# Statistical Convergence in Function Spaces

Authors: Agata Caserta; Giuseppe Di Maio; Ljubiša D. R. Kočinac
Journal: Abstract and Applied Analysis
 (2011)

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## 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.


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## Body

## 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, YX and C(X,Y) the sets of all in all continuous mappings from X to Y. The pointwise (resp., uniform) topology on YX and C(X,Y) will be denoted by τp(resp.τu). 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 ɛ>0, we write S(x0,ɛ) for the open ɛ-ball with center x0, and Aɛ=⋃x∈ASɛ(x) for the ɛ-enlargement of A.Recall that a bornology𝔅 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 onB if for each ɛ>0 there is δ>0 such that if d(x,z)<δ and {x,z}∩B≠∅, then ρ(f(x),f(z))<ɛ.If𝔅 is a family of nonempty subsets of X and (Y,ρ) a metric space, a function f∈YX 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)𝔅s the set of all strongly uniformly continuous functions on 𝔅.Given a bornology𝔅 with closed base on X, Beer and Levi presented a new uniformizable topology on the set YX.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 topologyτ𝔅sof strong uniform convergence is determined by the uniformity on YX having as a base all sets of the form
(2.1)[B;ɛ]s:={(f,g):∃δ>0foreachx∈Bδρ(f(x),g(x))<ɛ},(B∈B,ɛ>0).
OnC(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(fn)n∈ℕ be a sequence of real-valued continuous functions defined on an arbitrary set X, and let f:X→ℝ. The sequence (fn)n∈ℕ is said to converge tofquasiuniformly on X if it pointwise converges to f, and for each ɛ>0 and n0∈ℕ, there exists a finite number of indices n1,n2,…,nk≥n0 such that for each x∈X at least one of the following inequalities holds:
(2.2)|fni(x)-f(x)|<ɛ,i=1,…,k.Definition 2.4 (Alexandroff [5]).
Let(fn)n∈ℕ be a sequence in C(X,Y) and f∈YX. Then (fn)n∈ℕ is Alexandroff convergent tof on X, provided it pointwise converges to f, and for each ɛ>0 and each n0∈ℕ, there exist a countable open cover {U1,U2,…} of X and a sequence (nk)k∈ℕ, of positive integers greater than n0 such that for each x∈Uk we have ρ(fnk(x),f(x))<ɛ.Theorem 2.5 (see [6]).
If a net(fα)α∈D in C(X,Y) pointwise converges to f∈YX, then the following are equivalent: (i)
f is continuous;(ii)
(fα)α∈D Alexandroff converges to f;(iii)
(fα)α∈D converges to f quasiuniformly on compacta;(iv)
(fα)α∈Dτℱs-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:
(2.3)∂(A)=lim⁡n→∞⁡|{k∈A:k≤n}|n.
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 inℕ 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(xn)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:xn∉U})=0 [20]. This will be denoted by (xn)n∈ℕ→st-τx, 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 subsetA of ℕ with ∂(A)=1 such that the sequence (xn)n∈A converges to x.Facts1 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(fn)n∈ℕ in C(X,Y) is said to be statistically Alexandroff convergent tof∈YX, denoted by (fn)n∈ℕ→st-Alf, provided (fn)n∈ℕ→st-τpf, and for each ɛ>0 and each statistically dense set A⊂ℕ, there exist an open cover 𝒰={Un:n∈A} and an infinite set MA={n1<n2<⋯nk<⋯}⊂A such that for each x∈Uk we have ρ(fnk(x),f(x))<ɛ.Below, a statistical version of the celebrated Arzelà’s quasiuniform convergence is given.Definition 3.2.
A sequence(fn)n∈ℕ in C(X,Y) is said to be statistically Arzelà convergent tof∈YX, denoted by (fn)n∈ℕ→st-Arzf, if (fn)n∈ℕ→st-τpf, and for each ɛ>0 and each statistically dense set A⊂ℕ there exists a finite set {n1,n2,…,nk}⊂A such that for each x∈X it holds that ρ(fni(x),f(x))<ɛ for at least one i≤k.Theorem 3.3.
For a sequence(fn)n∈ℕ in C(X,Y) such that (fn)n∈ℕ→st-τpf∈YX, the following is equivalent: (1)
f is continuous;(2)
(fn)n∈ℕ→st-Arzf on compacta;(3)
(fn)n∈ℕ→st-τℱsf;(4)
(fn)n∈ℕ→st-Alf.Proof.
