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On Vitali Sets and Their Unions

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On Vitali Sets and Their Unions MATEMATIQKI VESNIK originalni nauqni rad 63, 2 (2011), 87–92 research paper June 2011 ON VITALI SETS AND THEIR UNIONS Vitalij A. Chatyrko Abstract. It is well known that any Vitali set on the real line R does not possess the Baire property. In this article we prove the following: Let S be a Vitali set, Sr be the image of S under the translation of R by a rational number 0 0 r and F= fSr : r is rationalg. Then for each non-empty proper subfamily F of F the union [F does not possess the Baire property. Our starting point is the classical theorem by Vitali [3] stating the existence of subsets of the real line R, called below Vitali sets, which are non-Lebesgue measurable. Recall that if one considers a Vitali set S on the real line R then each its image Sx under the translation of R by a real number x is also a Vitali set. Moreover, the family F= fSr : r is rationalg is disjoint and its union is R. Such decompositions of the real line R are used in measure theory to show the nonexistence of a certain type of measure. Recall that a subset A of R possesses the Baire property if and only if there is an open set O and two meager sets M; N such that A = (O n M) [ N. Note that the Vitali sets do not possess the Baire property. This can be easily seen from the following known properties of Vitali sets: (A) each Vitali set is not meager; (B) each Vitali set does not contain the set O n M for any non-empty open subset O of R and any meager set M. Really, if a Vitali set S possesses the Baire property then there is an open set O and two meager sets M; N such that S = (O n M) [ N. If O = ; then S = N, i.e. S is meager. But this is impossible by (A). So O 6= ; and S ⊃ O n M. However this inclusion is impossible by (B). Hence S does not possess the Baire property. The first proof of (B) belongs to Vitali (cf. [2]), another one the reader can find in Proposition 3.1. Let us say now that a subset A of R possesses Vitali property if there exist a non-empty open set O and a meager set M such that A ⊃ O n M. 2010 AMS Subject Classification: 03E15, 03E20 Keywords and phrases: Vitali set, Baire property. 88 V. A. Chatyrko We continue with the following question. S For what proper subfamilies F 0 of F do the unions F 0 not possess the Baire property (resp. the Vitali property)? Here we present characterizations for countable union of nonmeager sets of a hereditarily Lindel¨ofspace R: (1) to be the union (O nM)[N, where O is an open set and M; N are meager sets of R (Theorem 2.1); as a corollaryS we show that for each non-empty proper subfamily F 0 of F the union F 0 does not possess the Baire property (see Theorem 3.1); and (2) to have a complement in R which contains the set O n M, where O is a non- empty open set and M is a meager set of R (Theorem 2.2); as a corollary we 0 get a characterizationS of those non-empty proper subfamilies F of F which unions F 0 possess the Vitali property (see Theorem 3.2); 1. Preliminary notations Let R be the real line. For x 2 R denote by Tx the translation of R by x, i.e. Tx(y) = y + x for each y 2 R. The equivalence relation E on R is defined as follows. For x; y 2 R, let xEy if and only if x ¡ y 2 Q, where Q is the set of rational numbers. Let us denote the equivalence classes by E®; ® 2 I. Note that jIj = c. It is evident that for each ® 2 I and each x 2 E®, E® = Tx(Q). Hence every equivalence class E® is dense in R. A Vitali set here is any subset S of R such that jS \ E®j = 1 for each ® 2 I. Let S be a Vitali set. For each x 2 R put Sx = Tx(S). Note that Sx is a Vitali set. Observe also that S0 = S and Sx2 = Sx2¡x1 (Sx1 ) for any x1; x2 2 R. Moreover, if r1; r2 2 Q and r1 6= r2 then Sr1 \ Sr2 = ; and [ Sr = R: (*) r2Q Recall also that a subset A of a topological space R is called meager in R if A is the union of countably many nowhere dense in R sets, otherwise A is called nonmeager in R. It is evident that if B ½ A ½ R then B is meager in R whenever A is meager in R, and A is nonmeager in R whenever B is nonmeager in R. Our terminology follows [1] and [2]. 