primary: 03E75, 28A05; Secondary: 03E17, 54H05.
Key words and phrases.
Smital property, Steinhaus property, ccc, Fubini product, maximal invariantideal, orthogonal ideals.
SP implies Steinhaus Property. Assume that A − B has an empty interior for A, B ∈ Bor \I . Then there is a countable dense set D ⊆ ( A − B ) c . It follows from SP that( B + D ) ∩ A = ∅ , a contradiction. (cid:3) Let F ⊆ X × Y . Then for x ∈ X and y ∈ YF x = { y ∈ Y : ( x, y ) ∈ F } and F y = { x ∈ X : ( x, y ) ∈ F } . Definition 3.
Let I and J be σ -ideals on Polish spaces X and Y respectively. Then wedefine a Fubini product of these ideals as follows: A ∈ I ⊗ J ⇔ ( ∃ B ∈ Bor ( X × Y ))( A ⊆ B ∧ { x ∈ X : B x / ∈ J } ∈ I ) . Notice that the above definition ensures the existence of Borel base for
I ⊗ J .The following definition is a variation of the ones found in [4] (Definition 18.5) and [6].
Definition 4.
Let X and Y be Polish spaces and let A ⊆ P ( X ) , B ⊆ P ( Y ) , C ⊆ P ( X × Y ) be families of sets. Then we say that B is C -on- A if for each set C ∈ C{ x ∈ X : C x ∈ B} ∈ A . We will be mainly interested in the case where B = J ⊆ P ( Y ) is a σ -ideal, A ∈{
Bor ( X ) , σ ( Bor ( X ) ∪ I ) } and C ∈ {
Bor ( X × Y ) , σ ( Bor ( X × Y ) ∪ I ⊗ J ) } . Here σ ( F ) is the σ -algebra generated by the family F . We will write briefly, for example, that J is Borel-on-measurable instead of Bor ( X × Y )-on- σ ( Bor ( X ) ∪ I ) if the context is clear. Example 5.
Measurable-on-measurable not necessarily implies Borel-on-Borel.Proof.
Let J = {∅} and take a Borel set B projection of which is analytic and notBorel. (cid:3) Proposition 6.
Borel-on-measurable implies measurable-on-measurable.Proof.
Assume that J is Borel on measurable. Let C ⊆ X × Y be measurable withrespect to I ⊗ J . Then C = ( B \ A ) ∪ A , where B is Borel and A , A ∈ I ⊗ J . Clearly { x ∈ X : A x / ∈ J } ∈ I , hence this set measurable. See that { x ∈ X : ( B \ A ) x / ∈ J } = { x ∈ X : B x \ A x / ∈ J } == { x ∈ X : B x / ∈ J , A x ∈ J } ∪ { x ∈ X : B x / ∈ J , A x / ∈ J , B x \ A x / ∈ J } . Since { x ∈ X : B x / ∈ J } is measurable, { x ∈ X : A x ∈ J } ∈ I ⋆ and { x ∈ X : B x / ∈ J , A x / ∈ J , B x \ A x / ∈ J } ⊆ { x ∈ X : A x / ∈ J } ∈ I . Therefore { x ∈ X : C x / ∈ J } is measurable. (cid:3) Smital and ccc
We start this section with the following Theorem.
Theorem 7.
Let I be a σ -ideal possessing WSP. Then I has ccc or cov( I ) = ω .Proof. Let I be σ -ideal with WSP and let D witness it. Let { B α : α < ω } be afamily of pairwise disjoint Borel I -positive sets. WSP implies that for each α < ω a set D + B α is I -residual. If T α<ω ( D + B α ) = ∅ , then cov( I ) = ω . On the other hand, if T α<ω ( D + B α ) = ∅ then for x ∈ T α<ω ( D + B α ) we have( ∀ α ∈ ω )( ∃ d ∈ D )( x ∈ d + B α ) . DEALS WITH SMITAL PROPERTIES 3 D is countable, hence there exist a set W ⊆ ω of cardinality ω and d ∈ D such that( ∀ α ∈ W )( x ∈ d + B α ) , which gives x − d ∈ T α ∈ W B α , a contradiction. (cid:3) We have the following remark improving the result obtained in [2].
