REMARKS ON SMALL SETS ON THE REAL LINE

Transcription

1 Tatra Mt. Math. Publ , 7 80 DOI: /v t m Mathematical Publicatios REMARKS ON SMALL SETS ON THE REAL LINE Ma lgorzata Filipczak Elżbieta Wager-Bojakowska ABSTRACT. We cosider two kids of small subsets of the real lie: the sets of strog measure zero ad the microscopic sets. There are ivestigated the properties of these sets. The example of a microscopic set, which is ot a set of strog measure zero, is give. The otio of strog measure zero set was itroduced by E. B o r e l [Bo]. Properties of these sets were ivestigated by W. S i e r p i ński, A. S. Besicovitch, F. Galvi, J. Mycielski, R. M. Solovay, ad others. Defiitio. AsetE R is a strog measure zero set if for each sequece {ɛ } N of positive real umbers there exists a sequece of itervals {I } N such that E = I ad m I <ɛ for N. Sometimes, i the defiitio of strog measure zero set, istead of E = I, oe demads that E lim sup I,where lim sup I = I. p= =p The followig theorem shows that both coditios are equivalet. Theorem. The followig coditios are equivalet: i E is a strog measure zero set; ii for each sequece {η } N of positive real umbers there exists a sequece {J } N of itervals such that E lim sup J ad k= m J k <η for N; iii for each sequece {δ } N of positive real umbers there exists a sequece {I } N of itervals such that E lim sup I ad m I <δ for N. Proof. i = ii. Let E be a strog measure zero set ad let {η } N be a arbitrary sequece of positive real umbers. Put θ m =mi{η,...,η m } for m N M a t h e m a t i c s Subject Classificatio: 28A05, 28A75. K e y w o r d s: microscopic set, set of strog measure zero. 7

2 MA LGORZATA FILIPCZAK ELŻBIETA WAGNER-BOJAKOWSKA Obviously, the sequece {θ m } m N is oicreasig ad θ m η m for m N. Let [a k ] k N, N be a arbitrary ifiite matrix of positive real umbers such that a k k= = for example, a k = 2 k+ for k, N. Let us cosider a fuctio ψ : N N N defied as follows: { ψ k, =2 k 2 for k, N N. It is ot difficult to prove that ψ is a bijectio. Put ɛ k = a k θ ψk, for k, N. 2 Let k be a fixed positive iteger. From i it follows that for the sequece ɛ k } N there exists a sequece { E = I k } N I k ad m of itervals such that I k <ɛ k for N. Letm N. Siceψ is a oe-to-oe correspodece betwee N N ad N, there exists exactly oe pair k, N N such that ψ k, =m. The put J m = I k. Obviously, E lim sup J m, sice each poit of E belogs to ifiitely may of m itervals I k,, k N. Let p N. PutA p = { k, N N : ψ k, p }. Usig, 2, ad we obtai m J m = k, A p m = I k < a k θ ψk, k, A p = a ψ mθ m θ p ɛ k k, A p a ψ m θ p η p. The other implicatios are obvious. The otio of microscopic set was itroduced by J. Appell i [A]. The propertiesofthesesetswereivestigatedby J. Appell, E. D Aiello,ad M. V ä t h i [AAV] ad [A2]. 74

3 REMARKS ON SMALL SETS ON THE REAL LINE Defiitio 2. A set E R is microscopic if for each ɛ > 0thereexists a sequece of itervals {I } N such that E I ad m I <ɛ for N. = Theorem 2. The followig coditios are equivalet: i E is a microscopic set; ii for each positive umber η there exists a sequece {J } N of itervals such that E lim sup J ad m J k <η for N; k= iii for each positive umber δ there exists a sequece {I } N of itervals such that E lim sup I ad m I <δ for N. Proof. i = ii. Suppose that E is a microscopic set ad η 0,. Let ψ : N N N be the fuctio cosidered i the proof of Theorem. Put ad θ = η +η ɛ k = θ 2k for k N. 4 Let k be a fixed positive iteger. From i it follows that there exists a sequece { I k } of itervals such that N E = I k ad m I k < ɛ k. 5 Let m N. There exists a uique pair k, N N such that ψ k, =m. Put J m = I k. The E lim sup m J m.letp N ad A p = {k, N N : ψ k, p}. 75

4 MA LGORZATA FILIPCZAK ELŻBIETA WAGNER-BOJAKOWSKA Usig 5 ad 4 we obtai m J m = k, A p m = k, A p θ 2 = k I k θ 2 k 2 k, A p = = k, A p θ ψk, θ m = θp θ < k, A p ɛ k θ 2 k 2 k, A p < p θ = η p, θ sice 0 < θ<, so θ p θ. The other implicatios are obvious. Deote by P, S, M, N the family of coutable sets, strog measure zero sets, microscopic sets, ad Lebesgue measure zero sets, respectively. It is easy to see compare [BJ] ad [AAV] that both families S ad M are the σ-ideals situated betwee coutable sets ad sets of Lebesgue measure zero. Obviously, each strog measure zero set is microscopic, so P S M N. We have M N, because the classical Cator set has Lebesgue measure zero but is ot microscopic see [AAV]. If we assume CH, the P S, sice every Luzi set is a strog measure zero set which is ucoutable see [BJ]. Recall that a Luzi set is a ucoutable subset of a real lie havig coutable itersectio with every set of the first category. The costructio of such a set usig the cotiuum hypothesis was give first by L u z i 94 ad M a h l o 9, idepedetly. It is easy to see that each Luzi set is a strog measure zero set. Ideed, suppose that A is a Luzi set. Let {r } N be a sequece of all ratioal umbers ad let {ɛ } N be a arbitrary sequece of positive real umbers. The the set r ɛ 2,r + ɛ 2 76 =