(1)⇒(2): Let a compact set K⊂X, a statistically dense set A⊂ℕ, and ɛ>0 be fixed. Since (fn)n∈ℕ→st-τpf, for each y∈X there is a statistically dense set Ay⊂ℕ such that ρ(fn(y),f(y))<ɛ for each n∈Ay. Choose ny∈Ay∩A and set
(3.1)Uy={x∈X:ρ(fny(x),f(x))<ɛ}.
Since all functions fn and f are continuous, the sets Uy are open, and thus {Uy:y∈K} is an open cover of K. By compactness of K there are y1,y2,…,yk∈K such that K=⋃i=1kUyi. The set {nyi:i=1,2,…,k} is a finite subset of A such that for each x∈K it holds ρ(fnyi(x),f(x))<ɛ for at least one i≤k, that is, (2) is true.
(2)⇒(3): It suffices to show that for each x∈X and each ɛ>0 we have ∂({n∈ℕ:fn∉[{x},ɛ]s(f)})=0. Since (fn)n∈ℕ→st-τpf, there is a set A⊂ℕ with ∂(A)=1 so that ρ(fn(x),f(x))<ɛ/4 for each n∈A. We are going to prove that for each n∈A there is δn>0 such that for each y∈S(x,δn), ρ(fn(y),f(y))<ɛ. Suppose, by contradiction, that this assumption fails. Then there is n0∈A and a sequence (xj)j∈ℕ converging to x such that ρ(fn0(xj),f(xj))≥ɛ for each j∈A. The set K={xj:j∈ℕ}∪{x} is compact so that, by (2), there are m1,⋯,mk∈A such that for each z∈K, ρ(fmi(z),f(z))<ɛ/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(fmi(z),f(z))<ɛ/4. For this mi, we have
(3.2)ρ(fmi(x),fn0(x))≤ρ(fmi(x),f(x))+ρ(f(x),fn0(x))<ɛ2.
Since fmi and f are continuous at x, there are δmi>0 and δ0>0 such that ρ(fmi(u),fmi(x))<ɛ/8 for each u∈S(x,δmi), and ρ(fn0(u),fn0(x))<ɛ/8 for each u∈S(x,δ0). If δ=min{δ0,δmi}, then for each z∈C∩S(x,δ) we have
(3.3)ρ(fmi(z),fn0(z))≤ρ(fmi(z),fmi(x))+ρ(fmi(x),fn0(x))+ρ(fn0(x),fn0(z))<3ɛ4.
Since (xj)j∈ℕ converges to x, there is j*∈A such that xj*∈C∩S(x,δ). For this j*, we have
(3.4)ρ(fn0(xj*),f(xj*))≤ρ(fn0(xj*),fmi(xj*))+ρ(fmi(xj*),f(xj*))<3ɛ4+ɛ4=ɛ,
which is a contradiction.
(3)⇒(4): Let ɛ>0 and a statistically dense set A⊂ℕ be given. Since (fn)n∈ℕ→st-τℱsf, given [{x},ɛ]s(f), there is Bx⊂ℕ statistically dense, such that for each n∈Bx we have fn∈[{x},ɛ]s(f). Hence there is a δn,x such that for each y∈S(x,δn,x) and each n∈Bx we have ρ(fn(y),f(y))<ɛ. Let B=⋃x∈XBx. For each n∈B∩A, define
(3.5)En:={x∈X:∀m∈Bx∩A,m≥n,ρ(fm(y),f(y))<ɛ∀y∈S(x,δn,x)}.
Note that X=⋃n∈AEn. For each n∈A, let Un be the following open set:
(3.6)Un={∅,ifn∈A∖B,⋃x∈EnS(x,δn,x),ifn∈A∩B.
Then {Un:n∈A} is an open cover of X. Thus for each k∈A and each x∈Uk, there is some m∈B such that it holds that ρ(fm(x),f(x))<ɛ, that is, the set A∩B={n1<n2<⋯}⊂A and the cover {Un: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(fn)n∈ℕ in C(X,Y) is said to be statistically Dini convergent tof∈YX, denoted by (fn)n∈ℕ→st-Dif if (fn)n∈ℕ→st-τpf and for each ɛ>0 and each statistically dense set A⊂ℕ there exists an increasing sequence m1<m2<⋯ in A such that ρ(fmi(x),f(x))<ɛ for each x∈X and each i∈ℕ.Theorem 3.5.
If a sequence(fn)n∈ℕ in C(X,Y)  statistically Dini converges to f∈YX, then f is continuous.Proof.