Remark 8.
Let I be a σ -ideal possessing WSP. Then the following statements are equiv-alent:(i) For each family of sets { B α : α < ω } ⊆ B\I there exists a set W ⊆ ω of cardinality ω such that T α ∈ W B α = ∅ ;(ii) cov( I ) > ω .Proof. (ii) ⇒ (i) is a part of Theorem 7. To prove (i) ⇒ (ii) let us suppose thatcov( I ) = ω . Then there is a family of sets { A α : α < ω } ⊆ B ∩ I , for which S α<ω A α = X . Let us set for each α < ω a set e A α = S β ≤ α A α . A family { e A α : α < ω } is an ascending family of sets from B\I , which covers X . Then { e A cα : α <ω } is a descending family of I -residual sets such that for every W ⊆ ω of cardinality ω we have T α ∈ W e A cα = ∅ , which contradicts (i) . (cid:3) Preserving Smital properties via products
In [1] the authors present some results on various Smital properites in product spaces.Their setup is, in their words, as general as possible, concerned with algebras and ideals.It is not clear if they intended their results to hold for σ -algebras and σ -ideals or algebrasand ideals only. The formulation of [1, Theorem 4.2] suggests that they should since it isconcerned with the Borel algebra and the families of meager and null sets. However, inthe proof they rely on a certain property that does not hold for pairs σ -algebra - σ -idealin general, including the relevant here pair of Borel σ -algebra and the family of null sets.In this section we show explicitly that this property holds for pairs algebra-ideal andgive examples of failure of the said property in relevant contexts. Definition 9.
Let A be a product ( σ -)algebra. A pair ( A , I ) has a positive rectangleproperty (PRP) if every I -positive set A ∈ A contains an I -positive rectangle R satisfying R ⊆ A ∪ I for some I ∈ I . See that for a product ideal a positive rectangle R has positive sides. Example 10.
Assume ( ¬ CH ) . Then the pair (Bor( R ) , [ R ] ≤ ω ⊗ [ R ] ≤ ω ) does not havePRP.Proof. Let P ⊆ R be a perfect set such that P ∩ ( P + x ) is at most 1-point for x = 0 (see[5]). Let us set B = { ( x, y ) : x ∈ P ∧ y ∈ P − x }\ ( R × { } ) .B is Borel and B / ∈ [ R ] ≤ ω ⊗ [ R ] ≤ ω . If x ∈ B y , then x ∈ P and x ∈ P − y , therefore B y isat most 1-point.Let us suppose that there are sets A , A ∈ Bor( R ) \ [ R ] ≤ ω and a set K with countablymany vertical sections equal R and all other sections countable such that ( A × A ) \ K ⊆ B . We may assume that we removed big sections from A × A already and work with K that has all vertical sections countable. Since for each y ∈ A we have ( A × A ) y = A it follows that for the same points y the set K y is co-countable in A . Take Y ⊆ A ofcardinality ω < c . Then T y ∈ Y K y = ∅ , therefore there exists x such that | K x | ≥ ω , acontradiction. (cid:3) MARCIN MICHALSKI, ROBERT RA LOWSKI, AND SZYMON ˙ZEBERSKI
It is clear that the pair (Bor( R ) , M ) has PRP, since every nonmeager set possessingthe property of Baire is nonempty and open, modulo a set of the first category. Whatabout (Bor( R ) , N )? As a warm up let us recall the following folklore result. Proposition 11.
Every set E ⊆ [0 , of positive measure contains a full subset whichdoes not contain a rectangle of positive measure.Proof. Let E ⊆ [0 , have positive measure. Consider E ′ = { ( x, y ) ∈ E : x − y ∈ Q c } .To see that E ′ is full in E let us observe that E ′ x = E x ∩ ( Q c + x ) for every x ∈ [0 , Q c is co-null. Now, if A × B ⊆ E ′ , A and B of positive measure, then A − B shouldcontain a nonempty open set (Steinhaus Theorem), but clearly Q ∩ ( A − B ) = ∅ . Acontradiction completes the proof. (cid:3) This result may be improved with the following Lemma.
