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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 420419, 11 pages
Statistical Convergence in Function Spaces
1Department of Mathematics, SUN, 81100 Caserta, Italy
2Faculty of Sciences and Mathematics, University of Niš, 18000 Niš, Serbia
Received 26 April 2011; Accepted 2 October 2011
Academic Editor: Jean Pierre Gossez
Copyright © 2011 Agata Caserta et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study statistical versions of several classical kinds of convergence of sequences of functions between metric spaces (Dini, Arzelà, and Alexandroff) in different function spaces. Also, we discuss a statistical approach to recently introduced notions of strong uniform convergence and exhaustiveness.
One of central questions in analysis is what precisely must be added to pointwise convergence of a sequence of continuous functions to preserve continuity of the limit function? In 1841, Weierstrass discovered that uniform convergence yields continuity of the limit function. Dini had given in 1878 a sufficient condition, weaker than uniform convergence, for continuity of the limit function. In 1883/1884, Arzelà  found out a necessary and sufficient condition under which the pointwise limit of a sequence of real-valued continuous functions on a compact interval is continuous. He called this condition “uniform convergence by segments” (“convergenza uniforme a tratti”) , and his work initiated a study that led to several outstanding papers. In 1905, Borel in  introduced the term “quasiuniform convergence” for the Arzelà condition, and Bartle in  extended Arzelà’s result to nets of real-valued continuous functions on a topological space. In 1948, Alexandroff studied the question for a sequence of continuous functions from a topological space , not necessarily compact, to a metric space . The reader may consult [6, 7] for the literature concerning the preservation of continuity of the limit function.
In 2009, Beer and Levi  proposed a new approach to this investigation, in the realm of metric spaces, through the notion of strong uniform convergence on bornologies, when this bornology reduces to that of all nonempty finite subsets of . In , a direct proof of the equivalence of Arzelà, Alexandroff, and Beer-Levi conditions was offered.
In , Caserta and Kočinac proposed a new model to investigate convergence in function spaces: the statistical one. Actually they obtained results parallel to the classical ones in spite of the fact that statistical convergence has a mild control of the whole set of functions. One of the main goals of this paper is to continue their analysis. In Section 3, we prove that continuity of the limit of a sequence of functions is equivalent to several modes of statistical convergence which are similar to but weaker than the classical ones, namely, Arzelà, Alexandroff, and Beer-Levi. Moreover, we state the novel notion of statistically strong Arzelà convergence, the appropriate tool to investigate strong uniform continuity of the limit of a sequence of strongly uniformly continuous functions, a concept introduced in .
In 2008, the definition of exhaustiveness, closely related to equicontinuity , for a family of functions (not necessarily continuous), was introduced by Gregoriades and Papanastassiou in . Exhaustiveness describes convergence of a net of functions in terms of properties of the whole net and not of properties of the functions as single members. Thus, statistical versions of exhaustiveness and its variations are natural and the investigation in this direction was initiated by Caserta and Kočinac in . In Section 4, we continue this study and provide additional information about exhaustiveness and its variations. First, we analyze the exact location of exhaustiveness. In fact, in  it was shown that equicontinuity implies exhaustiveness. We prove that exhaustiveness lies between equicontinuity and even continuity , a classical property weaker than equicontinuity. Furthermore, we propose a notion of statistical uniform exhaustiveness of a sequence of functions which is the appropriate device to study uniform convergence.
2. Notation and Preliminaries
Throughout the paper, and will be metric spaces, and the sets of all in all continuous mappings from to . The pointwise (resp., uniform) topology on and will be denoted by . We denote by the family of all nonempty subsets of , and by , or simply , the family of all nonempty finite subsets of . If , , and , we write for the open -ball with center , and for the -enlargement of .
Recall that a bornology on a space is a hereditary family of subsets of which covers and is closed under taking finite unions (see [12, 13]). By a base for a bornology , we mean a subfamily of that is cofinal with respect to inclusion. The smallest bornology on is the family , and the largest is the family .
In , as mentioned above, the notions of strong uniform continuity of a function on a bornology and the topology of strong uniform convergence on for function spaces were introduced.
Definition 2.1 (see ). Let and be metric spaces, and let be a subset of . A function is strongly uniformly continuous on if for each there is such that if and , then .
If is a family of nonempty subsets of and a metric space, a function is called uniformly continuous (resp., strongly uniformly continuous) on if for each , is uniformly continuous (resp., strongly uniformly continuous) on . We denote by the set of all strongly uniformly continuous functions on .
Given a bornology with closed base on , Beer and Levi presented a new uniformizable topology on the set .
Definition 2.2 (see ). Let and be metric spaces, and let be a bornology with a closed base on . The topology of strong uniform convergence is determined by the uniformity on having as a base all sets of the form
On , this topology is in general finer than the classical topology of uniform convergence on . This new function space has been intensively studied in [6, 8, 14–16].
Let us recall some classical definitions and results.
