MICROSCOPIC SETS WITH RESPECT TO SEQUENCES OF FUNCTIONS

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1 t m Mathematical Publications DOI: 0.478/tmmp Tatra Mt. Math. Publ , MICROSCOPIC SETS WITH RESPECT TO SEQUENCES OF FUNCTIONS Grażyna Horbaczewska ABSTRACT. Consequences of replacing the geometric sequence with another in the definition of microscopic sets are considered. The notion of a microscopic set on the real line was introduced by J. A p p e l l in [] at the beginning of the st century. Thereafter, some papers were devoted to this topic [7], [9] []. Lately, a chapter on microscopic sets was published in a monography [8]. Definition. A set E R is microscopic if for each ɛ > 0thereexists a sequence of intervals {I n } I n and λi n ɛ n for n N. The family of all microscopic sets will be denoted by M. A special role in this definition is played by a geometric sequence. A question about the consequences of replacing this specific sequence with another one was raised by J. Appell, E. D Aniello and M. Väth in [3]. If we consider an arbitrary sequence here, we get a definition of a family of strong measure zero sets [5], denoted by S. Definition. AsetE R is of strong measure zero if for each sequence of positive reals {ɛ n } there exists a sequence of intervals {I n} I n and λi n ɛ n for n N. Obviously, S M. An example of a microscopic set which is not a strong measure zero set is given in [6]. c 04 Mathematical Institute, Slovak Academy of Sciences. 00 M a t h e m a t i c s Subject Classification: 8A05, 54H05, 03E5. K e y w o r d s: microscopic sets, null sets, strong measure zero sets, Cantor-type sets, comparison of σ-ideals. 37 Download Date 9//8 9:3 PM

2 GRAŻYNA HORBACZEWSKA Let f n be a sequence of increasing functions f n :0, 0, lim x 0 + f n x = 0, and there exists x 0 0, for every x 0,x 0 the series f nx is convergent and the sequence f n x is nonincreasing. Definition 3. AsetE R belongs to M fn if for each x 0, there exists a sequence of intervals {I n } I n and λi n f n x for n N. For f n x =x n, n N, wehavem fn = M. If H denotes the family of all sequences of functions with properties described above, then M fn = S. f n H Now, we may ask several questions. We would like to know for which sequences we get microscopic sets, when different sequences give different families of sets, under which condition M fn is a σ-ideal. The next theorem gives a sufficient condition for getting microscopic sets. Theorem 4. If there exists k N for every n N and for every x 0, we have x n+k f n x x n,then M fn = M. P r o o f. Since the inclusion M fn Mis obvious, we have to justify only the inverse one. Let E M.Fixx 0,. Since E Mfor x 0 = x k+, there exists a sequence of intervals {I n } I n and λi n < x 0 n for n N. Therefore λi n < x k+ n = x nk+n x n+k f n x, so E M fn. The above condition is not necessary, for example, for a sequence f n x =x n, n N, the assumption of the above theorem is not satisfied, but M fn = M. More examples of sequences of functions leading to microscopic sets can be given for: x n, f n x = where a>0, a f n x =x an, where a>0, f n x = xn, where a, na the family M fn is exactly the family of microscopic sets. 38 Download Date 9//8 9:3 PM

3 MICROSCOPIC SETS WITH RESPECT TO SEQUENCES OF FUNCTIONS For reader s convenience we show that in the last case every microscopic set belongs to M fn.leta M.Fixx 0, and find n 0 N /n 0 <x. For ɛ := there exists a sequence of intervals {I n a+ n } 0 I n and λi n ɛ n for n N. Then λi n ɛ n = n a+ 0 = n n a 0 n n n < 0 n n 0 a xn < xn n a since n n 0 >nfor n 0 >. Therefore, a M fn. One can observe an obvious sufficient condition for an inclusion between the families M gn and M fn for sequences f n,g n from the family H : x 0, g nx f n x M gn M fn. The next theorem gives a not so obvious sufficient condition. Theorem 5. If for every x 0, there exists y 0, there exists asequencep m m N of pairwise disjoint, nonempty subsets of N g m y f i x for every m N, then i P m M gn M fn. Proof. Let E M gn. Let x 0,. By our assumption, there exists y 0, and a sequence {P m } of pairwise disjoint nonempty subsets of N g m y i P m f i x. Since E M gn, for y there exists a sequence of intervals I m m N I m and λi m g m y. m N Since λi m g m y i P m f i x, we may divide the interval I m into nonoverlapping intervals J m i, i P m, λj m i f i x for i P m. Let n N. Ifn belongs to none of P m, m N, thenj n :=. Ifn P m, then J n := J n m. Therefore, I m = J n and λj n f n x, so E M fn. m N 39 Download Date 9//8 9:3 PM

