QUALITATIVE PROPERTIES OF IDEAL CONVERGENT SUBSEQUENCES AND REARRANGEMENTS. 1. Introduction

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1 QUALITATIVE PROPERTIES OF IDEAL CONVERGENT SUBSEQUENCES AND REARRANGEMENTS MAREK BALCERZAK, SZYMON G LA B, AND ARTUR WACHOWICZ Abstract. We ivestigate the Baire category of I-coverget subsequeces ad rearragemets of a diverget sequece s = (s ) of reals, if I is a ideal o N havig the Baire property. We also discuss the measure of the set of I-coverget subsequeces for some classes of ideals o N. Our results geeralize theorems due to H. Miller ad C. Orha (2001). 1. Itroductio Deote N := 1, 2,.... Let s = (s ) R N be a give sequece. Firstly, we are iterested i subsequeces of s. Let T 0, 1 N deote the set of all sequeces with ifiite umber of oes. For x = (x ) T we geerate a subsequece sx of s i such a way that, if t is a positio of the th 1 i the sequece x, the (sx) := s t for N. Clearly, all subsequeces of s ca be coded i this maer (ad the codig is oe-to-oe). It is kow that the Cator space 0, 1 N equipped with the product topology is complete. Observe that T is a G δ subset of 0, 1 N, therefore by the Alexadrov theorem, it is completely metrizable. So, we may apply the Baire category theorem to the space T while studyig sets of subsequeces of s. Secodly, we are iterested i rearragemets of the sequece s. Recall that N N equipped with the product topology is a complete space, ad the set P N N of all bijectios from N oto itself (permutatios of N) is a G δ subspace [19, p. 66], so it is completely metrizable. Hece the Baire category theorem works i P. Rearragemets of s are sequeces of the form (s p() ) for p P. Let us recall some useful defiitios ad facts o ideals of subsets of N (cf. [14],[8]). We say that a ideal I P(N) is admissible if N / I ad the ideal Fi of all fiite subsets of N is cotaied i I. From ow o, we will cosider oly admissible ideals; we will simply call them ideals o N. Ideals o N ca be treated (via the characteristic fuctios) as subsets of the Polish space 0, 1 N, so they ca have the Baire property, be Borel, aalytic, coaalytic, ad so o. The followig importat fact is due to Jalali-Naii [12] ad Talagrad [20]; see also [21, Theorem 1, Sectio 8]. Lemma 1.1. Let I be a ideal o N. The followig coditios are equivalet: I has the Baire property; I is meager; there is a ifiite sequece 1 < 2 <... of itegers i N such that o member of I cotais ifiitely may itervals [ i, i+1 ) i N Mathematics Subject Classificatio. 40A35, 40A05, 54E52, 28A05. Key words ad phrases. Ideal covergece, Baire category, Lebesgue measure, subsequece, rearragemet. 1

2 2 MAREK BALCERZAK, SZYMON G LA B, AND ARTUR WACHOWICZ A ideal I o N is called maximal if there is o ideal o N which is a proper superset of I. It is well kow that I is maximal if ad oly if, for every partitio A, B of N ito ifiite sets, exactly oe of the coditios A / I, B / I is true. If I is a ideal o N ad (s ) is a sequece of reals, we say (cf. [14]) that (s ) is I-coverget to t R (ad write I-lim s = t) if for every ε > 0 we have N: s t ε I. It is easy to see that if I = Fi, we get the usual covergece of (s ) to t. Also, let us metio the case whe I equals I d, the desity ideal which cosists of sets A N with asymptotic desity zero. (Recall that the asymptotic desity of A N is give by d(a) := lim A [1, ] /, provided the limit exists.) I this case we speak about statistical covergece (see [9], [11]). For several iterestig applicatios of statistical ad ideal covergece, see for istace [10],[5],[7],[1],[4]. 