Statistical Convergence in Function Spaces

Abstract

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.

1. Introduction

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 [1] 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”) [2], and his work initiated a study that led to several outstanding papers. In 1905, Borel in [3] introduced the term “quasiuniform convergence” for the Arzelà condition, and Bartle in [4] 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 𝑋, not necessarily compact, to a metric space [5]. The reader may consult [6, 7] for the literature concerning the preservation of continuity of the limit function.

In 2009, Beer and Levi [8] 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 𝑋. In [6], a direct proof of the equivalence of Arzelà, Alexandroff, and Beer-Levi conditions was offered.

In [9], 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 [8].

In 2008, the definition of exhaustiveness, closely related to equicontinuity [10], for a family of functions (not necessarily continuous), was introduced by Gregoriades and Papanastassiou in [11]. 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 [9]. 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 [11] it was shown that equicontinuity implies exhaustiveness. We prove that exhaustiveness lies between equicontinuity and even continuity [10], 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, (𝑋,𝑑) and (𝑌,𝜌) will be metric spaces, 𝑌^𝑋 and 𝖢(𝑋,𝑌) the sets of all in all continuous mappings from 𝑋 to 𝑌. The pointwise (resp., uniform) topology on 𝑌^𝑋 and 𝖢(𝑋,𝑌) will be denoted by 𝜏_𝑝(resp.𝜏_𝑢). We denote by 𝒫₀(𝑋) the family of all nonempty subsets of 𝑋, and by ℱ(𝑋), or simply ℱ, the family of all nonempty finite subsets of 𝑋. If 𝑥₀∈(𝑋,𝑑), 𝐴⊂𝑋⧵{∅}, and 𝜀>0, we write 𝑆(𝑥₀,𝜀) for the open 𝜀-ball with center 𝑥₀, and 𝐴^𝜀=⋃_𝑥∈𝐴𝑆_𝜀(𝑥) for the 𝜀-enlargement of 𝐴.

Recall that a bornology 𝔅 on a space (𝑋,𝑑) is a hereditary family of subsets of 𝑋 which covers 𝑋 and is closed under taking finite unions (see [12, 13]). By a base for a bornology 𝔅, we mean a subfamily 𝔅₀ of 𝔅 that is cofinal with respect to inclusion. The smallest bornology on 𝑋 is the family ℱ(𝑋), and the largest is the family 𝒫₀(𝑋).

In [8], as mentioned above, the notions of strong uniform continuity of a function on a bornology 𝔅 and the topology of strong uniform convergence on 𝔅 for function spaces were introduced.

Definition 2.1 (see [8]). Let (𝑋,𝑑) and (𝑌,𝜌) be metric spaces, and let 𝐵 be a subset of 𝑋. A function 𝑓∶𝑋→𝑌 is strongly uniformly continuous on 𝐵 if for each 𝜀>0 there is 𝛿>0 such that if 𝑑(𝑥,𝑧)<𝛿 and {𝑥,𝑧}∩𝐵≠∅, then 𝜌(𝑓(𝑥),𝑓(𝑧))<𝜀.

If 𝔅 is a family of nonempty subsets of 𝑋 and (𝑌,𝜌) a metric space, a function 𝑓∈𝑌^𝑋 is called uniformly continuous (resp., strongly uniformly continuous) on 𝔅 if for each 𝐵∈𝔅, 𝑓↾𝐵 is uniformly continuous (resp., strongly uniformly continuous) on 𝐵. We denote by 𝖢(𝑋,𝑌)^𝑠_𝔅 the set of all strongly uniformly continuous functions on 𝔅.

Given a bornology 𝔅 with closed base on 𝑋, Beer and Levi presented a new uniformizable topology on the set 𝑌^𝑋.

Definition 2.2 (see [8]). Let (𝑋,𝑑) and (𝑌,𝜌) be metric spaces, and let 𝔅 be a bornology with a closed base on 𝑋. The topology 𝜏^𝑠_𝔅 of strong uniform convergence is determined by the uniformity on 𝑌^𝑋 having as a base all sets of the form [𝐵;𝜀]^𝑠∶=𝑓,𝑔∶∃𝛿>0foreach𝑥∈𝐵^𝛿𝜌𝑓(𝑥),𝑔(𝑥)<𝜀,(𝐵∈𝔅,𝜀>0). (2.1)
On 𝖢(𝑋,𝑌), this topology is in general finer than the classical topology of uniform convergence on 𝔅. 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 [1], [7, page  268])). Let (𝑓_𝑛)_𝑛∈ℕ be a sequence of real-valued continuous functions defined on an arbitrary set 𝑋, and let 𝑓∶𝑋→ℝ. The sequence (𝑓_𝑛)_𝑛∈ℕ is said to converge to 𝑓 quasiuniformly on 𝑋 if it pointwise converges to 𝑓, and for each 𝜀>0 and 𝑛₀∈ℕ, there exists a finite number of indices 𝑛₁,𝑛₂,…,𝑛_𝑘≥𝑛₀ such that for each 𝑥∈𝑋 at least one of the following inequalities holds: 𝑓_𝑛𝑖(𝑥)−𝑓(𝑥)<𝜀,𝑖=1,…,𝑘. (2.2)

Definition 2.4 (Alexandroff [5]). Let (𝑓_𝑛)_𝑛∈ℕ be a sequence in 𝖢(𝑋,𝑌) and 𝑓∈𝑌^𝑋. Then (𝑓_𝑛)_𝑛∈ℕ is Alexandroff convergent to 𝑓 on 𝑋, provided it pointwise converges to 𝑓, and for each 𝜀>0 and each 𝑛₀∈ℕ, there exist a countable open cover {𝑈₁,𝑈₂,…} of 𝑋 and a sequence (𝑛_𝑘)_𝑘∈ℕ, of positive integers greater than 𝑛₀ such that for each 𝑥∈𝑈_𝑘 we have 𝜌(𝑓_𝑛𝑘(𝑥),𝑓(𝑥))<𝜀.