4 GRAŻYNA HORBACZEWSKA Theorem 6. M fn \S for every f n H. Proof. Let I := [0, ]. We will define by induction the sequence of open intervals {J n,i }, i {,..., n }, n N, in the following way. Let k := min { k : f k k+ < 3}. Put J, := f k k +, f k k +. Then, of course, λj, > 3. Let K,, K, denote successive components of the set I \ J,. Obviously, λk,i =f k k + for i {, }. Let k := min { k : f k k+ < 3 f k k +}. Let J,, J, be two open intervals concentric with K, and K,, respectively, λj, =λj, =λk, f k k +. Let K,, K,, K,3, and K,4 denote successive components of the set I \ J, J, J,. Notice that λk,i = f k k + for i {,, 3, 4}. Let m. Assume that we have constructed the open, nonempty intervals J l,,...,j l, l concentric with K l,,...,k l, l, respectively, λj l,i =λk l, f k l k l + for l {,...,m}, i {,..., l },where k l := min { k : f k k+ < 3 f k l k l +}, for l {,...,m}. Let K m,,...,k m, m be successive components of the set I \ m l l= i= J l,i. Notice that λk m,i =f km k m + for i {,..., m }. Now, let k m+ := min { k :f k k+ < 3 f k m k m +} and Jm+,,...,J m+, m be open intervals concentric with K m,,...,k m, m, respectively, λj m+,i =λk m, f k m+ k m+ + for i {,..., m }. Let K m+,,...,k m+, m+ be successive components of the set m+ l I \ J l,i. l= i= Obviously, λk m+,i =f k m+ k m+ + for i {,..., m+ }. Let us put M := m m N i= K m,i. Now, let x 0,. There exists m 0 N k m0 <x. Let I i := K m0,i for i {,..., m 0 } and I i := for i> m 0. Obviously, M m 0 i= K m 0,i = i N I i and λi i =λk m0,i=f km 0 k m0 + f m 0 k m0 + < f m 0 x f i x for i {,..., m 0 }. Hence, M is a Cantor-type set from M fn. As a perfect set, M cannot be a strong measure zero set compare [5, Corollary 8..5], so M M fn \S. 40 Download Date 9//8 9:3 PM

5 MICROSCOPIC SETS WITH RESPECT TO SEQUENCES OF FUNCTIONS Theorem 7. Let f n, g n Hand let k n be a sequence of natural numbers chosen as in the proof of the previous theorem. Suppose that there exists δ>0 g n x <f n x for every n N and for every x 0,δ. Ifthereexists apointx 0 0, g n x 0 <f kn k n +, for every n N, then M gn M fn. P r o o f. Since the inclusion M gn M fn is obvious, we only need to show that these families are different. Consider the set M from the proof of the previous theorem. Let {I n } be any sequence of intervals λi n g n x 0. As λi g x 0 <f k, I cannot have common points with both sets K, and K,,sincef k < 3 <λj, =distk,,k,. Put P 0 := [0, ]. Let P := K,i,wherei {, } and I K,i =. Let n. We assume that for l {,...,n } we have already chosen an interval P l, P l := K l,il,wherei l {,..., l }, I l K l,il = and P l P l. The interval I n satisfies a condition λi n g n x 0 <f kn = λk n,i k n + for i {,..., n } and by the construction of M, precisely by the way of defining k n,wehaveλk n,i <λj l,j, where J l,j is a gap between the intervals K n,i and K n,i contained in P n, so one of them has no common points with I n. We denote it by P n. Therefore, we have inductively constructed a descending sequence of closed intervals P n, λp n =f kn k n + and Pn I n = for. ByCantor Theorem, there exists a point x P n.ofcoursex M and x/ I n, so M/ M gn. Theorem 8. For every f n H,thereexistsg n H S M gn M fn. P r o o f. Suffice it to put g n x :=f n xf kn k n +, for n N, wherekn is chosen as in the proof of Theorem 6. Then g n H and g n satisfies the condition from Theorem 7, since for every x 0, g n x =f n xf kn k n + <f kn k n + Therefore, by Theorem 6 and Theorem 7, we are done.. 4 Download Date 9//8 9:3 PM