2. Results o the Baire category Give a ideal I o N ad a diverget sequece s = (s ) R N, the Baire category (i the spaces T ad P, respectively) of the followig two sets will be ivestigated: S I s := x T : sx is I-coverget, R I s := p P : (s p() ) is I-coverget. The motivatio of such studies comes from the results by Miller ad Orha [16] who proved that for I := I d these sets are of the first category. From ow o, sets of the first category will be called meager, ad residual sets will be called comeager. Theorem 2.1. Let I be a ideal o N with the Baire property ad let s = (s ) be a diverget sequece of reals. The the sets S I s ad R I s are meager i T ad P, respectively. Proof. Note that, if s is diverget to or to, the S I s = = R I s. Ideed, if a subsequece s of s is I-coverget to t R, the a subsequece of s is coverget to t which is impossible. Also, if a rearragemet s = (s p() ) of s is I-coverget to t R, the a subsequece of s is coverget to t which is impossible. Cosequetly, from ow o we assume that s has at least two differet partial limits a, b where a < b. Pick α, β such that a < α < β < b. At first we will show that Ss I is meager. Sice I has the Baire property, by Lemma 1.1 fix a ifiite sequece 1 < 2 <... of itegers i N such that o member of I cotais ifiitely may itervals [ i, i+1 ). For ay m N, defie A m := x T : ( k N)( k > m & (sx) [k, k+1 ) α & (sx) [k+1, k+2 ) β). (Here (sx) [k, k+1 ) α meas that (sx) i α for each i [ k, k+1 ).) Note that m N A m T \ S I s. Ideed, if x m N A m, each of the sets N: (sx) α ad N: (sx) β cotais ifiitely may itervals of the form [ k, k+1 ), hece it does ot belog to I. Thus sx is ot I-coverget. Cosequetly, it suffices to show that every set A m is comeager i T. Fix m N. We will prove that every ope set U from a stadard coutable base of topology i T (iherited from 0, 1 N ) cotais a ope subset icluded i A m. This will demostrate that A m cotais a dese G δ set, hece it is comeager. So, cosider a basic ope set U := T x 0, 1 N : x exteds (x 1,..., x d ).

3 QUALITATIVE PROPERTIES OF IDEAL CONVERGENT SUBSEQUENCES 3 Without loss of geerality, we may assume that the sequece x := (x 1,..., x d ) cotais at least m oes (i fact, we may assume that every member of the base satisfies this property). Let the umber of oes i x be equal to t m. Put k := mii N: i > t. We ca exted x to a sequece (x 1,..., x q ) i which there are precisely k 1 oes ad x q := 1. The proceed as follows. Sice a is a limit poit of the sequece s, fid iductively idices i k < i k +1 < < i k+1 1, greater tha d, for which the respective terms of the sequece s are α. Let x i, for these idices i, be equal to 1, ad cosider the ext extesio of x to a sequece where the remaiig terms are filled up by zeros. (x 1,..., x q,... x ik,..., x ik+1 1 ) I the same maer, sice b is a limit umber of s, we ca fid the ext idices i k+1 < i k+1 +1 < i k+2 1, for which the respective terms of the sequece s are β. Let x i, for these idices i, be equal to 1, ad cosider the fial extesio of x to a sequece x := (x 1,..., x q,... x ik,..., x ik+1 1, x i k+1..., x ik+2 1 ) where the remaiig terms are filled up by zeros. Let V := T x 0, 1 N : x exteds x. The V is a ope subset of U. Moreover, V A m sice, if x V, the coditios k > m, (sx) [k, k+1 ) α ad (sx) [k+1, k+2 ) β are fulfilled. The proof for R I s is similar. For ay m N, let A m := p P : ( k N)( k > m & (s p( ) [k, k+1 ) α & s p( ) [k+1, k+2 ) β). We show that every set A m is comeager i P. So, fix m N ad cosider a basic ope set U := P p N N : p exteds x where x := (x 1,..., x d ) is a sequece with distict terms. We may assume that d m. Pick the smallest k N such that k > d. The we ca choose distict umbers p(d + 1),..., p( k ),..., p( k+1 1), p( k+1 ),..., p( k+2 1) take from N \ x 1,..., x d ad fulfillig the followig coditios: Put ad s p(i) α for i = k, k + 1,..., k+1 1; s p(i) β for i = k+1, k+1 + 1,..., k+2 1. x := (x 1,..., x d,... p( k ),..., p( k+1 1), p( k+1 )..., p( k+2 1) V := P p N N : x exteds x. The V is ope ad icluded i A m U. This implies that A m is comeager i P, ad so is A := m N A m. Hece Rs I icluded i P \ A is meager.