Theorem 2.5 (see [6]). If a net (𝑓_𝛼)_𝛼∈𝐷 in 𝖢(𝑋,𝑌) pointwise converges to 𝑓∈𝑌^𝑋, then the following are equivalent: (i)𝑓 is continuous; (ii)(𝑓_𝛼)_𝛼∈𝐷 Alexandroff converges to 𝑓; (iii)(𝑓_𝛼)_𝛼∈𝐷 converges to 𝑓 quasiuniformly on compacta; (iv)(𝑓_𝛼)_𝛼∈𝐷𝜏^𝑠_ℱ-converges to 𝑓.

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 [17] of Zygmund. Explicitly, the notion of statistical convergence of sequences of real numbers was introduced by Fast in [18] and Steinhaus in [19] and is based on the notion of asymptotic density 𝜕(𝐴) of a set 𝐴⊂ℕ: 𝜕(𝐴)=lim_𝑛→∞|{𝑘∈𝐴∶𝑘≤𝑛}|𝑛. (2.3) We recall that 𝜕(ℕ⧵𝐴)=1−𝜕(𝐴) for 𝐴⊂ℕ. A set 𝐴⊂𝑋 is said to be statistically dense if 𝜕(𝐴)=1.

Fact 1. The union and intersection of two statistically dense sets in ℕ 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 [20] (where statistical convergence was generalized to sequences in topological and uniform spaces) and references therein, and about some applications see [21, 22].

A sequence (𝑥_𝑛)_𝑛∈ℕ in a topological space 𝑋  statistically converges (or shortly, 𝗌𝗍-converges) to 𝑥∈𝑋 if for each neighborhood 𝑈 of 𝑥, 𝜕({𝑛∈ℕ∶𝑥_𝑛∉𝑈})=0 [20]. This will be denoted by (𝑥_𝑛)−−−→_𝑛∈ℕ^{𝗌𝗍-𝜏}𝑥, where 𝜏 is a topology on 𝑋.

It was shown in [20, Theorem  2.2] (see [23, 24] for 𝑋=ℝ) that for first countable spaces this definition is equivalent to the following statement.

Fact 2. There exists a subset 𝐴 of ℕ with 𝜕(𝐴)=1 such that the sequence (𝑥_𝑛)_𝑛∈𝐴 converges to 𝑥.

Facts 1 and 2 will be used in the sequel without special mention.

The reader is referred to [7, 10, 25–27] for standard notation and terminology.

3. Statistical Arzelà and Alexandroff Convergence

In [9], a statistical version of the Alexandroff convergence was defined.

Definition 3.1. A sequence (𝑓_𝑛)_𝑛∈ℕ in 𝖢(𝑋,𝑌) is said to be statistically Alexandroff convergent to 𝑓∈𝑌^𝑋, denoted by (𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝖠𝗅}𝑓, provided (𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝜏𝑝}𝑓, and for each 𝜀>0 and each statistically dense set 𝐴⊂ℕ, there exist an open cover 𝒰={𝑈_𝑛∶𝑛∈𝐴} and an infinite set 𝑀_𝐴={𝑛₁<𝑛₂<⋯𝑛_𝑘<⋯}⊂𝐴 such that for each 𝑥∈𝑈_𝑘 we have 𝜌(𝑓_𝑛𝑘(𝑥),𝑓(𝑥))<𝜀.

Below, a statistical version of the celebrated Arzelà’s quasiuniform convergence is given.

Definition 3.2. A sequence (𝑓_𝑛)_𝑛∈ℕ in 𝖢(𝑋,𝑌) is said to be statistically Arzelà convergent to 𝑓∈𝑌^𝑋, denoted by (𝑓_𝑛)−−−−−→_𝑛∈ℕ^{𝗌𝗍-𝖠𝗋𝗓}𝑓, if (𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝜏𝑝}𝑓, and for each 𝜀>0 and each statistically dense set 𝐴⊂ℕ there exists a finite set {𝑛₁,𝑛₂,…,𝑛_𝑘}⊂𝐴 such that for each 𝑥∈𝑋 it holds that 𝜌(𝑓_𝑛𝑖(𝑥),𝑓(𝑥))<𝜀 for at least one 𝑖≤𝑘.

Theorem 3.3. For a sequence (𝑓_𝑛)_𝑛∈ℕ in 𝖢(𝑋,𝑌) such that (𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝜏𝑝}𝑓∈𝑌^𝑋, the following is equivalent: (1)𝑓 is continuous; (2)(𝑓_𝑛)−−−−−→_𝑛∈ℕ^{𝗌𝗍-𝖠𝗋𝗓}𝑓 on compacta; (3)(𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝜏𝑠ℱ}𝑓; (4)(𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝖠𝗅}𝑓.