Letx0∈X and ɛ>0 be given. Since (fn(x0))n∈ℕ→stf(x0), there is a statistically dense set A⊂ℕ such that ρ(fn(x0),f(x0))<ɛ/3 for each n∈A. Because (fn)n∈ℕ→st-Dif, there exists an increasing sequence m1<m2<⋯ in A such that ρ(fmi(x),f(x))<ɛ for each x∈X and each i∈ℕ. Take some nk. Since fnk is continuous at x0, there is δ>0 such that ρ(fnk(x),fnk(x0))<ɛ/3 whenever d(x,x0)<δ. So, for each x∈S(x0,δ), we have
(3.7)ρ(f(x),f(x0))≤ρ(f(x),fmk(x))+ρ(fmk(x),fmk(x0))+ρ(fmk(x0),f(x0))<ɛ,
that is, f is continuous at x0, hence on X.Definition 3.6.
A sequence(fn)n∈ℕ in C(X,Y)statistically strongly Arzelà converges to a function f∈YX on a bornology 𝔅 on X, denoted by (fn)n∈ℕ→st-s.Arzf, if (fn)n∈ℕ→st-τ𝔅f and for each B∈𝔅, ɛ>0 and each statistically dense set A in ℕ, there are finitely many n1,…,nk∈A such that fni∈[B,ɛ]s(f) for at least one i≤k.Theorem 3.7.
If a sequence(fn)n∈ℕ in C𝔅s(X,Y) statistically strongly Arzelà converges to f∈YX on a bornology 𝔅 with closed base on X, then f is a strongly uniformly continuous on 𝔅.Proof.
LetB∈𝔅 and ɛ>0. As (fn)n∈ℕ→st-τ𝔅f, there is a statistically dense set A⊂ℕ such that for each n∈A we have fn∈[B,ɛ/3](f), that is, for each x∈B, ρ(fn(x),f(x))<ɛ/3. By assumption, there are n1,…,nk∈A with fnm∈[B,ɛ/3]s(f) for some m≤k, that is, there exists δm>0 such that ρ(fnm(x),f(x))<ɛ/3 for each x∈Bδm. Since fnm is strongly uniformly continuous on B, there is δ0>0 so that for each x∈B and each y∈Bδ0 with d(x,y)<δ0, we have ρ(fnm(x),fnm(y))<ɛ/3. Set δ=min{δm,δ0}. Then for each x∈B and y∈X with d(x,y)<δ by the above relations, it follows
(3.8)ρ(f(x),f(y))≤ρ(f(x),fnm(x))+ρ(fnm(x),fnm(y))+ρ(fnm(y),f(y))<ɛ,
that is, f is strongly uniformly continuous on B, hence on 𝔅.Theorem 3.8.
LetX be a compact space, 𝔅 a bornology on X with closed base, and (fn)n∈ℕ a sequence in C𝔅s(X,Y). If (fn)n∈ℕ→st-τ𝔅f and f is strongly uniformly continuous on 𝔅, then (fn)n∈ℕ statistically strongly Arzelà converges to f on 𝔅.Proof.
LetB∈𝔅, ɛ>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∈Bδ0 with d(x,y)<δ0 we have ρ(f(x),f(y))<ɛ/3. From (fn)n∈ℕ→st-τ𝔅f, it follows that there is a statistically dense set B⊂ℕ such that for each n∈C, C=A∩B, we have fn∈[B,ɛ/3](f), that is, for each n∈C, and each x∈B it holds that ρ(fn(x),f(x))<ɛ/3. For each n∈C set Un={x∈X:ρ(fn(x),f(x))<ɛ/3}. Since f and fn’s are continuous, each Un is open in X, so that {Un:n∈ℕ} is an open cover of X. By compactness of X, there are finitely many n1,…,nk∈C such that X=⋃i=1kUni. But each fni is strongly uniformly continuous on B, so that for each i≤k there is δi>0 such that ρ(fni(x),fni(y))<ɛ/3 whenever x∈B and y∈Bδi, ρ(x,y)<δi. Let δ=min{δ0,δ1,…,δk}. Then for x∈B and y∈Bδ with d(x,y)<δ, since x∈Unm for some m≤k, we have
(3.9)ρ(fnm(y),f(y))≤ρ(fnm(y),fnm(x))+ρ(fnm(x),f(x))+ρ(f(x),f(y))<ɛ.
So, fnm∈[B,ɛ]s(f) which completes the proof.Theorem 3.9.