4 4 MAREK BALCERZAK, SZYMON G LA B, AND ARTUR WACHOWICZ Theorem 2.1 geeralizes results of [16, Theorems 3.1, 5.1(b)] dealig with a special case whe I := I d. Note that, i [16], a differet (but equivalet) approach was proposed to code subsequeces (istead of x 0, 1 N, biary expasios of reals x (0, 1] are used). Let us show some cosequeces of Theorem 2.1. We will formulate several equivalet coditios for the usual covergece of a sequece i the laguage of the Baire category of ideal coverget subsequeces ad rearragemets. Corollary 2.2. Let s = (s ) be a sequece of reals ad let I be a ideal o N with the Baire property. The followig coditios are equivalet: (i) s is coverget; (ii) Ss I = T ; (iii) Ss I is comeager i T ; (iv) Ss I is omeager i T. Proof. To show (i) (iii) recall that, if (i) holds, say s is coverget to t R, the all subsequeces of s are coverget to t, ad also I-coverget to t. Hece Ss I = T ad so, (ii) is true. Implicatios (ii) (iii) ad (iii) (iv) are trivial, ad Theorem 2.1 yields implicatio (iv) (i). Aalogously, we obtai Corollary 2.3. Let s = (s ) be a sequece of reals ad let I be a ideal o N with the Baire property. The followig coditios are equivalet: (i) s is coverget; (ii) Rs I = P ; (iii) Rs I is comeager i P ; (iv) Rs I is omeager i P. Observe that, if I is a maximal ideal o N, the assertio of Theorem 2.1 is false. Ideed, it is kow that every bouded sequece of reals is I-coverget (cf. [14, Lemma 5.2]). Hece it suffices to cosider a bouded diverget sequece s ad the Ss I = T, Rs I = P. Note that maximal ideals do ot have the Baire property (cf. [13, 8.50]). 3. Some results i the measure case It is atural to ask whether a measure couterpart of Theorem 2.1 is true i the case of the set Ss I. Namely, cosider the uiform probability measure o 0, 1 ad let µ deote the respective product measure o 0, 1 N which sometimes is called Lebesgue measure o 0, 1 N (cf. [19, Example ]). I fact, µ is strictly associated with liear Lebesgue measure o [0, 1] (whe oe uses the Cator cotiuous fuctio from 0, 1 N oto [0, 1]). The, by measurable subsets of 0, 1 N we mea sets belogig to the µ-completio of the respective product σ-algebra o 0, 1 N. Of course, T as a cocoutable subset of 0, 1 is of full µ-measure (that is µ(t ) = 1). If we treat 0, 1 as the group Z 2, the space 0, 1 N ca be treated as the compact metric group (Z 2 ) N, ad the µ is the respective Haar measure. We propose a prelimiary observatio (Propositio 3.1) which shows a dichotomy for Ss I, provided that I is a aalytic or a coaalytic ideal o N (thus it is a special measurable subset of 0, 1 N ).