Lemma 12.
There exists a set F ⊆ R such that λ ( F ∩ U ) > and λ ( F c ∩ U ) > forevery nonempty open set U .Proof. Let ( B n : n ∈ ω ) be an enumeration of the basis of R . At the step 0 let C and C be two disjoint ”fat” Cantor subsets of B . At the step n + 1 let assume that we havetwo sequences of pairwise disjoint ”fat” Cantor sets ( C k : k ≤ n ) and ( C k : k ≤ n ) whichfor all i ∈ { , } and k ≤ n satisfy λ ( C ik ∩ B k ) >
0. A set B n +1 \ [ k ≤ n ( C k ∪ C k )is nonempty and open, hence it contains two disjoint ”fat” Cantor sets which we denoteby C n +1 and C n +1 . This completes the construction and F = S n ∈ ω C k is the desiredset. (cid:3) The above Lemma will serve as a tool to prove the result from [3].
Example 13 (Erd¨os, Oxtoby) . There is a set E ⊆ R such that E ∩ A × B and E c ∩ A × B have positive measure for each A, B ⊆ R of positive measure.Proof. Let F be as in the formulation of Lemma 12 and set E = { ( x, y ) ∈ R : x − y ∈ F } .Let A, B ⊆ R have a positive measure. Then λ ( E ∩ A × B ) = Z Z χ F ( x − y ) χ A ( x ) χ B ( y ) dxdy = Z Z χ F ( x ) χ A ( x + y ) χ B ( y ) dxdy == Z F λ (( A − x ) ∩ B ) dx.λ (( A − x ) ∩ B ) is a continuous non-negative function. Furthermore, it is positive on someinterval, since R R λ (( A − x ) ∩ B ) dx = λ ( A × B ), so R F λ (( A − x ) ∩ B ) dx > (cid:3) Corollary 14. (Bor( R ) , N ) does not have PRP. We may pose the following question.
Question 15.
What is the characterization of pairs of σ -ideals whose positive sets in theFubini product contain rectangles modulo a set from the ideal? Let us recall that a family A is called algebra of sets if X ∈ A and A is closed underfinite unions and complements. Proposition 16.
Let A and A be algebras and A an algebra generated by rectanglesof the form A × B , where A ∈ A and B ∈ A . Then A consists of finite unions ofrectangles. DEALS WITH SMITAL PROPERTIES 5
Proof.
First let us observe that complements of rectangles are finite unions of rectangles:( A × B ) c = ( A × B c ) ∪ ( A c × B ) ∪ ( A c × B c ) . Next, see that finite intersection of a finite union of rectangles is again a finite unionof rectangles. Let T , T ⊆ ω be finite. Let A nk , B nk be rectangles from A and A respectively for every n ∈ T and k ∈ T . Then \ n ∈ T [ k ∈ T A nk × B nk = [ f ∈ T T \ n ∈ T A nf ( n ) × B nf ( n ) .T T is finite and finite intersections of rectangles are also rectangles, hence the proof iscomplete. (cid:3) For algebra A and ideal I let A [ I ] denote the algebra generated by A ∪ I . Corollary 17.
Let A be as previously and let I be an ideal in X × Y . Then A ∈ A [ I ] \I if and only if A contains an I -positive rectangle modulo a set from I .Proof. It follows from the fact that for every set AA ∈ A [ I ] ⇔ A = [ n ∈ F R n △ I for some finite set F ⊆ ω , rectangles R n , n ∈ F , and I ∈ I . (cid:3) From now on let
A ⊆ P ( X ) , B ⊆ P ( Y ) be σ -algebras, and I ⊆ P ( X ), J ⊆ P ( Y ) σ -ideals. Definition 18.
We say that a pair ( A ⊗ B , I ⊗ J ) has Tall Rectangle Hull Property(TRHP) if for every set C ∈ A ⊗ B ( ∃ e C ∈ A , I ∈ I , J ∈ J )(( e C \ I ) × ( Y \ J ) ⊆ C ⊆ ( e C × Y ) ∪ ( I × Y ) ∪ ( X × J )) . If a set C fulfills the above condition we will say that it has TRHP witnessed by the triple ( e C, I, J ) .Analogously we define Wide Rectangle Hull Property (WRHP): ( ∃ e C ∈ B , I ∈ I , J ∈ J )(( X \ I ) × ( e C \ J ) ⊆ C ⊆ ( X × e C ) ∪ ( I × Y ) ∪ ( X × J )) . Proposition 19.