Definition 2.3 (Arzelà (see , [7, page 268])). Let be a sequence of real-valued continuous functions defined on an arbitrary set , and let . The sequence is said to converge to quasiuniformly on if it pointwise converges to , and for each and , there exists a finite number of indices such that for each at least one of the following inequalities holds:
Definition 2.4 (Alexandroff ). Let be a sequence in and . Then is Alexandroff convergent to on , provided it pointwise converges to , and for each and each , there exist a countable open cover of and a sequence , of positive integers greater than such that for each we have .
Theorem 2.5 (see ). If a net in pointwise converges to , then the following are equivalent: (i) is continuous; (ii) Alexandroff converges to ; (iii) converges to quasiuniformly on compacta; (iv) -converges to .
In the next section, we will show that similar results about continuity of the limit function are true for statistical pointwise convergence of sequences of functions between two metric spaces.
The idea of statistical convergence appeared, under the name almost convergence, in the first edition (Warsaw, 1935) of the celebrated monograph  of Zygmund. Explicitly, the notion of statistical convergence of sequences of real numbers was introduced by Fast in  and Steinhaus in  and is based on the notion of asymptotic density of a set : We recall that for . A set is said to be statistically dense if .
Fact 1. The union and intersection of two statistically dense sets in are also statistically dense.
Statistical convergence has many applications in different fields of mathematics: number theory, summability theory, trigonometric series, probability theory, measure theory, optimization, approximation theory, and so on. For more information, see  (where statistical convergence was generalized to sequences in topological and uniform spaces) and references therein, and about some applications see [21, 22].
A sequence in a topological space statistically converges (or shortly, converges) to if for each neighborhood of , . This will be denoted by , where is a topology on .
Fact 2. There exists a subset of with such that the sequence converges to .
3. Statistical Arzelà and Alexandroff Convergence
In , a statistical version of the Alexandroff convergence was defined.
Definition 3.1. A sequence in is said to be statistically Alexandroff convergent to , denoted by , provided , and for each and each statistically dense set , there exist an open cover and an infinite set such that for each we have .
Below, a statistical version of the celebrated Arzelà’s quasiuniform convergence is given.
Definition 3.2. A sequence in is said to be statistically Arzelà convergent to , denoted by , if , and for each and each statistically dense set there exists a finite set such that for each it holds that for at least one .
Theorem 3.3. For a sequence in such that , the following is equivalent: (1) is continuous; (2) on compacta; (3); (4).
Proof. : Let a compact set , a statistically dense set , and be fixed. Since , for each there is a statistically dense set such that for each . Choose and set
Since all functions and are continuous, the sets are open, and thus is an open cover of . By compactness of there are such that . The set is a finite subset of such that for each it holds for at least one , that is, (2) is true.
: It suffices to show that for each and each we have . Since , there is a set with so that for each . We are going to prove that for each there is such that for each , . Suppose, by contradiction, that this assumption fails. Then there is and a sequence converging to such that for each . The set is compact so that, by (2), there are such that for each , holds for at least one . Therefore, we found such that there is an infinite set with the property that for each , . For this , we have Since and are continuous at , there are and such that for each , and for each . If , then for each we have Since converges to , there is such that . For this , we have which is a contradiction.
: Let and a statistically dense set be given. Since , given , there is statistically dense, such that for each we have . Hence there is a such that for each and each we have . Let . For each , define Note that . For each , let be the following open set: Then is an open cover of . Thus for each and each , there is some such that it holds that , that is, the set and the cover witness that (4) is true.
: It is proved in [9, Theorem 4.7].
The following two theorems use other kinds of statistical convergence, related to Dini convergence [29, pages 105-106] and Arzelà convergence, which imply continuity and strong uniform continuity of the limit function.
Definition 3.4. A sequence in is said to be statistically Dini convergent to , denoted by if and for each and each statistically dense set there exists an increasing sequence in such that for each and each .
Theorem 3.5. If a sequence in statistically Dini converges to , then is continuous.
Proof. Let and be given. Since , there is a statistically dense set such that for each . Because , there exists an increasing sequence in such that for each and each . Take some . Since is continuous at , there is such that whenever . So, for each , we have that is, is continuous at , hence on .
Definition 3.6. A sequence in statistically strongly Arzelà converges to a function on a bornology on , denoted by , if and for each , and each statistically dense set in , there are finitely many such that for at least one .
Theorem 3.7. If a sequence in statistically strongly Arzelà converges to on a bornology with closed base on , then is a strongly uniformly continuous on .
Proof. Let and . As , there is a statistically dense set such that for each we have , that is, for each , . By assumption, there are with for some , that is, there exists such that for each . Since is strongly uniformly continuous on , there is so that for each and each with , we have . Set . Then for each and with by the above relations, it follows that is, is strongly uniformly continuous on , hence on .
Theorem 3.8. Let be a compact space, a bornology on with closed base, and a sequence in . If and is strongly uniformly continuous on , then statistically strongly Arzelà converges to on .