5 QUALITATIVE PROPERTIES OF IDEAL CONVERGENT SUBSEQUENCES 5 We eed oe more otio. Give a ideal I o N, we say that a sequece (s ) of reals satisfies I-Cauchy coditio (cf. [6]) if for every ε > 0 there is N N such that N: s s N ε I. I a similar way, oe ca defie I-covergece ad I-Cauchy coditio i a metric space. Recall that (cf. [6]), i a complete metric space, the classes of I-coverget ad of I-Cauchy sequeces are equal. So, the sets Ss I ca be expressed i the form (1) Ss I = x T : i N: (sx) i (sx) N ε I; ε>0 N N Propositio 3.1. Let s R N ad let I be a ideal o N which is a aalytic or a coaalytic subset of 0, 1 N. The S I s is aalytic or coaalytic i T, ad either of measure 0 or 1. Proof. A set E 0, 1 N (see [17]) is called a tail set if, wheever x E ad y 0, 1 N differs from x i a fiite umber of coordiates, the y E. By [17, Theorem 21.3] if a tail set is measurable the it is either of measure 0 or 1. Observe that S I s, treated as a subset of 0, 1 N, is a tail set. If we show that Ss I is measurable, we will get the secod assertio (ote that 0, 1 N \ T is coutable). We cosider the expressio (1) with ε take from the set Q + of positive ratioals. For ε Q + ad N N defie f ε,n : T 0, 1 N as the sequece of characteristic fuctios f ε,n (x) := (χ i N: (sx)i (sx) N ε(j)) j N, x T. The Ss I = ε Q + N N f 1 ε,n [I]. To fiish the proof we eed to show that every fuctio f ε,n is cotiuous sice the (by the respective properties of aalytic ad coaalytic sets; see [19]) the set S I s will be aalytic or coaalytic, hece it will be measurable. Let f := f ε,n. It suffices to prove that every coordiate f j := χ i N: (sx)i (sx) N ε(j) of f is cotiuous. Fix j N. Let x T for N ad assume that x x T. We will show that f j (x ) f j (x). Let for example f j (x) = 1 (the secod case is aalogous). Hece (sx) j (sx) N ε. Deote by k the maximum of the Nth ad the jth positios of oes i the sequece x. Sice x x, there is 0 N such that, for ay 0, the first k terms of x ad x are equal. That is why (sx ) j (sx ) N ε, ad fially f j (x ) = f j (x) = 1 for ay 0. Remark 1. I a aalogous way, oe ca prove that, if I is aalytic or coaalytic, the for ay sequece s R N ad ay t R, the set x T : I- lim(sx) i = t = x T : i N: (sx) i t ε I i ε Q + is aalytic or coaalytic, ad either of measure 0 or 1 (observe that this is a tail set). I [15], [16], the measure of S I s was ivestigated, i the case I := I d. Note that I d is a F σδ subset of 0, 1 N (cf. [8]). We summarize the respective results of [15] ad [16] i the followig theorem. Theorem 3.2. ([15],[16]) For I := I d ad a sequece s = (s ) of reals, the followig coditios are equivalet: (i) s is I-coverget; (ii) µ(x T : I- lim (sx) = t) = 1 for some t R; (iii) µ(s I s ) = 1.

6 6 MAREK BALCERZAK, SZYMON G LA B, AND ARTUR WACHOWICZ Proof. The equivalece (i) (ii) was show i [15, Theorem 3], with I-lim s = t i (i). Implicatio (ii) (iii) is obvious. By [16, Theorem 3.5], if s is ot I-coverget, the µ(t \S I s ) = 1. Cosequetly, if µ(s I s ) = 1, the s is I-coverget. This yields (iii) (i). Theorem 3.2 shows that the measure aalogue of Theorem 2.1 is false. Ideed, it suffices to cosider a diverget sequece s which is I d -coverget ad the µ(s I d s ) = 1 by Theorem 3.2. Now, we are goig to exted Theorem 3.2 to a wider class of ideals. Let