If a pair ( A ⊗ B , I ⊗ J ) have TRHP or WRHP then it has PRP.Proof. Let C ∈ A ⊗ B has TRHP witnessed by ( e C, I, J ):( e C \ I ) × ( Y \ J ) ⊆ C ⊆ ( e C × Y ) ∪ ( I × Y ) ∪ ( X × J ))and assume that C / ∈ I ⊗ J . Then e C × Y is the desired rectangle. Clearly e C × Y ⊆ C modulo a set from I ⊗ J . It is also
I ⊗ J -positive, otherwise( e C × Y ) ∪ ( I × Y ) ∪ ( X × J )) ∈ I ⊗ J and also C ∈ I ⊗ J .The proof of WRHP case is almost identical. (cid:3) Lemma 20.
The family of sets possessing TRHP is closed under countable unions andcomplements. The same is true for the family of sets possessing WRHP.Proof.
Proofs for both cases follow the same pattern, so without loss of generality let usfocus on the case of TRHP.Let C = S n ∈ ω C n and ( e C n , I n , J n ) witness TRHP for C n , n ∈ ω . Then for each n ∈ ω ( e C n \ I n ) × ( Y \ J n ) ⊆ C n ⊆ ( e C n × Y ) ∪ ( I n × Y ) ∪ ( X × J n ) . MARCIN MICHALSKI, ROBERT RA LOWSKI, AND SZYMON ˙ZEBERSKI
Then( [ n ∈ ω e C n \ [ n ∈ ω I n ) × ( Y \ [ n ∈ ω J n ) ⊆ [ n ∈ ω ( e C n \ I n ) × ( Y \ J n ) ⊆ [ n ∈ ω C n = C ⊆ [ n ∈ ω (cid:0) ( e C n × Y ) ∪ ( I n × Y ) ∪ ( X × J n ) (cid:1) ⊆ (( [ n ∈ ω C n ) × Y ) ∪ (( [ n ∈ ω I n ) × Y ) ∪ ( X × ( [ n ∈ ω J n )) . Hence, setting e C = S n ∈ ω e C n , I = S n ∈ ω I n , J = S n ∈ ω J n completes this part of the proof.Now let C = D c for D witnessing TRHP with ( e D, I, J ). We have( e D \ I ) × ( Y \ J ) ⊆ D ⊆ ( e D × Y ) ∪ ( I × Y ) ∪ ( X × J ) . Through complementation (cid:0) ( e D × Y ) ∪ ( I × Y ) ∪ ( X × J ) (cid:1) c ⊆ C ⊆ (cid:0) ( e D \ I ) × ( Y \ J ) (cid:1) c . Let us focus on the right-hand side (cid:0) ( e D \ I ) × ( Y \ J ) (cid:1) c = (( e D \ I ) × ( Y \ J ) c ) ∪ (( e D \ I ) c × Y ) ⊆ (( e D ) c × Y ) ∪ ( X × J ) ∪ ( I × Y ) . Now the left-hand side. It is an intersection of the following sets( e D × Y ) c = ( e D ) c × Y, ( I × Y ) c = ( X \ I ) × Y, ( X × J ) c = X × ( Y \ J ) , which is equal to (( e D ) c \ I ) × ( Y \ J ) . In summary (( e D ) c \ I ) × ( Y \ J ) ⊆ C ⊆ (( e D ) c × Y ) ∪ ( X × J ) ∪ ( I × Y ) . Then (( e D ) c , I, J ) witnesses TRHP for C . The proof is complete. (cid:3) Theorem 21.