Proof. (1)⇒(2): Let a compact set 𝐾⊂𝑋, a statistically dense set 𝐴⊂ℕ, and 𝜀>0 be fixed. Since (𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝜏𝑝}𝑓, for each 𝑦∈𝑋 there is a statistically dense set 𝐴_𝑦⊂ℕ such that 𝜌(𝑓_𝑛(𝑦),𝑓(𝑦))<𝜀 for each 𝑛∈𝐴_𝑦. Choose 𝑛_𝑦∈𝐴_𝑦∩𝐴 and set 𝑈_𝑦=𝑥∈𝑋∶𝜌𝑓_𝑛𝑦(𝑥),𝑓(𝑥)<𝜀. (3.1) Since all functions 𝑓_𝑛 and 𝑓 are continuous, the sets 𝑈_𝑦 are open, and thus {𝑈_𝑦∶𝑦∈𝐾} is an open cover of 𝐾. By compactness of 𝐾 there are 𝑦₁,𝑦₂,…,𝑦_𝑘∈𝐾 such that 𝐾=⋃^𝑘_𝑖=1𝑈_𝑦𝑖. The set {𝑛_𝑦𝑖∶𝑖=1,2,…,𝑘} is a finite subset of 𝐴 such that for each 𝑥∈𝐾 it holds 𝜌(𝑓_𝑛𝑦𝑖(𝑥),𝑓(𝑥))<𝜀 for at least one 𝑖≤𝑘, that is, (2) is true.
(2)⇒(3): It suffices to show that for each 𝑥∈𝑋 and each 𝜀>0 we have 𝜕({𝑛∈ℕ∶𝑓_𝑛∉[{𝑥},𝜀]^𝑠(𝑓)})=0. Since (𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝜏𝑝}𝑓, there is a set 𝐴⊂ℕ with 𝜕(𝐴)=1 so that 𝜌(𝑓_𝑛(𝑥),𝑓(𝑥))<𝜀/4 for each 𝑛∈𝐴. We are going to prove that for each 𝑛∈𝐴 there is 𝛿_𝑛>0 such that for each 𝑦∈𝑆(𝑥,𝛿_𝑛), 𝜌(𝑓_𝑛(𝑦),𝑓(𝑦))<𝜀. Suppose, by contradiction, that this assumption fails. Then there is 𝑛₀∈𝐴 and a sequence (𝑥_𝑗)_𝑗∈ℕ converging to 𝑥 such that 𝜌(𝑓_𝑛0(𝑥_𝑗),𝑓(𝑥_𝑗))≥𝜀 for each 𝑗∈𝐴. The set 𝐾={𝑥_𝑗∶𝑗∈ℕ}∪{𝑥} is compact so that, by (2), there are 𝑚₁,⋯,𝑚_𝑘∈𝐴 such that for each 𝑧∈𝐾, 𝜌(𝑓_𝑚𝑖(𝑧),𝑓(𝑧))<𝜀/4 holds for at least one 𝑖≤𝑘. Therefore, we found 𝑖≤𝑘 such that there is an infinite set 𝖢⊂𝐾 with the property that for each 𝑧∈𝐶, 𝜌(𝑓_𝑚𝑖(𝑧),𝑓(𝑧))<𝜀/4. For this 𝑚_𝑖, we have 𝜌𝑓_𝑚𝑖(𝑥),𝑓_𝑛0(𝑥)≤𝜌𝑓_𝑚𝑖(𝑥),𝑓(𝑥)+𝜌𝑓(𝑥),𝑓_𝑛0(𝑥)<𝜀2. (3.2) Since 𝑓_𝑚𝑖 and 𝑓 are continuous at 𝑥, there are 𝛿_𝑚𝑖>0 and 𝛿₀>0 such that 𝜌(𝑓_𝑚𝑖(𝑢),𝑓_𝑚𝑖(𝑥))<𝜀/8 for each 𝑢∈𝑆(𝑥,𝛿_𝑚𝑖), and 𝜌(𝑓_𝑛0(𝑢),𝑓_𝑛0(𝑥))<𝜀/8 for each 𝑢∈𝑆(𝑥,𝛿₀). If 𝛿=min{𝛿₀,𝛿_𝑚𝑖}, then for each 𝑧∈𝖢∩𝑆(𝑥,𝛿) we have 𝜌𝑓_𝑚𝑖(𝑧),𝑓_𝑛0(𝑧)≤𝜌𝑓_𝑚𝑖(𝑧),𝑓_𝑚𝑖(𝑥)+𝜌𝑓_𝑚𝑖(𝑥),𝑓_𝑛0(𝑥)+𝜌𝑓_𝑛0(𝑥),𝑓_𝑛0(𝑧)<3𝜀4. (3.3) Since (𝑥_𝑗)_𝑗∈ℕ converges to 𝑥, there is 𝑗^∗∈𝐴 such that 𝑥_𝑗^∗∈𝐶∩𝑆(𝑥,𝛿). For this 𝑗^∗, we have 𝜌𝑓_𝑛0𝑥_𝑗^∗,𝑓𝑥_𝑗^∗≤𝜌𝑓_𝑛0𝑥_𝑗^∗,𝑓_𝑚𝑖𝑥_𝑗^∗+𝜌𝑓_𝑚𝑖𝑥_𝑗^∗,𝑓𝑥_𝑗^∗<3𝜀4+𝜀4=𝜀, (3.4) which is a contradiction.
(3)⇒(4): Let 𝜀>0 and a statistically dense set 𝐴⊂ℕ be given. Since (𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝜏𝑠ℱ}𝑓, given [{𝑥},𝜀]^𝑠(𝑓), there is 𝐵_𝑥⊂ℕ statistically dense, such that for each 𝑛∈𝐵_𝑥 we have 𝑓_𝑛∈[{𝑥},𝜀]^𝑠(𝑓). Hence there is a 𝛿_𝑛,𝑥 such that for each 𝑦∈𝑆(𝑥,𝛿_𝑛,𝑥) and each 𝑛∈𝐵_𝑥 we have 𝜌(𝑓_𝑛(𝑦),𝑓(𝑦))<𝜀. Let 𝐵=⋃_𝑥∈𝑋𝐵_𝑥. For each 𝑛∈𝐵∩𝐴, define 𝐸_𝑛∶=𝑥∈𝑋∶∀𝑚∈𝐵_𝑥∩𝐴,𝑚≥𝑛,𝜌𝑓_𝑚𝑦,𝑓𝑦<𝜀∀𝑦∈𝑆𝑥,𝛿_𝑛,𝑥. (3.5) Note that 𝑋=⋃_𝑛∈𝐴𝐸_𝑛. For each 𝑛∈𝐴, let 𝑈_𝑛 be the following open set: 𝑈_𝑛=∅,if𝑛∈𝐴⧵𝐵,_{𝑥∈𝐸𝑛}𝑆𝑥,𝛿_𝑛,𝑥,if𝑛∈𝐴∩𝐵. (3.6) Then {𝑈_𝑛∶𝑛∈𝐴} is an open cover of 𝑋. Thus for each 𝑘∈𝐴 and each 𝑥∈𝑈_𝑘, there is some 𝑚∈𝐵 such that it holds that 𝜌(𝑓_𝑚(𝑥),𝑓(𝑥))<𝜀, that is, the set 𝐴∩𝐵={𝑛₁<𝑛₂<⋯}⊂𝐴 and the cover {𝑈_𝑛∶𝑛∈𝐴} witness that (4) is true.
(4)⇒(1): It is proved in [9, Theorem  4.7].