I be a ideal o N. A fuctio f : N N is called bi-i-ivariat if A I f[a] I for every A N (if is true, we say that f is I-ivariat). We will eed the followig fact (cf. [3, Propositio 24]). Propositio 3.3 ([3]). Let f, g : N N be bi-i-ivariat ijectios such that f[n] g[n] = ad f[n] g[n] = N. The, for ay sequece (s ) of reals ad a poit t R, we have I- lim s = t (I- lim s f() = t ad I- lim s g() = t). For a 0-1 sequece x T let 1 < 2 <... := k N: x k = 1. Defie f x : N N by f x (k) = k, k N, ad let T I := x T : f x is bi-i-ivariat. We will say that a ideal I o N has property (G) if µ(t I ) = 1. Theorem 3.4. Let I be a ideal o N. For a sequece s = (s ) of reals, cosider the followig coditios: (i) s is I-coverget; (ii) µ(x T : I- lim (sx) = t) = 1 for some t R; (iii) µ(s I s ) = 1. The (i) ad (ii) are equivalet, provided that I has property (G). Implicatio (ii) (iii) is always true. Implicatio (iii) (i) holds provided that I is aalytic or coaalytic ad I has property (G). Cosequetly, uder these two assumptios o I, coditios (i),(ii),(iii) are equivalet. Proof. Assume that I has property (G). To show (i) (ii), assume that I-lim s = t R. x T I. Hece f x is bi-i-ivariat. Let 1 < 2 <... := k N: x k = 1. Fix ε > 0. The Let i N: (sx) i t ε = i N: s i t ε = fx 1 [i N: s i t ε]. We have i N: s i t ε I. Thus f 1 x [i N: s i t ε] I by the bi-i-ivariace of f x. Cosequetly, sx is I-coverget to t. By property (G) ad the choice of x, we obtai (ii). To show (ii) (i), let t R be such that µ(b) = 1 where B := x T : I- lim (sx) = t. If x T, deote by x c := 1 x, the coverse of x i the group (Z 2 ) N, where 1 := (1, 1,... ). By property (G) we have µ(t I ) = 1, hece µ(1 T I ) = 1 sice µ is the Haar measure. Pick x B T I (1 T I ). Let 1 < 2 <... := k N: x k = 1 = f x [N] ad m 1 < m 2 <... := k N: x c k = 1 = f x c[n]. The f x [N] f x c[n] =, f x [N] f x c[n] = N ad I- lim (sx) = t I- lim s i = t I- lim s fx() = t, I- lim (sx c ) = t I- lim s mi = t I- lim s fx c () = t. Hece, usig the bi-ivariace of f x ad f x c, we obtai I-lim s = t by Propositio 3.3. Implicatio (ii) (iii) is obvious (o extra assumptio o I is eeded).

7 QUALITATIVE PROPERTIES OF IDEAL CONVERGENT SUBSEQUENCES 7 Now, let I be aalytic or coaalytic, with property (G). To prove (iii) (i), we follow some ideas from [16, Theorem 3.5]. Assume that µ(s I s ) = 1. Pick x 0 from the set H := T I (1 T I ) S I s (1 S I s ) which is of µ-measure 1. The f x0 ad f x c 0 are bi-i-ivariat, ad the subsequeces sx 0 ad sx c 0 are I- coverget to t ad t, respectively. If t = t, the (i) follows from Propositio 3.3. Assume that t t. Fix ay x H ad suppose that sx is I-coverget to some u / t, t. Take ε > 0 such that the sets (t ε, t + ε), (t ε, t + ε) ad (u ε, u + ε) are pairwise disjoit. The : (sx 0 ) t ε I ad sice f x0 is I-ivariat, we have f x0 (): (sx 0 ) t ε I. Thus f x0 (): (sx 0 ) u < ε I. Similarly f x c 0 (): (sx c 0 ) u < ε I. Sice : (sx) u ε I, the f x (): (sx) u ε I. Moreover, f x (): (sx) u < ε f x0 (): (sx 0 ) u < ε f x c 0 (): (sx c 0 ) u < ε. Therefore f x maps N oto a set from I which cotradicts the bi-i-ivariace of f x. µ-almost every x T, the sequece sx is I-coverget either to t or to t. Hece, for Sice I is aalytic or coaalytic, by Remark 1 we ifer that x T : (sx) is I-coverget to t ad x