2. Countable unions of nonmeager sets in a hereditarily Lindel¨ofspace Let R be a topological space. Definition 2.1. Let X be a nonmeager subset in R. Put OX = IntR(ClR(X)). Note that OX 6= ;. Define further 0 X = fx 2 X \ OX : there is an open nbd V of x such that V \ X is meagerg 00 0 and X = (X \ OX ) n X . On Vitali sets and their unions 89 Lemma 2.1. Let X be a nonmeager set in R. Then the set Y = X \ (R n ClR OX ) is nowhere dense in R. Proof. Since Y ½ X (resp. Y ½ R n ClR OX ), we have IntR(ClR(Y )) ½ OX (resp. IntR(ClR(Y )) ½ R n OX ). Hence, IntR(ClR(Y )) = ;. Proposiiton 2.1. Let X be a nonmeager set in R and X0 be Lindel¨of.Then (i) the set X0 is meager; (ii) the set X n X00 is meager and X00 6= ;; (iii) for any open subset V of R such that V \X00 6= ;, the set V \X00 is nonmeager; 00 00 In particular, the set X = X \ R is nonmeager and so OX00 6= ;. 0 Proof. (i) For each x 2 X there is an open neighbourhood Vx of x such that 0 S 0 0 0 Vx \ X is meager. Note that X = fVx \ X : x 2 X g. Since X is Lindel¨off, 1 0 S 0 0 there exists a sequence fxigi=1 of points of X such that fVx \ X : x 2 X g = S1 0 0 i=1(Vxi \ X ). Note that the set Vxi \ X ½ Vxi \ X is meager for each i. So the set X0 is also meager. 00 0 (ii) Let us note that X n X = (X \ (R n ClR OX )) [ (X \ BdR OX ) [ X and each term of the union is meager. Hence, the union X n X00 is meager. Since X is nonmeager, we have X00 6= ;. (iii) Since V \ X00 6= ;, the set V \ X = V \ (X00 [ (X n X00)) = (V \ X00) [ (V \ (X n X00)) is nonmeager. By (ii) the set V \ (X n X00) is meager. Hence, the set (V \ X00) is nonmeager. Corollary 2.1. Let X be a nonmeager set in R and X0 be Lindel¨of.Then 00 (i) X ½ ClR(OX00 ); (ii) the set X n OX00 is meager; 00 (iii) for any non-empty open subset V of OX00 the set V \ X is nonmeager. Proof. Let us note that by Proposition 2.1 (iii) the set X00 is nonmeager and OX00 6= ;. 00 00 (i) Assume that Y = X n ClR(OX00 ) = X \ (R n ClR(OX00 ) 6= ;. It follows from Lemma 2.1 that the set Y is nowhere dense in R. Observe also that the set R n ClR(OX00 ) is open. Hence by Proposition 2.1 (iii), Y must be nonmeager. This is a contradiction. 00 (ii) By (i) we have X \ (R n ClR(OX00 )) ½ X n X . Now it follows from Proposition 2.1 (ii) that X \ (R n ClR(OX00 )) is meager. Hence, the set X n OX00 = (X \ (R n ClR(OX00 ))) [ (X \ BdR(OX00 )) is also meager. (iii) Since V \ X00 6= ;, it follows from Proposition 2.1 (iii) that V \ X00 is nonmeager. Theorem 2.1. Let R be a hereditarily Lindel¨oftopological space, A be a non- empty set with jAj · @0 and X(®) be nonmeager in R for each ® 2 A. Then the 90 V. A. Chatyrko S set U = f(X(®)) : ® 2 Ag is equal to the union (O n M) [ N for some open set O and some meager sets M; N iff OX00(®) n U is meager for each ® 2 A. Proof.“)”. Let us assume that U = (O n M) [ N for some open set O and ¤ some meager sets M; N, and the set A(® ) = OX00(®¤) n U is nonmeager for some ®¤ 2 A. Since U is a nonmeager set, we have O 6= ;. It follows from Proposition 2.1 (iii) that A00(®¤) is nonmeager and so the 00 ¤ ¤ set V = OA00(®¤) is non-empty. Since A (® ) ½ A(® ) ½ OX00(®¤), we have 00 ¤ V = IntR(ClR(A (® ))) ½ IntR(ClR(OX00(®¤)). But IntR(ClR(OX00(®¤))) = 00 ¤ 00 ¤ IntR(ClR(IntR(ClR(X (® ))))) = IntR(ClR(X (® ))) = OX00(®¤). So we get that 00 ¤ V ½ OX00(®¤). It follows from Corollary 2.1 (iii) that the set X (® ) \ V is non- meager. Put W = O \ V . Note that W 6= ;. Really, if O \ V = ; then we have that X00(®¤) \ V ½ X(®¤) \ V ½ U \ V = ((O n M) [ N) \ V ½ N.
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