The following two theorems use other kinds of statistical convergence, related to Dini convergence [28][29, pages 105-106] and Arzelà convergence, which imply continuity and strong uniform continuity of the limit function.

Definition 3.4. A sequence (𝑓_𝑛)_𝑛∈ℕ in 𝖢(𝑋,𝑌) is said to be statistically Dini convergent to 𝑓∈𝑌^𝑋, denoted by (𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝖣𝗂}𝑓 if (𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝜏𝑝}𝑓 and for each 𝜀>0 and each statistically dense set 𝐴⊂ℕ there exists an increasing sequence 𝑚₁<𝑚₂<⋯ in 𝐴 such that 𝜌(𝑓_𝑚𝑖(𝑥),𝑓(𝑥))<𝜀 for each 𝑥∈𝑋 and each 𝑖∈ℕ.

Theorem 3.5. If a sequence (𝑓_𝑛)_𝑛∈ℕ in 𝖢(𝑋,𝑌)  statistically Dini converges to 𝑓∈𝑌^𝑋, then 𝑓 is continuous.

Proof. Let 𝑥₀∈𝑋 and 𝜀>0 be given. Since (𝑓_𝑛(𝑥₀))−→_𝑛∈ℕ^𝗌𝗍𝑓(𝑥₀), there is a statistically dense set 𝐴⊂ℕ such that 𝜌(𝑓_𝑛(𝑥₀),𝑓(𝑥₀))<𝜀/3 for each 𝑛∈𝐴. Because (𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝖣𝗂}𝑓, there exists an increasing sequence 𝑚₁<𝑚₂<⋯ in 𝐴 such that 𝜌(𝑓_𝑚𝑖(𝑥),𝑓(𝑥))<𝜀 for each 𝑥∈𝑋 and each 𝑖∈ℕ. Take some 𝑛_𝑘. Since 𝑓_𝑛𝑘 is continuous at 𝑥₀, there is 𝛿>0 such that 𝜌(𝑓_𝑛𝑘(𝑥),𝑓_𝑛𝑘(𝑥₀))<𝜀/3 whenever 𝑑(𝑥,𝑥₀)<𝛿. So, for each 𝑥∈𝑆(𝑥₀,𝛿), we have 𝜌𝑓(𝑥),𝑓𝑥₀≤𝜌𝑓(𝑥),𝑓_𝑚𝑘(𝑥)+𝜌𝑓_𝑚𝑘(𝑥),𝑓_𝑚𝑘𝑥₀+𝜌𝑓_𝑚𝑘𝑥₀,𝑓𝑥₀<𝜀, (3.7) that is, 𝑓 is continuous at 𝑥₀, hence on 𝑋.

Definition 3.6. A sequence (𝑓_𝑛)_𝑛∈ℕ in 𝖢(𝑋,𝑌)statistically strongly Arzelà converges to a function 𝑓∈𝑌^𝑋 on a bornology 𝔅 on 𝑋, denoted by (𝑓_𝑛)−−−−−−→_𝑛∈ℕ^{𝗌𝗍-𝗌.𝖠𝗋𝗓}𝑓, if (𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝜏𝔅}𝑓 and for each 𝐵∈𝔅, 𝜀>0 and each statistically dense set 𝐴 in ℕ, there are finitely many 𝑛₁,…,𝑛_𝑘∈𝐴 such that 𝑓_𝑛𝑖∈[𝐵,𝜀]^𝑠(𝑓) for at least one 𝑖≤𝑘.

Theorem 3.7. If a sequence (𝑓_𝑛)_𝑛∈ℕ in 𝖢^𝑠_𝔅(𝑋,𝑌) statistically strongly Arzelà converges to 𝑓∈𝑌^𝑋 on a bornology 𝔅 with closed base on 𝑋, then 𝑓 is a strongly uniformly continuous on 𝔅.