Let C = A ⊗ B . Then(i) if A = I ∪ I ⋆ then ( C , I ⊗ J ) has WRHP.(ii) if B = J ∪ J ⋆ then ( C , I ⊗ J ) has TRHP.Proof. Without loss of generality consider the case ( ii ). Let us notice that rectanglesfrom C have TRHP. Indeed, if C = A × B then set e C = A, I = ∅ , J = B c if B ∈ J ⋆ e C = ∅ , I = ∅ , J = B if B ∈ J . The rest of the proof relies on Lemma 20 which allows us to perform an inductive proofover the hierarchy of sets making up the σ -algebra C . (cid:3) Proposition 22. If I ⊗ J has SP (WSP, VWSP), then I and J also have it.Proof. Let us consider the case of WSP for I . Let D be a witness that I ⊗ J has WSP.We will show that π X ( D ) witnesses WSP for I . Let A be a Borel I -positive set. Let usconsider A × Y . R = D + ( A × Y ) is I ⊗ J -residual, therefore e R = { x ∈ X : R x is J -residual } is I -residual, so R is I ⊗ J -residual. Clearly, A + π X ( D ) = e R , hence we are done. (cid:3) Theorem 23.
Let I and J possess WSP and assume one of the following properties DEALS WITH SMITAL PROPERTIES 7 (i) J is Borel-on-Borel;(ii) J measurable-on-measurable;(iii) (Bor( X × Y ) , I ⊗ J ) has PRP.Then I ⊗ J also has WSP.Proof.
Let D and D witness WSP for I and J respectively. Let B ∈ Bor( X × Y ) \I ⊗J .If any of the properties (i)-(iii) holds then a set e B = { x ∈ X : B x / ∈ J } contains a Borel, I -positive set and D + e B is I -residual. Let us observe that( D × D ) + B ⊇ [ d ∈ D [ x ∈ e B ( { d + x } × ( D + B x )) , therefore for every x ∈ D + e B the set (( D × D ) + B ) x is J -residual. Since D + e B is I -residual, the proof is complete. (cid:3) There is a straightforward generalization of Smital Property for any pair σ -algebra - σ -ideal ( A , I ), i.e. a pair ( A , I ) has SP if for every dense set D and every set B ∈ A\I the set B + D is I -residual. Theorem 24.
Let C = A ⊗ B and assume that(i) ( C , I ⊗ J ) has TRHP and ( A , I ) has SP, or(ii) ( C , I ⊗ J ) has WRHP and ( B , J ) has SP.Then ( C , I ⊗ J ) has SP.Proof. Without loss of generality assume ( i ). Let D ⊆ X × Y be dense, set D = π ( D )and let B ∈ C be I ⊗ J -positive. Then there are e B ∈ A\I and J ∈ J such that e B × ( Y \ J ) ⊆ B . It follows that D + B ⊇ D + ( e B × ( Y \ J )) ⊇ [ d ∈ D [ d ∈ D d ( d + e B ) × ( d + Y \ J )) . Therefore for every x ∈ D + e B the set ( D + B ) x contains a translation of Y \ J . By SP D + e B ∈ I ⋆ thus D + B is I ⊗ J -residual. (cid:3) Maximal invariant σ -ideals with Borel bases It has occurred that there is a surprising connection between maximal invariant σ -idealswith Borel bases and Smital properties. Proposition 25.
The following are equivalent:(i) I has VWSP;(ii) I is maximal among invariant proper σ -ideals with Borel base.Proof. ( i ) ⇒ ( ii ) : Let us suppose that J ) I is such an ideal. Let A be a Borel setfrom J \I . Then there exists a countable set D such that D + A is I -residual, therefore J -residual, hence J is not proper.( ii ) ⇒ ( i ) : Let us suppose that there is a set B ∈ Bor \I for which B + D is not I -residual for every countable set D . Consider a family I ′ = { A ∪ C : A ∈ I ∧ ( ∃ D )( C ⊆ D + B ∧ | D | ≤ ω ) } . It is an invariant proper σ -ideal satisfying I ( I ′ , which brings a contradiction. (cid:3) Proposition 26.
Let {I n : n ∈ ω } be a countable family of pairwise distinct maximalinvariant σ -ideals on X with Borel bases. Then for each n ∈ ω the σ -ideal I n is orthogonalto T k ∈ ω \{ n } I k . MARCIN MICHALSKI, ROBERT RA LOWSKI, AND SZYMON ˙ZEBERSKI
Proof.