Let𝔅 be a bornology on X with closed base, and let (fn)n∈ℕ be a sequence in C𝔅s(X,Y) such that (fn)n∈ℕ→st-τ𝔅f. Then f is strongly uniformly continuous on 𝔅 if and only if (fn)n∈ℕ→st-τ𝔅sf.Proof.
By [8, Proposition  6.5], we have τ𝔅s=τ𝔅. So, it suffices to prove that (fn)n∈ℕ→st-τ𝔅sf implies strong uniform continuity of f on 𝔅.
Assume thatf is not strongly uniformly continuous on 𝔅. There are a B∈𝔅 and ɛ>0 such that for each n∈ℕ, there are points xn,zn∈B1/n with d(xn,zn)<1/n such that ρ(f(xn),f(zn))≥ɛ. Since (fn)n∈ℕ→st-τ𝔅sf, the density of the set A={n∈ℕ:fn∉[B,ɛ/3]s(f)} is 0. Let M=ℕ∖A. Then M is statistically dense in ℕ, and there exist m∈M, xm,zm∈B1/m, d(xm,zm)<1/m, such that ρ(f(xm),fm(xm))<ɛ/3, and ρ(f(zm),fm(zm))<ɛ/3. Thus
(3.10)ɛ≤ρ(f(xm),f(zm))≤ρ(f(xm),fm(xm))+ρ(fm(xm),fm(zm))+ρ(fm(zm),f(zm)),
and so
(3.11)ρ(fm(xm),fm(zm))≥ɛ-ρ(f(xm),fm(xm))-ρ(fm(zm),f(zm))>ɛ3,
that is, fm 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.
Letℳ be a family and (fn)n∈ℕ a sequence in YX. 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 ɛ>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))<ɛ. The sequence (fn)n∈ℕ is exhaustive at x if the family {fn:n∈ℕ} is exhaustive at x. The family ℳ (sequence (fn)n∈ℕ) is exhaustive onX 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(fn)n∈ℕ in YX is said to be statistically exhaustive (shortly, st-exhaustive) at a point x∈X if for each ɛ>0 there are δ>0 and a statistically dense set M⊂ℕ such that for each y∈S(x,δ) we have ρ(fn(y),fn(x))<ɛ for each n∈M. The sequence (fn)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 familyℳ⊂YX is evenly continuous if for each net (fα,xα)α∈Λ in ℳ×X such that (xα)α∈Λ converges to x∈X and (fα(x))α∈Λ converges to y∈Y, the net (fα(xα))α∈Λ converges to y.Definition 4.4 (see [10]).
A familyℳ⊂YX is equicontinuous at a pointx if and only if for each ɛ>0 there is a neighborhood U of x such that f(U)⊂S(f(x),ɛ) for each member f of ℳ. A family ℳ is equicontinuous if it is equicontinuous at each x∈X.Theorem 4.5.
If a familyℳ⊂YX is exhaustive, then ℳ is evenly continuous.Proof.
Ifℳ is finite there is nothing to prove, and thus we assume that ℳ is infinite. Let (fα,xα)α∈Λ be a net in ℳ×X satisfying (xα)α∈Λ  converges to x and (fα(x))α∈Λ  converges to y∈Y, and let ɛ>0. As ℳ is exhaustive at x, there exist δx>0 and {fβ1,fβ2,…,fβk}⊂ℳ such that for each z∈S(x,δx) and each f∈ℳ∖{fβ1,fβ2,…,fβk} we have ρ(f(x),f(z))<ɛ/2. Because the set {fβ1,fβ2,…,fβk} is finite, there is some α*∈Λ such that fα≠fβi for each α≥α* and each i≤k.
Letδ=min{δx,ɛ/2}. Since (xα)α∈Λ converges to x and (fα(x))α∈Λ converges to y there is α0≥α* in Λ such that xα∈S(x,δ) and fα(x)∈S(y,δ) for each α≥α0. Then fα(xα)∈S(fα(x),ɛ/2) and fα(x)∈S(y,δ)⊂S(y,ɛ/2) for each α≥α0. So for each α≥α0, we have
(4.1)ρ(fα(xα),y)≤ρ(fα(xα),fα(x))+ρ(fα(x),y)<ɛ,
that is, (fα(xα))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 (fn)n∈ℕ in YX is strongly exhaustive on X if for each ɛ>0 there are δ>0 and n0∈ℕ such that for all x,y∈X with d(x,y)<δ, ρ(fn(x),fn(y))<ɛ for each n≥n0.The novel notion of statistical uniform exhaustiveness for a sequence is related to uniform convergence.Definition 4.6.