Proof. Let 𝐵∈𝔅 and 𝜀>0. As (𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝜏𝔅}𝑓, there is a statistically dense set 𝐴⊂ℕ such that for each 𝑛∈A we have 𝑓_𝑛∈[𝐵,𝜀/3](𝑓), that is, for each 𝑥∈𝐵, 𝜌(𝑓_𝑛(𝑥),𝑓(𝑥))<𝜀/3. By assumption, there are 𝑛₁,…,𝑛_𝑘∈𝐴 with 𝑓_𝑛𝑚∈[𝐵,𝜀/3]^𝑠(𝑓) for some 𝑚≤𝑘, that is, there exists 𝛿_𝑚>0 such that 𝜌(𝑓_𝑛𝑚(𝑥),𝑓(𝑥))<𝜀/3 for each 𝑥∈𝐵^𝛿𝑚. Since 𝑓_𝑛𝑚 is strongly uniformly continuous on 𝐵, there is 𝛿₀>0 so that for each 𝑥∈𝐵 and each 𝑦∈𝐵^𝛿0 with 𝑑(𝑥,𝑦)<𝛿₀, we have 𝜌(𝑓_𝑛𝑚(𝑥),𝑓_𝑛𝑚(𝑦))<𝜀/3. Set 𝛿=min{𝛿_𝑚,𝛿₀}. Then for each 𝑥∈𝐵 and 𝑦∈𝑋 with 𝑑(𝑥,𝑦)<𝛿 by the above relations, it follows 𝜌𝑓(𝑥),𝑓𝑦≤𝜌𝑓(𝑥),𝑓_𝑛𝑚(𝑥)+𝜌𝑓_𝑛𝑚(𝑥),𝑓_𝑛𝑚𝑦+𝜌𝑓_𝑛𝑚𝑦,𝑓𝑦<𝜀, (3.8) that is, 𝑓 is strongly uniformly continuous on 𝐵, hence on 𝔅.

Theorem 3.8. Let 𝑋 be a compact space, 𝔅 a bornology on 𝑋 with closed base, and (𝑓_𝑛)_𝑛∈ℕ a sequence in 𝖢^𝑠_𝔅(𝑋,𝑌). If (𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝜏𝔅}𝑓 and 𝑓 is strongly uniformly continuous on 𝔅, then (𝑓_𝑛)_𝑛∈ℕ statistically strongly Arzelà converges to 𝑓 on 𝔅.

Proof. Let 𝐵∈𝔅, 𝜀>0, and a statistically dense set 𝐴⊂ℕ be given. Since 𝑓 is strongly uniformly continuous on 𝐵, there is 𝛿₀>0 such that for each 𝑥∈𝐵 and each 𝑦∈𝐵^𝛿0 with 𝑑(𝑥,𝑦)<𝛿₀ we have 𝜌(𝑓(𝑥),𝑓(𝑦))<𝜀/3. From (𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝜏𝔅}𝑓, it follows that there is a statistically dense set 𝐵⊂ℕ such that for each 𝑛∈𝐶, 𝐶=𝐴∩𝐵, we have 𝑓_𝑛∈[𝐵,𝜀/3](𝑓), that is, for each 𝑛∈𝐶, and each 𝑥∈𝐵 it holds that 𝜌(𝑓_𝑛(𝑥),𝑓(𝑥))<𝜀/3. For each 𝑛∈𝐶 set 𝑈_𝑛={𝑥∈𝑋∶𝜌(𝑓_𝑛(𝑥),𝑓(𝑥))<𝜀/3}. Since 𝑓 and 𝑓_𝑛’s are continuous, each 𝑈_𝑛 is open in 𝑋, so that {𝑈_𝑛∶𝑛∈ℕ} is an open cover of 𝑋. By compactness of 𝑋, there are finitely many 𝑛₁,…,𝑛_𝑘∈𝖢 such that 𝑋=⋃^𝑘_𝑖=1𝑈_𝑛𝑖. But each 𝑓_𝑛𝑖 is strongly uniformly continuous on 𝐵, so that for each 𝑖≤𝑘 there is 𝛿_𝑖>0 such that 𝜌(𝑓_𝑛𝑖(𝑥),𝑓_𝑛𝑖(𝑦))<𝜀/3 whenever 𝑥∈𝐵 and 𝑦∈𝐵^𝛿𝑖, 𝜌(𝑥,𝑦)<𝛿_𝑖. Let 𝛿=min{𝛿₀,𝛿₁,…,𝛿_𝑘}. Then for 𝑥∈𝐵 and 𝑦∈𝐵^𝛿 with 𝑑(𝑥,𝑦)<𝛿, since 𝑥∈𝑈_𝑛𝑚 for some 𝑚≤𝑘, we have 𝜌𝑓_𝑛𝑚𝑦,𝑓𝑦≤𝜌𝑓_𝑛𝑚𝑦,𝑓_𝑛𝑚(𝑥)+𝜌𝑓_𝑛𝑚(𝑥),𝑓(𝑥)+𝜌𝑓(𝑥),𝑓𝑦<𝜀. (3.9) So, 𝑓_𝑛𝑚∈[𝐵,𝜀]^𝑠(𝑓) which completes the proof.

Theorem 3.9. Let 𝔅 be a bornology on 𝑋 with closed base, and let (𝑓_𝑛)_𝑛∈ℕ be a sequence in 𝖢^𝑠_𝔅(𝑋,𝑌) such that (𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝜏𝔅}𝑓. Then 𝑓 is strongly uniformly continuous on 𝔅 if and only if (𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝜏𝑠𝔅}𝑓.