T : (sx) is I-coverget to t are 0-1 sets with respect to µ. However, their uio is of full µ-measure. Hece oe of them has full µ-measure. Now, we ca use (ii) (i). Let NR := x T : d( N: x = 1) = 1/2. Recall that µ(nr) = 1, by the Borel theorem o ormal umbers. If I = I d, the set x T : f x is bi-i-ivariat cotais NR ad therefore it is of full µ-measure. (This was used i the proofs of [15, Theorem 3] ad [16, Theorem 3.5].) We will preset two classes of ideals I that fulfil the iclusio NR T I. This will witess that the class of ideals with property (G) is quite rich. Let (a ) be a sequece of oegative real umbers with N a =. By I (a) deote the ideal of all sets A N with A a <. This is called the summable ideal associated with (a ); cf. [8]. Aother class cosists of desity-like ideals cosidered i [2]. For ay α (0, 1] let Note that I 1 = I d. I α := A [1, ] A N : lim sup α = 0. Propositio 3.5. (I) Let (a ) be a oicreasig sequece of positive reals with N a =. Assume that there is C > 0 such that a /a 2 C for each N. The I (a) has property (G). (II) The ideal I α has property (G) for each α (0, 1]. Proof. Fix x NR. We claim that f x () 4 for all but fiitely may s. Suppose ot. The there are ifiitely may s such that f x () > 4. Fix 0 N such that i : x i = 1 > 1 3 for every 0. Take 1 0 with f x ( 1 ) > 4 1. Sice f x is icreasig, f x () > 4 1 for every 1. Thus i 4 1 : x i = , a cotradictio. From ow o, we fix m 0 N such that f x () 4 for all m 0. We will prove that f x is bi-i-ivariat for ideals I cosidered i statemets (I) ad (II).

8 8 MAREK BALCERZAK, SZYMON G LA B, AND ARTUR WACHOWICZ (I) Let A N. Sice (a ) is oicreasig, A I (a) implies f x [A] I (a). Assume that f x [A] I (a ). The A, m 0 a C 2 A, m 0 a 4 A, m 0 a fx() < which shows that A I (a). (II) Note that each icreasig ijectio from N to N is I α -ivariat. Let A N ad assume that f x [A] I α. Fix ε > 0 ad fid k 0 m 0 such that for all k k 0 we have The for all k k 0 we have which shows that A I α. k : A k α f x () k : A k α < ε 4 α. 4 α f x() 4k : A (4k) α Now, we preset a example of summable ideal ad a sequece of reals for which implicatio (i) (iii) i Theorem 3.4 is false. Example 2. Defie I as follows: for A N let A I A (2N + 1) Fi. Note that I is a summable ideal; amely I = I (a) where a 2 := 0 ad a 2+1 := 1 for all N. By the Borel-Catelli lemma, a sequece x T cotais ifiitely may blocks (1, 0, 1, 1, 0, 1) with probability 1 (i.e. with µ-measure 1). Defie E := x T : x k = x k+2 = x k+3 = x k+5 = 1, x k+1 = x k+4 = 0, for ifiitely may k N. Let s := (a ). Clearly, s is I-coverget to 1. Observe that every I-coverget subsequece of s with odd idices is coverget i the usual way. We will show that sx is ot I-coverget for every x E. Fix x E. By K deote the ifiite set of idices k such that (x k,..., x k+5 ) = (1, 0, 1, 1, 0, 1). Let k N: x k = 1 = 1 < 2 < 3 <.... To prove that sx is ot I-coverget, we eed to show that the sequece (a 2k+1 ) k N cotais ifiitely may zeros ad ifiitely may oes. This holds sice, for ifiitely may eve k s, the idices k are odd, ad for ifiitely may odd k s, the idices k are also odd. Ideed, fix k K. Cosider the followig four cases. Case 1. k is eve ad k is eve. The k + 2 is eve ad k+2 is odd, ad k + 5 is odd ad k+5 is odd. Case 2. k is eve ad k is odd. The