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Qualitative properties of ideal convergent subsequences and rearrangements. (English) Zbl 1399.40010
Summary: We investigate the Baire category of $$\mathcal{I}$$-convergent subsequences and rearrangements of a divergent sequence $$s = (s_n)$$ of reals if $$\mathcal{I}$$ is an ideal on $$\mathbb{N}$$ having the Baire property. We also discuss the measure of the set of $$\mathcal{I}$$-convergent subsequences for some classes of ideals on $$\mathbb{N}$$. Our results generalize theorems due to H. I. Miller and C. Orhan [Acta Math. Hung. 93, No. 1–2, 135–151 (2001; Zbl 0989.40002)].

##### MSC:
 40A35 Ideal and statistical convergence 40A05 Convergence and divergence of series and sequences 54E52 Baire category, Baire spaces 28A05 Classes of sets (Borel fields, $$\sigma$$-rings, etc.), measurable sets, Suslin sets, analytic sets
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##### References:
 [1] Balcerzak M., Dems K., Komisarski A.: Statistical convergence and ideal convergence for sequences of functions. J. Math. Anal. Appl. 328, 715–729 (2007) · Zbl 1119.40002 [2] Balcerzak M., Das P., Filipczak M., Swaczyna J.: Generalized kinds of density and the associated ideals. Acta Math. Hungar. 147, 97–115 (2015) · Zbl 1374.03032 [3] M. Balcerzak, Sz. Głąb and J. Swaczyna, Ideal invariant injections, J. Math. Anal. Appl., accepted. · Zbl 1359.40002 [4] Bartoszewicz A., Głąb Sz., Wachowicz A.: Remarks on ideal boundedness, convergence and variation of sequences. J. Math. Anal. Appl. 375, 431–435 (2011) · Zbl 1219.40003 [5] Connor J., Ganichev M., Kadets V.: A characterization of Banach spaces with separable duals via weak statistical convergence. J. Math. Anal. Appl. 244, 251–261 (2000) · Zbl 0982.46007 [6] K. Dems, On $${\mathcal{I}}$$ I -Cauchy sequences, Real Anal. Exchange, 30 (2004/2005), 123–128. · Zbl 1070.26003 [7] Di Maio G., Kočinac Lj. D. R.: Statistical convergence in topology. Topology Appl. 156, 28–45 (2008) · Zbl 1155.54004 [8] I. Farah, Analytic quotients. Theory of lifting for quotients over analytic ideals on integers, Mem. Amer. Math. Soc., 148 (2000). · Zbl 0966.03045 [9] Fast H.: Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951) · Zbl 0044.33605 [10] Filipów R., Mro\.zek N., Recław I., Szuca P.: Ideal convergence of bounded sequences. J. Symb. Logic 72, 501–512 (2007) · Zbl 1123.40002 [11] J. A. Fridy: On statistical convergence. Analysis. 5, 301–313 (1985) [12] S. A. Jalali-Naini, The monotone subsets of Cantor space, filters and descriptive set theory, PhD thesis (Oxford, 1976). [13] A. S. Kechris, Classical Descriptive Set Theory, Springer (New York, 1998). [14] P. Kostyrko, T. Šalát and W. Wilczyński, $${\mathcal{I}}$$ I -convergence, Real Anal. Exchange, 26 (2000-2001), 669–685. [15] Miller H.I.: A measure theoretical subsequence characterization of statistical convergence. Trans. Amer. Math. Soc. 347, 1811–1819 (1995) · Zbl 0830.40002 [16] Miller H.I., Orhan C.: On almost convergent and statistically convergent subsequences. Acta Math. Hungar. 93, 135–151 (2001) · Zbl 0989.40002 [17] J. C. Oxtoby, Measure and Category, Springer (New York, 1980). [18] Solecki S.: Analytic ideals and their applications. Ann. Pure Appl. Logic 99, 51–72 (1999) · Zbl 0932.03060 [19] S. M. Srivastava, A Course of Borel Sets, Springer (New York, 1998). · Zbl 0903.28001 [20] Talagrand M.: Compacts de fonctions mesurables et filtres non measurables. Studia Math. 67, 13–43 (1980) · Zbl 0435.46023 [21] S. Todorcevic, Topics in Topology, Lecture Notes in Math. 1652, Springer (New York, 1997). · Zbl 0953.54001
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