Proof. Let , , and a statistically dense set be given. Since is strongly uniformly continuous on , there is such that for each and each with we have . From , it follows that there is a statistically dense set such that for each , , we have , that is, for each , and each it holds that . For each set . Since and ’s are continuous, each is open in , so that is an open cover of . By compactness of , there are finitely many such that . But each is strongly uniformly continuous on , so that for each there is such that whenever and , . Let . Then for and with , since for some , we have So, which completes the proof.
Theorem 3.9. Let be a bornology on with closed base, and let be a sequence in such that . Then is strongly uniformly continuous on if and only if .
Proof. By [8, Proposition 6.5], we have . So, it suffices to prove that implies strong uniform continuity of on .
Assume that is not strongly uniformly continuous on . There are a and such that for each , there are points with such that . Since , the density of the set is 0. Let . Then is statistically dense in , and there exist , , , such that , and . Thus and so that is, is not strongly uniformly continuous on . A contradiction.
4. More on (Statistical) Exhaustiveness
Definition 4.1. Let be a family and a sequence in . If case is finite, we say that is exhaustive at if all functions in are continuous at . If is infinite, then is exhaustive at if for each there exist and a finite set such that for each and for each , we have . The sequence is exhaustive at if the family is exhaustive at . The family (sequence ) is exhaustive on if it is exhaustive at each .
In , it was shown that exhaustiveness for a net of functions at each point of the domain is the property that must be added to pointwise convergence to have uniform convergence on compacta.
The notion of weak exhaustiveness was also introduced in , and it was proved that it gives a necessary and sufficient condition under which the pointwise limit of a sequence of (not necessarily continuous) functions is continuous.
In , two of the authors investigated the continuity of the statistical pointwise limit of a sequence of functions via the notion of statistical exhaustiveness.
Definition 4.2 (see ). A sequence in is said to be statistically exhaustive (shortly, exhaustive) at a point if for each there are and a statistically dense set such that for each we have for each . The sequence is exhaustive if it is -exhaustive at each .
In this section, we continue this study and provide some additional information about exhaustiveness and its variations.
First, we show that exhaustiveness is a property between equicontinuity and even continuity. It is well known that equicontinuity implies even continuity, and in  it was shown that equicontinuity implies exhaustiveness.
Definition 4.3 (see [10, L p. 241]). A family is evenly continuous if for each net in such that converges to and converges to , the net converges to .
Definition 4.4 (see ). A family is equicontinuous at a point if and only if for each there is a neighborhood of such that for each member of . A family is equicontinuous if it is equicontinuous at each .
Theorem 4.5. If a family is exhaustive, then is evenly continuous.
Proof. If is finite there is nothing to prove, and thus we assume that is infinite. Let be a net in satisfying converges to and converges to , and let . As is exhaustive at , there exist and such that for each and each we have . Because the set is finite, there is some such that for each and each .
Let . Since converges to and converges to there is in such that and for each . Then and for each . So for each , we have that is, converges to .
Recall that the concept of uniform exhaustiveness was defined in [15, Definition 4.1] under the name strong exhaustiveness: a sequence in is strongly exhaustive on if for each there are and such that for all with , for each .
The novel notion of statistical uniform exhaustiveness for a sequence is related to uniform convergence.
Definition 4.6. A sequence in is uniformly exhaustive on if for each there are and a statistically dense set such that for all with , for all .
Theorem 4.7. Let be a compact space, and let be a -exhaustive sequence in such that . Then (a) is statistically uniformly exhaustive; (b)there is a statistically dense set such that is uniformly exhaustive and .
Proof. (a) Let and be fixed. Since, by hypothesis, is statistically exhaustive at , there are and a statistically dense set such that for each and each , it holds . From and compactness of , it follows the existence of finitely many points in such that . Let and . The set is statistically dense in . We claim that and witness that (a) is true.
Let such that . There is such that . Therefore, for each and all with , we have , that is, (a) is true.
(b) By (a) for each , there are a statistically dense set and such that and imply for each . Then is a family of density zero sets, that is, this family is contained in the ideal of all subsets of having density zero. It is known that is a -ideal (i.e., for each countable collection there is some such that is finite for each ), so there exists such that the set is finite for each . Let . Then is a statistically dense subset of .Claim 1. The sequence is uniformly exhaustive.
Let be fixed. Choose such that . Let be such that . It follows that for each , we have . Thus there is such that for all with we have for each greater than .Claim 2. There is with such that .
From assumptions, according to [9, Theorem 3.5], it follows that the function is continuous on , and so uniformly continuous since is compact. Fix . There is such that for all satisfying . By Claim 1, is uniformly exhaustive on , so that there exist and such that implies for all with . Let . Using compactness of choose a finite set such that . Since , for each there is a statistically dense set such that for each we have , . Set . Then is statistically dense and the sequence is still uniformly exhaustive. Each belongs to for some , and thus for each with we have which completes the proof of (Claim 2 and) the theorem.
Agata Caserta and Giuseppe Di Maio supported by GNSAGA, and Ljubiša D. R. Kočinac supported by MN RS, Grant 174025, and GNSAGA.
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