k + 3 is odd ad k+3 is odd. Case 3. k is odd ad k is eve. The k + 2 is odd ad k+2 is odd, ad k + 5 is eve ad k+5 is odd. Case 4. k is odd ad k is odd. The k + 3 is eve ad k+2 is odd. Sice sx is ot I-coverget for every x E, we have µ(s I c ) = 0. Note that summable ideals ad ideals of the form I α, α (0, 1], are aalytic P-ideals. A ideal I o N is called a P-ideal if for ay sequece (A ) of sets i I there exists A I such that A \ A Fi for every N. We leave usolved the problem how to describe the class of aalytic P-ideals for which the equivalece of coditios (i),(ii),(iii) i Theorem 3.2 is true. Recall a importat theorem ε

9 QUALITATIVE PROPERTIES OF IDEAL CONVERGENT SUBSEQUENCES 9 of Solecki [18]: I is a aalytic P-ideal o N if ad oly if I = Exh(ϕ) for some lower semicotiuous submeasure ϕ o N (where Exh(ϕ) := A N: lim ϕ(a \ [1, ]) = 0). Refereces [1] M. Balcerzak, K. Dems, A. Komisarski, Statistical covergece ad ideal covergece for sequeces of fuctios, J. Math. Aal. Appl. 328 (2007), [2] M. Balcerzak, P. Das, M. Filipczak, J. Swaczya, Geeralized kids of desity ad the associated ideals, Acta Math. Hugar. 147 (2015), [3] M. Balcerzak, Sz. G l ab, J. Swaczya, Ideal ivariat ijectios, submitted. [4] A. Bartoszewicz, Sz. G l ab, A. Wachowicz, Remarks o ideal boudedess, covergece ad variatio of sequeces, J. Math. Aal. Appl. 375 (2011), [5] J. Coor, M. Gaichev, V. Kadets, A characterizatio of Baach spaces with separable duals via weak statistical covergece, J. Math. Aal. Appl. 244 (2000), [6] K. Dems, O I-Cauchy sequeces, Real Aal. Exchage 30 (2004/2005), [7] G. Di Maio, Lj.D.R. Kočiac, Statistical covergece i topology, Topology Appl. 156 (2008), [8] I. Farah, Aalytic quotiets. Theory of liftig for quotiets over aalytic ideals o itegers, Mem. Amer. Math. Soc. 148 (2000). [9] H. Fast, Sur la covergece statistique, Colloq. Math. 2 (1951), [10] R. Filipów, N. Mrożek, I. Rec law, P. Szuca, Ideal covergece of bouded sequeces, J. Symb. Logic 72 (2007), [11] J.A. Fridy, O statistical covergece, Aalysis 5 (1985), [12] S.A. Jalali-Naii, The mootoe subsets of Cator space, filters ad descriptive set theory, PhD thesis, Oxford [13] A.S. Kechris, Classical Descriptive Set Theory, Spriger, New York [14] P. Kostyrko, T. Šalát, W. Wilczyński, I-covergece, Real Aal. Exchage 26 ( ), [15] H.I. Miller, A measure theoretical subsequece characterizatio of statistical covergece, Tras. Amer. Math. Soc. 347 (1995), [16] H.I. Miller, C. Orha, O almost coverget ad statistically coverget subsequeces, Acta Math. Hugar. 93 (2001), [17] J.C. Oxtoby, Measure ad category, Spriger, New York [18] S. Solecki, Aalytic ideals ad their applicatios, A. Pure Appl. Logic 99 (1999), [19] S.M. Srivastava, A course of Borel sets, Spriger, New York [20] M. Talagrad, Compacts de foctios mesurables et filtres o measurables, Studia Math. 67 (1980), [21] S. Todorcevic, Topics i topology, Lecture Notes i Math. 1652, Spriger, New York Istitute of Mathematics, Lódź Uiversity of Techology, Wólczańska 215, Lódź, Polad address: marek.balcerzak@p.lodz.pl Istitute of Mathematics, Lódź Uiversity of Techology, ul. Wólczańska 215, Lódź, Polad address: szymo.glab@p.lodz.pl Istitute of Mathematics, Lódź Uiversity of Techology, Wólczańska 215, Lódź, Polad address: artur.wachowicz@p.lodz.pl