6 GRAŻYNA HORBACZEWSKA l+ Remark 9. If f < 3 and f l f l l < 3 for l, then k n = n and for such a sequence f n the condition from the Theorem 7 is as follows x0 0, g nx 0 <f n. n + The sequence f n x =x n satisfies assumptions from the above remark, so for the sequence g n x = xn n+ n see the proof of Theorem 8, we have M gn M. Another sequence satisfying condition for f n x = x n is the sequence g n x =x nn, more precisely, for x 0 = 4 and for every n>, we have n g n x 0 <, so M n + x n n M. The next theorem gives a sufficient condition for a family M fn to be a σ-ideal. Theorem 0. Let f n Hfor n N. If there exists a bijection Φ: N N N for a fixed k N afunctionφ k n :=Φn, k is increasing and for every x>0 and for every k N lim n + fn fφn,k x > 0, thenm fn is a σ-ideal. P r o o f. Suffice it to show that a countable sum of sets from M fn belongs to M fn.let{a k } k N be a sequence of sets, A k M fn for k N. Fix k N and x > 0. A sequence { fn f Φn,kx } is monotonous, so denoting a k := lim n + fn fφn,k x,wehavefn fφn,k x a k > a k for every n N. By monotonocity of f n,weget ak f Φn,k x f n. For a k there exists a cover {I n k } of the set A k λi n k < f ak k n. Arrange intervals {I n }, k N in a sequence {I m} m N, where m =Φn, k. Then k N A k m N I m and λi m =λ I φn,k = λ I k n Therefore, k N A k M fn. <fn ak f Φn,k x =f m x. It is known that if I is a σ-ideal containing all singletons and G δ -generated then a family I := {A R : B Fσ IA B} is also a σ-ideal [4]. We do not know if M x n is a σ-ideal, but we can proove that M a family of sets x n which can be covered by F σ sets from M x n is a σ-ideal. 4 Download Date 9//8 9:3 PM

7 MICROSCOPIC SETS WITH RESPECT TO SEQUENCES OF FUNCTIONS Lemma. Let A k M x n and A k be compact for every k N. Then, A k M x n. k N Proof. Let x 0,. For x := x there exists a cover {I n } of A such that λi n < x n for every n N. By compactness of A,wecanchoose a finite subcover {I n } n n=.forx := x n there exists a cover {I n } of A such that λi n < x n for every n N. By compactness of A,wecanchoose a finite subcover {I n } n n=. And so on, for x k := x k n k we find a finite cover {I n k } n k n= of A k λi n k < x k n for every n N. Now, for every k N, thereexistst N Σ t i=0 n i <k Σ t+ i=0 n i, where n 0 := 0, and there exists l {,...,n t+ } k = n n t + l. Let J k = I t+ l. Then, k N A k k N J k and λj k =λ I t+ l < xt+ l = x n +n + +n t l = x n +n + +n t +l = x k, so k N A k M x n. Lemma. If A n M for every, then there exists a sequence of compact sets {D k } k N M x n x n A n D k. Proof. SinceA n M for every n N then, for every n N, there exists x n B n F σ M x n A n B n. Then, B n = l N Bn l,whereb n l is closed, hence it is a countable union of compact sets D l,n t for t N, so A n = D k. l N t N k N D l,n t k N Obviously, {D k } k N M x n. Theorem 3. M x n is a σ-ideal. P r o o f. It suffices to show that M is closed under a countable union. x n Let A n M x n for n N. By Lemma, there exists a sequence {D k} k N D k M x n and D k is compact for A n k N D k and for every k N. ByLemma,wehave k N D k M x n and, of course, k N D k F σ,so A n M x n. 43 Download Date 9//8 9:3 PM

8 GRAŻYNA HORBACZEWSKA REFERENCES [] APPELL, J.: Insiemi ed operatori piccoli in analisi funzionale, Rend. Instit. Mat. Univ. Trieste 33 00, [] APPELL, J.: A short story on microscopic sets, Atti. Sem. Mat. Fis. Univ. Modena Reggio Emilia 5 004, [3] APPELL, J. D ANIELLO, E. VÄTH, M.: Some remarks on small sets, Ric. Mat , [4] BALCERZAK, M. BAUMGARTNER, J. E. HEJDUK, J.: On certain σ-ideals of sets, Real Anal. Exchange , [5] BARTOSZYŃSKI, T. JUDAH, H.: Set Theory: On the Structure of the Real Line. A. K. Peters, Ltd., Wellesley, MA, 995. [6] FILIPCZAK, M. WAGNER-BOJAKOWSKA, E.: Remarks on small sets on the real line, Tatra Mt. Math. Publ , [7] HORBACZEWSKA, G. WAGNER-BOJAKOWSKA, E.: Some kinds of convergence with respect to small sets, Reports on Real Analysis, Conference at Rowy 003, [8] HORBACZEWSKA, G. KARASIŃSKA, A. WAGNER-BOJAKOWSKA, E.: Properties of the σ-ideal of microscopic sets, in: Traditional and Present-Day Topics in Real Analysis, Lódź University Press, Lódź, 03, pp [9] KARASIŃSKA, A. POREDA, W. WAGNER-BOJAKOWSKA, E.: Duality principle for microscopic sets, in: Monograph Real Functions, Density Topology and Related Topics, Lódź University Press, Lódź, 0, pp [0] KARASIŃSKA, A. WAGNER-BOJAKOWSKA, E.: Nowhere monotone functions and microscopic sets, Acta Math. Hungar , [] KARASIŃSKA, A. WAGNER-BOJAKOWSKA, E.: Homeomorphisms of linear and planar sets of the first category into microscopic sets, Topology Appl. 59 0, Received January, 04 Department of Mathematics and Computer Science University of Lódź Banacha PL Lódź POLAND grhorb@math.uni.lodz.pl 44 Download Date 9//8 9:3 PM