Fix n ∈ ω . There are sets A k ∈ (Bor( X ) ∩ I n ) \I k , k ∈ ω \{ n } . By Proposition 25for each of them there is countable set C k such that A k + C k ∈ I n ∩ I ⋆k . Therefore [ k ∈ ω \{ n } ( A k + C k ) ∈ I n ∩ \ k ∈ ω \{ n } I ⋆k . (cid:3) Corollary 27.
Let I be a maximal invariant σ -ideal with a Borel base different from M and N . Then there is A ∈ I ∩ ( M ∩ N ) ⋆ . Let us now focus on X = 2 ω . The following result incorporates techniques similar tothese used in [7, Theorem 3.1]. Theorem 28.
There are c many maximal invariant σ -ideals on ω .Proof. Let { A α : α < c } be an AD family on ω , i.e. for all distinct α, β < c the set A α ∩ A β is finite. For every α < c set I α = { A ⊆ ω : ( ∃ B ∈ Bor(2 ω ) A ⊆ B ∧ ( ∃ M ∈ M (2 A α ))( ∀ x ∈ A α )( x / ∈ M → B αx ∈ N (2 ω \ A α )) } , where B αx = { y ↾ ω \ A α : y ∈ B, y ↾ A α = x } .Let us show that I α = I β if α = β . Set M ∈ M (2 A α \ A β ) \N (2 A α \ A β ) ,C = { x ∈ ω : x ↾ ( A α \ A β ) ∈ M } . Notice that C ∈ I α . The set M α = [ t ∈ Aα ∩ Aβ { y ∈ A α : y ↾ ( A α ∩ A β ) = t ∧ y ↾ ( A α \ A β ) ∈ M } is a finite union of meager sets, hence meager. For each x ∈ A α \ M α the set C αx is empty.On the other hand for each x ∈ A β C βx = { y ∈ ω \ A β : y ↾ ( A α \ A β ) ∈ M } . The above set may be considered as a product M × ω \ ( A α ∪ A β ) of non-null set and thewhole space, which is not null.Now we will show that every I α is maximal among invariant σ -ideals on 2 ω with Borelbases. Each I α is essentially M (2 A α ) ⊗ N (2 ω \ A α ). It follows that I α has WSP, thus byProposition 25 the proof is complete. (cid:3) The reasoning in the above Theorem does not translate for the case of R . We may askthe following question. Question 29.
Are M and N the only maximal invariant σ -ideals with Borel bases in R ? Question 30.
Is it true that for every set G ∈ ( M ∩ N ) ⋆ there is a countable set C suchthat C + G = R ? Positive answer to this question would also answer positively Question 29.
DEALS WITH SMITAL PROPERTIES 9
References [1] Bartoszewicz A., Filipczak M., Natkaniec T., On Smital Properties, Topology and its Applications,vol. 158 (2011), 2066-2075.[2] Cicho´n J., Szyma´nski A., Weglorz B., On intersections of sets of positive Lebesgue measure, Collo-quium Mathematicum, vol. 52, no. 2 (1987), 173-174.[3] Erd¨os P., Oxtoby J. C., Partitions of the plane into sets having positive measure in every non-nullmeasurable product set, Transactions of the American Mathematical Society, vol. 79, no. 1 (1955),91-102.[4] Kechris A. S., Classical descriptive set theory, Graduate Texts in Mathematics 156 (1995), Springer-Verlag New York, Inc.[5] Michalski M., ˙Zeberski Sz., Some properties of I -Luzin sets, Topology and its Applications, vol. 189(2015), 122-135.[6] Srivastava S. M., A course on Borel sets, Graduate Texts in Mathematics 180 (1998), Springer-VerlagNew York, Inc.[7] Zakrzewski P., On invariant ccc σ -ideals on 2 N , Acta Mathematica Hungarica, vol. 143 (2014),367–377. Email address : marcin.k.michalski@pwr,edu.pl Email address : [email protected] Email address : [email protected]@pwr.edu.pl