Proof. By [8, Proposition  6.5], we have 𝜏^𝑠_𝔅=𝜏_𝔅. So, it suffices to prove that (𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝜏𝑠𝔅}𝑓 implies strong uniform continuity of 𝑓 on 𝔅.
Assume that 𝑓 is not strongly uniformly continuous on 𝔅. There are a 𝐵∈𝔅 and 𝜀>0 such that for each 𝑛∈ℕ, there are points 𝑥_𝑛,𝑧_𝑛∈𝐵^1/𝑛 with 𝑑(𝑥_𝑛,𝑧_𝑛)<1/𝑛 such that 𝜌(𝑓(𝑥_𝑛),𝑓(𝑧_𝑛))≥𝜀. Since (𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝜏𝑠𝔅}𝑓, the density of the set 𝐴={𝑛∈ℕ∶𝑓_𝑛∉[𝐵,𝜀/3]^𝑠(𝑓)} is 0. Let 𝑀=ℕ⧵𝐴. Then 𝑀 is statistically dense in ℕ, and there exist 𝑚∈𝑀, 𝑥_𝑚,𝑧_𝑚∈𝐵^1/𝑚, 𝑑(𝑥_𝑚,𝑧_𝑚)<1/𝑚, such that 𝜌(𝑓(𝑥_𝑚),𝑓_𝑚(𝑥_𝑚))<𝜀/3, and 𝜌(𝑓(𝑧_𝑚),𝑓_𝑚(𝑧_𝑚))<𝜀/3. Thus 𝜀≤𝜌𝑓𝑥_𝑚,𝑓𝑧_𝑚≤𝜌𝑓𝑥_𝑚,𝑓_𝑚𝑥_𝑚+𝜌𝑓_𝑚𝑥_𝑚,𝑓_𝑚𝑧_𝑚+𝜌𝑓_𝑚𝑧_𝑚,𝑓𝑧_𝑚, (3.10) and so 𝜌𝑓_𝑚𝑥_𝑚,𝑓_𝑚𝑧_𝑚≥𝜀−𝜌𝑓𝑥_𝑚,𝑓_𝑚𝑥_𝑚−𝜌𝑓_𝑚𝑧_𝑚,𝑓𝑧_𝑚>𝜀3, (3.11) that is, 𝑓_𝑚 is not strongly uniformly continuous on 𝐵. A contradiction.

4. More on (Statistical) Exhaustiveness

As we mentioned in Introduction, in 2008 the notion of exhaustiveness was introduced in [11]. We recall the definition for both families and nets of functions [11].

Definition 4.1. Let ℳ be a family and (𝑓_𝑛)_𝑛∈ℕ a sequence in 𝑌^𝑋. If case ℳ is finite, we say that ℳ is exhaustive at 𝑥∈𝑋 if all functions in ℳ are continuous at 𝑥. If ℳ is infinite, then ℳ is exhaustive at 𝑥∈𝑋 if for each 𝜀>0 there exist 𝛿>0 and a finite set 𝒜⊂ℳ such that for each 𝑦∈𝑆(𝑥,𝛿) and for each 𝑓∈ℳ⧵𝒜, we have 𝜌(𝑓(𝑥),𝑓(𝑦))<𝜀. The sequence (𝑓_𝑛)_𝑛∈ℕ is exhaustive at 𝑥 if the family {𝑓_𝑛∶𝑛∈ℕ} is exhaustive at 𝑥. The family ℳ (sequence (𝑓_𝑛)_𝑛∈ℕ) is exhaustive on 𝑋 if it is exhaustive at each 𝑥∈𝑋.

In [15], 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 [11], 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 [9], 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 [9]). A sequence (𝑓_𝑛)_𝑛∈ℕ in 𝑌^𝑋 is said to be statistically exhaustive (shortly, 𝗌𝗍-exhaustive) at a point 𝑥∈𝑋 if for each 𝜀>0 there are 𝛿>0 and a statistically dense set M⊂ℕ such that for each 𝑦∈𝑆(𝑥,𝛿) we have 𝜌(𝑓_𝑛(𝑦),𝑓_𝑛(𝑥))<𝜀 for each 𝑛∈M. The sequence (𝑓_𝑛)_𝑛∈ℕ is 𝗌𝗍-exhaustive if it is 𝑠𝑡-exhaustive at each 𝑥∈𝑋.

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 [11] it was shown that equicontinuity implies exhaustiveness.

Definition 4.3 (see [10, L p. 241]). A family ℳ⊂𝑌^𝑋 is evenly continuous if for each net (𝑓_𝛼,𝑥_𝛼)_𝛼∈Λ in ℳ×𝑋 such that (𝑥_𝛼)_𝛼∈Λ converges to 𝑥∈𝑋 and (𝑓_𝛼(𝑥))_𝛼∈Λ converges to 𝑦∈𝑌, the net (𝑓_𝛼(𝑥_𝛼))_𝛼∈Λ converges to 𝑦.

Definition 4.4 (see [10]). A family ℳ⊂𝑌^𝑋 is equicontinuous at a point 𝑥 if and only if for each 𝜀>0 there is a neighborhood 𝑈 of 𝑥 such that 𝑓(𝑈)⊂𝑆(𝑓(𝑥),𝜀) for each member 𝑓 of ℳ. A family ℳ is equicontinuous if it is equicontinuous at each 𝑥∈𝑋.

Theorem 4.5. If a family ℳ⊂𝑌^𝑋 is exhaustive, then ℳ is evenly continuous.