5 REMARKS ON SMALL SETS ON THE REAL LINE is ope ad dese, so its complemet is a set of the first category. Cosequetly, the set B = A \ r ɛ 2,r + ɛ 2 = is coutable. Let {x } N be a sequece of all elemets of B. The A r ɛ 2,r + ɛ 2 x ɛ 2,x + ɛ 2, = = so A is a strog measure zero set. Now we will costruct a example of some set A M\S. Example. Let {r k } k N be a followig sequece of all ratioal umbers from the iterval 0, : 2,, 2, 4, 2 4, 4,... Let I k be a closed iterval cetered at a poit r k with a legth equal to for k N. k + 2k Put ad The A = I A A = = lim sup A. p= =p I A = I. p= =p for N FirstwewillprovethatA is a microscopic set. Let ɛ be a arbitrary positive umber. There exists k 0 N such that <ɛ. k 0 + 2k 0 Obviously, A I = I k0 +. =k 0 = We will show that m I k0 + <ɛ for N. 77

6 MA LGORZATA FILIPCZAK ELŻBIETA WAGNER-BOJAKOWSKA We have 2 k 0+ 2 =2 k k 0 for N, so m I k0 + = <ɛ for N. k 0 + 2k k 0 + 2k 0 Cosequetly, A is a microscopic set. Now we will prove that A is ot a strog measure zero set. We will costruct a sequece {ɛ } N of positive umbers such that for each sequece {J } N of itervals with m J <ɛ for N, wehave A \ J. ad Put Obviously, so = δ =mi{m I i :I i A } { } A = +, 2 +,..., + δ = m for N, I = for N is the ɛ-et of the iterval [0, ] for ɛ =/ +. ad the set A We will defie by iductio two sequeces {ɛ } N ad {i } N i a followig way. Put ɛ = 2 δ. There exists a positive iteger i > such that i + < 7 ɛ. Put ɛ 2 = δ i. Now suppose that the positive real umbers ɛ,...,ɛ ad positive itegers i,...,i are chose. There exists i >i such that i + < 7 ɛ. Let us put ɛ + = δ i. 78

7 REMARKS ON SMALL SETS ON THE REAL LINE Now, let {J } N be a arbitrary sequece of itervals such that m J <ɛ for N. We will fid the descedig subsequece {I k } N of the sequece {I k } k N such that for each N I k =. J j j= We have m J <ɛ = 2 δ 2 4 = 8. Simultaeously, A = I 4 I 5 I 6 ad the distace betwee I 4 ad I 6 is greater tha 4 because m I 6 <mi 4 < 8. So, there exists the compoet iterval of the set A which is disjoit with J.DeoteitbyI k if there is more tha oe such compoet iterval, we take the logest oe. Let us cosider the iterval I k ad the set A i.sicemi k δ,ithe iterval I k we ca fid at least six poits of the set A i.wehave m J 2 <ɛ 2 = δ i i +, so, card J 2 A i adj2 has o-empty itersectio with at most three compoet itervals of A i. Cosequetly, there exists a compoet iterval I k2 of the set A i such that I k2 I k ad I k2 J 2 =. Suppose that the itervals I k,i k2,...,i k such that I k I k2 I k, I kp is a compoet iterval of the set A ip for p =,..., ad I k = J j j= are chose. Let us cosider the iterval I k A i ad the set A i.wehave m J + <ɛ + = δ i i + so card J + A i. Hece, there exists a compoet iterval Ik+ of the set A i such that I k+ I k ad I k+ J + =. Cosequetly, + j= J j I k+ =. 79

8 MA LGORZATA FILIPCZAK ELŻBIETA WAGNER-BOJAKOWSKA Let x 0 = I k.thex 0 lim sup I = A. O the other had, x 0 I k,so x 0 / j= J j for each N. Cosequetly,x 0 / j= J j, i.e., x 0 A\ = J. I the previous example we have show straightly from the defiitio that A is ot a strog measure zero set. Remark that it is sufficiet to observe that A is a perfect set because o perfect set has strog measure zero compare [BJ]. REFERENCES [A] APPELL, J.: Isiemi ed operatori piccoli i aalisi fuzioale, Red. Istit. Mat. Uiv. Trieste 200, [A2] APPELL, J.: A short story o microscopic sets, Atti Sem. Mat. Fis. Uiv. Modea , [AAV] APPELL, J. D ANIELLO, E. VÄTH, M.: Some remarks o small sets, Ricerche Mat , [Bo] BOREL, E.: Sur la classificatio des esembles de mesure ulle, Bull. Soc. Math. Frace 47 99, [BJ] BARTOSZYŃSKI, T. JUDAH, H.: Set Theory: O the Structure of the Real Lie, A. K. Peters, Ltd., Wellesley, MA, 995. [M] MILLER, A. W.: Special subsets of the real lie, i: Hadbook of Set-Theoretic Topology, K. Kue, J. Vaugha, eds. North-Hollad, Amsterdam, 984, pp Received September 26, 2007 Ma lgorzata Filipczak Uiversity of Lódź Faculty of Mathematics ad Computer Sciece Baacha 22 PL Lódź POLAND malfil@math.ui.lodz.pl Elżbieta Wager-Bojakowska The College of Computer Sciece Chair of Mathematics Rzgowska 7A PL Lódź POLAND Uiversity of Lódź Faculty of Mathematics ad Computer Sciece Baacha 22 PL Lódź POLAND wager@ui.lodz.pl 80