A sequence(fn)n∈ℕ in YX is st-uniformly exhaustive on X if for each ɛ>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(fn(x),fn(y))<ɛ for all n∈A.Theorem 4.7.
Let(X,d) be a compact space, and let (fn)n∈ℕ be a st-exhaustive sequence in YX such that (fn)n∈ℕ→st-τpf. Then (a)
(fn)n∈ℕ is statistically uniformly exhaustive;(b)
there is a statistically dense setM⊂ℕ such that (fm)m∈M is uniformly exhaustive and (fm)m∈M→τuf.Proof.
(a) Letɛ>0 and x∈X be fixed. Since, by hypothesis, (fn)n∈ℕ is statistically exhaustive at x, there are δx>0 and a statistically dense set Ax⊂ℕ such that for each z∈S(x,δx) and each n∈Ax, it holds ρ(fn(x),fn(z))<ɛ/2. From X=⋃x∈XS(x,δx/2) and compactness of X, it follows the existence of finitely many points x1,…,xk in X such that X=⋃i=1kS(xi,δxi/2). Let δ*=min{δxi/2:i≤k} and A=⋂i=1kAxi. The set A is statistically dense in ℕ. We claim that δ* and A witness that (a) is true.
Letx,z∈X such that d(x,z)<δ*. There is j≤k such that x,z∈S(xj,δxj). Therefore, for each n∈A and all x,z∈X with d(x,z)<δ*, we have ρ(fn(x),fn(z))<ρ(fn(x),fn(xj))+ρ(fn(xj),fn(z))<ɛ, that is, (a) is true.
(b) By (a) for eachj∈ℕ, there are a statistically dense set Aj⊂ℕ and δj>0 such that x,y∈X and d(x,y)<δj imply ρ(fn(x),fn(y))<1/j for each n∈Aj. Then {ℕ∖Aj:j∈ℕ} is a family of density zero sets, that is, this family is contained in the ideal ℐd of all subsets of ℕ having density zero. It is known that ℐd is a P-ideal (i.e., for each countable collection 𝒥⊂ℐd there is some L∈ℐd such that J∖L is finite for each J∈𝒥), so there exists D∈ℐd such that the set (ℕ∖Aj)∖D is finite for each j∈ℕ. Let N1=ℕ∖D:={n1<n2<⋯<nk<⋯}. Then N1 is a statistically dense subset of ℕ.Claim 1.
The sequence(fn)n∈N1 is uniformly exhaustive.
Letɛ>0 be fixed. Choose j∈ℕ such that 1/j<ɛ. Let tɛ∈ℕ be such that (ℕ∖Aj)∖D⊂{n1,n2,…,ntɛ}. It follows that for each t>tɛ, we have nt∈Aj. Thus there is n*>ntɛ such that for all x,y∈X with d(x,y)<1/j we have ρ(fn(x),fn(y))<ɛ for each n∈N1 greater than n*.Claim 2.
There isM⊂ℕ with ∂(M)=1 such that (fm)m∈M→τuf.
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 ɛ>0. There is δ>0 such that ρ(f(x),f(y))<ɛ/4 for all x,y∈X satisfying d(x,y)<δ. By Claim 1, (fn)n∈N1 is uniformly exhaustive on X, so that there exist δ0>0 and nk0∈N1 such that d(x,y)<δ0 implies ρ(fnk(x),fnk(y))<ɛ/4 for all nk∈N1 with nk≥nk0. Let δ*=min{δ,δ0}. Using compactness of X choose a finite set {x1,…,xs}⊂X such that X=⋃i=1sS(xi,δ*). Since (fn)n∈ℕ→st-τpf, for each i≤s there is a statistically dense set Ai⊂ℕ such that for each n∈A0=⋂i=1sAi we have ρ(fn(xi),f(xi))<ɛ/4, i≤s. Set M=A0∩N1. Then M is statistically dense and the sequence (fn)n∈ℕ is still uniformly exhaustive. Each y∈X belongs to S(xi,δ*) for some i≤s, and thus for each nk∈M with nk≥nk0 we have
(4.2)ρ(fnk(y),f(y))≤ρ(fnk(y),fnk(xi))+ρ(fnk(xi),f(xi))+ρ(f(xi),f(y))<ɛ,
which completes the proof of (Claim 2 and) the theorem.



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*Source: 420419-2011-12-15.xml*