Proof. If ℳ is finite there is nothing to prove, and thus we assume that ℳ is infinite. Let (𝑓_𝛼,𝑥_𝛼)_𝛼∈Λ be a net in ℳ×𝑋 satisfying (𝑥_𝛼)_𝛼∈Λ converges to 𝑥 and (𝑓_𝛼(𝑥))_𝛼∈Λ converges to 𝑦∈𝑌, and let 𝜀>0. As ℳ is exhaustive at 𝑥, there exist 𝛿_𝑥>0 and {𝑓_𝛽1,𝑓_𝛽2,…,𝑓_𝛽𝑘}⊂ℳ such that for each 𝑧∈𝑆(𝑥,𝛿_𝑥) and each 𝑓∈ℳ⧵{𝑓_𝛽1,𝑓_𝛽2,…,𝑓_𝛽𝑘} we have 𝜌(𝑓(𝑥),𝑓(𝑧))<𝜀/2. Because the set {𝑓_𝛽1,𝑓_𝛽2,…,𝑓_𝛽𝑘} is finite, there is some 𝛼^∗∈Λ such that 𝑓_𝛼≠𝑓_𝛽𝑖 for each 𝛼≥𝛼^∗ and each 𝑖≤𝑘.
Let 𝛿=min{𝛿_𝑥,𝜀/2}. Since (𝑥_𝛼)_𝛼∈Λ converges to 𝑥 and (𝑓_𝛼(𝑥))_𝛼∈Λ converges to 𝑦 there is 𝛼₀≥𝛼^∗ in Λ such that 𝑥_𝛼∈𝑆(𝑥,𝛿) and 𝑓_𝛼(𝑥)∈𝑆(𝑦,𝛿) for each 𝛼≥𝛼₀. Then 𝑓_𝛼(𝑥_𝛼)∈𝑆(𝑓_𝛼(𝑥),𝜀/2) and 𝑓_𝛼(𝑥)∈𝑆(𝑦,𝛿)⊂𝑆(𝑦,𝜀/2) for each 𝛼≥𝛼₀. So for each 𝛼≥𝛼₀, we have 𝜌𝑓_𝛼𝑥_𝛼,𝑦≤𝜌𝑓_𝛼𝑥_𝛼,𝑓_𝛼(𝑥)+𝜌𝑓_𝛼(𝑥),𝑦<𝜀, (4.1) that is, (𝑓_𝛼(𝑥_𝛼))_𝛼∈Λconverges to 𝑦.

Recall that the concept of uniform exhaustiveness was defined in [15, Definition  4.1] under the name strong exhaustiveness: a sequence (𝑓_𝑛)_𝑛∈ℕ in 𝑌^𝑋 is strongly exhaustive on 𝑋 if for each 𝜀>0 there are 𝛿>0 and 𝑛₀∈ℕ such that for all 𝑥,𝑦∈𝑋 with 𝑑(𝑥,𝑦)<𝛿, 𝜌(𝑓_𝑛(𝑥),𝑓_𝑛(𝑦))<𝜀 for each 𝑛≥𝑛₀.

The novel notion of statistical uniform exhaustiveness for a sequence is related to uniform convergence.

Definition 4.6. A sequence (𝑓_𝑛)_𝑛∈ℕ in 𝑌^𝑋 is 𝗌𝗍-uniformly exhaustive on 𝑋 if for each 𝜀>0 there are 𝛿>0 and a statistically dense set 𝐴⊂ℕ such that for all 𝑥,𝑦∈𝑋 with 𝑑(𝑥,𝑦)<𝛿, 𝜌(𝑓_𝑛(𝑥),𝑓_𝑛(𝑦))<𝜀 for all 𝑛∈𝐴.

Theorem 4.7. Let (𝑋,𝑑) be a compact space, and let (𝑓_𝑛)_𝑛∈ℕ be a 𝑠𝑡-exhaustive sequence in 𝑌^𝑋 such that (𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝜏𝑝}𝑓. Then (a)(𝑓_𝑛)_𝑛∈ℕ is statistically uniformly exhaustive; (b)there is a statistically dense set 𝑀⊂ℕ such that (𝑓_m)_𝑚∈𝑀 is uniformly exhaustive and (𝑓_m)−−→^𝜏_𝑚∈𝑀^𝑢𝑓.

Proof. (a) Let 𝜀>0 and 𝑥∈𝑋 be fixed. Since, by hypothesis, (𝑓_𝑛)_𝑛∈ℕ is statistically exhaustive at 𝑥, there are 𝛿_𝑥>0 and a statistically dense set 𝐴_𝑥⊂ℕ such that for each 𝑧∈𝑆(𝑥,𝛿_𝑥) and each 𝑛∈𝐴_𝑥, it holds 𝜌(𝑓_𝑛(𝑥),𝑓_𝑛(𝑧))<𝜀/2. From 𝑋=⋃_𝑥∈𝑋𝑆(𝑥,𝛿_𝑥/2) and compactness of 𝑋, it follows the existence of finitely many points 𝑥₁,…,𝑥_𝑘 in 𝑋 such that 𝑋=⋃^𝑘_𝑖=1𝑆(𝑥_𝑖,𝛿_𝑥𝑖/2). Let 𝛿^∗=min{𝛿_𝑥𝑖/2∶𝑖≤𝑘} and 𝐴=⋂^𝑘_𝑖=1𝐴_𝑥𝑖. The set 𝐴 is statistically dense in ℕ. We claim that 𝛿^∗ and 𝐴 witness that (a) is true.
Let 𝑥,𝑧∈𝑋 such that 𝑑(𝑥,𝑧)<𝛿^∗. There is 𝑗≤𝑘 such that 𝑥,𝑧∈𝑆(𝑥_𝑗,𝛿_𝑥𝑗). Therefore, for each 𝑛∈𝐴 and all 𝑥,𝑧∈𝑋 with 𝑑(𝑥,𝑧)<𝛿^∗, we have 𝜌(𝑓_𝑛(𝑥),𝑓_𝑛(𝑧))<𝜌(𝑓_𝑛(𝑥),𝑓_𝑛(𝑥_𝑗))+𝜌(𝑓_𝑛(𝑥_𝑗),𝑓_𝑛(𝑧))<𝜀, that is, (a) is true.
(b) By (a) for each 𝑗∈ℕ, there are a statistically dense set 𝐴_𝑗⊂ℕ and 𝛿_𝑗>0 such that 𝑥,𝑦∈𝑋 and 𝑑(𝑥,𝑦)<𝛿_𝑗 imply 𝜌(𝑓_𝑛(𝑥),𝑓_𝑛(𝑦))<1/𝑗 for each 𝑛∈𝐴_𝑗. Then {ℕ⧵𝐴_𝑗∶𝑗∈ℕ} is a family of density zero sets, that is, this family is contained in the ideal ℐ_𝑑 of all subsets of ℕ having density zero. It is known that ℐ_𝑑 is a 𝑃-ideal (i.e., for each countable collection 𝒥⊂ℐ_𝑑 there is some 𝐿∈ℐ_𝑑 such that 𝐽⧵𝐿 is finite for each 𝐽∈𝒥), so there exists 𝐷∈ℐ_𝑑 such that the set (ℕ⧵𝐴_𝑗)⧵𝐷 is finite for each 𝑗∈ℕ. Let 𝑁₁=ℕ⧵𝐷∶={𝑛₁<𝑛₂<⋯<𝑛_𝑘<⋯}. Then 𝑁₁ is a statistically dense subset of ℕ.Claim 1. The sequence (𝑓_𝑛)_𝑛∈𝑁1 is uniformly exhaustive.
Let 𝜀>0 be fixed. Choose 𝑗∈ℕ such that 1/𝑗<𝜀. Let 𝑡_𝜀∈ℕ be such that (ℕ⧵𝐴_𝑗)⧵𝐷⊂{𝑛₁,𝑛₂,…,𝑛_𝑡𝜀}. It follows that for each 𝑡>𝑡_𝜀, we have 𝑛_𝑡∈𝐴_𝑗. Thus there is 𝑛^∗>𝑛_𝑡𝜀 such that for all 𝑥,𝑦∈𝑋 with 𝑑(𝑥,𝑦)<1/𝑗 we have 𝜌(𝑓_𝑛(𝑥),𝑓_𝑛(𝑦))<𝜀 for each 𝑛∈𝑁₁ greater than 𝑛^∗.Claim 2. There is 𝑀⊂ℕ with 𝜕(𝑀)=1 such that (𝑓_𝑚)−−→^𝜏_𝑚∈𝑀^𝑢𝑓.
From assumptions, according to [9, Theorem  3.5], it follows that the function 𝑓 is continuous on 𝑋, and so uniformly continuous since 𝑋 is compact. Fix 𝜀>0. There is 𝛿>0 such that 𝜌(𝑓(𝑥),𝑓(𝑦))<𝜀/4 for all 𝑥,𝑦∈𝑋 satisfying 𝑑(𝑥,𝑦)<𝛿. By Claim 1, (𝑓_𝑛)_𝑛∈𝑁1 is uniformly exhaustive on 𝑋, so that there exist 𝛿₀>0 and 𝑛_𝑘0∈𝑁₁ such that 𝑑(𝑥,𝑦)<𝛿₀ implies 𝜌(𝑓_𝑛𝑘(𝑥),𝑓_𝑛𝑘(𝑦))<𝜀/4 for all 𝑛_𝑘∈𝑁₁ with 𝑛_𝑘≥𝑛_𝑘0. Let 𝛿^∗=min{𝛿,𝛿₀}. Using compactness of 𝑋 choose a finite set {𝑥₁,…,𝑥_𝑠}⊂𝑋 such that 𝑋=⋃^𝑠_𝑖=1𝑆(𝑥_𝑖,𝛿^∗). Since (𝑓_𝑛)−−−−→_𝑛∈ℕ^{𝗌𝗍-𝜏𝑝}𝑓, for each 𝑖≤𝑠 there is a statistically dense set 𝐴_𝑖⊂ℕ such that for each 𝑛∈𝐴₀=⋂^𝑠_𝑖=1𝐴_𝑖 we have 𝜌(𝑓_𝑛(𝑥_𝑖),𝑓(𝑥_𝑖))<𝜀/4, 𝑖≤𝑠. Set 𝑀=𝐴₀∩𝑁₁. Then 𝑀 is statistically dense and the sequence (𝑓_𝑛)_𝑛∈ℕ is still uniformly exhaustive. Each 𝑦∈𝑋 belongs to 𝑆(𝑥_𝑖,𝛿^∗) for some 𝑖≤𝑠, and thus for each 𝑛_𝑘∈𝑀 with 𝑛_𝑘≥𝑛_𝑘0 we have 𝜌𝑓_𝑛𝑘𝑦,𝑓𝑦≤𝜌𝑓_𝑛𝑘𝑦,𝑓_𝑛𝑘𝑥_𝑖+𝜌𝑓_𝑛𝑘𝑥_𝑖,𝑓𝑥_𝑖+𝜌𝑓𝑥_𝑖,𝑓𝑦<𝜀, (4.2) which completes the proof of (Claim 2 and) the theorem.

Acknowledgments

Agata Caserta and Giuseppe Di Maio supported by GNSAGA, and Ljubiša D. R. Kočinac supported by MN RS, Grant 174025, and GNSAGA.