#### Abstract

We give necessary and sufficient conditions for exchange of limits of double-indexed families, taking values in sets endowed with an abstract structure of convergence, and for preservation of continuity or semicontinuity of the limit family, with respect to filter convergence. As a consequence, we give some filter limit theorems and some characterization of continuity and semicontinuity of the limit of a pointwise convergent family of set functions. Furthermore, we pose some open problems.

#### 1. Introduction

A widely investigated problem in convergence theory and topology is to find necessary and/or sufficient conditions for continuity and/or semicontinuity of the limit of a pointwise convergent net of functions or measures. There have been many recent related studies in abstract structures, like topological spaces, lattice groups, metric semigroups, and cone metric spaces, with respect to usual, statistical, or filter/ideal convergence and associated with the notions of equicontinuity, filter exhaustiveness, and filter continuous convergence (see also [1–9]). The study of semicontinuous functions is associated with quasimetric spaces, that is, spaces endowed with an asymmetric distance function (for a related literature, see, e.g., [3–5, 10–13]).

A concept associated with these topics is that of* strong uniform continuity*, which is used to study the problem of finding a topology with respect to which the set of the continuous functions is closed, and pointwise convergence of continuous functions implies convergence in this topology (see also [1, 14, 15]).

Another related field is the study of convergence theorems for measures taking values in abstract structures. When dealing with the classical convergence, it is possible to prove -additivity, -boundedness, and absolute continuity of the limit measure directly from pointwise convergence (with respect to a single order sequence of regulator) of the involved measures, without requiring additional hypotheses. This is not always true in the setting of filter convergence. A historical comprehensive overview, together with a survey on the most recent results and developments, can be found in [16] (see also its bibliography).

In this paper we present a unified axiomatic approach and extend results of this kind to double-indexed families, taking values in abstract structures, whose particular cases are lattice groups, topological groups, (quasi)metric semigroups, and cone (quasi)metric spaces. To include both continuity and semicontinuity, we assume the existence of a “generalized distance” function, which is assumed to satisfy only the triangular property and takes values in a group endowed with a suitable system of “intervals” or “half lines” containing its neutral element . Thus, both topological groups and lattice groups endowed with -, -, or order convergence are particular cases of these abstract structures. We prove some results on exchange of limits in the setting of filter convergence. Observe that the involved “distance” can be symmetric or asymmetric (for a literature, see also [3, 5, 10] and their bibliographies). Furthermore, in our setting, both sequences and nets of functions/measures are included, and note that it is possible to consider them as families endowed with filters (see also [17–19]).

As applications, we give some necessary and sufficient conditions for continuity from above/below and absolute continuity and semicontinuity of the limit measure in the context of filter convergence, which include the cases of -additivity and -boundedness, showing, by means of related examples, that they are not always satisfied, differently from the classical case. For a literature on measures satisfying upper/lower semicontinuity conditions or similar properties and related applications, see, for instance, [20] and the bibliography therein. Finally, we pose some open problems.

#### 2. Assumptions and Examples

We begin with giving our axiomatic approach, which deals with abstract convergence with respect to filters, without using necessarily nets. For a literature about these topics, see, for instance, [16, 17, 19, 21–24] and their bibliographies.

*Definition 1. *(a) Let be any nonempty set, and let be the class of all subsets of . A family of sets is called a* filter* of iff , , and for each , , and whenever and .

Some examples are the filter of all subsets of whose complement is finite and the filter of all subsets of having asymptotic density one. Some other classes of filters can be found in [16].

(b) Let be a nonempty set, and let be an abelian group with neutral element . Given and , , put , and ( times).

(c) Let be a nonempty set. A *-system * is a class of families of subsets of , with for each , such that for every and there is such that for every . Let be a function, and suppose that for every , and for each and , if and , then .(d) Fix a -system on and a filter of . A family , , of elements of is said to -*backward* (resp., -*forward*)* converge* to iff there is a family , such that for every there is a set with (resp., ) for any . We say that -*converges* to iff it -converges both backward and forward to , and in this case we write .

(e) Let be a nonempty set. Given two families and of elements of , we say that *-backward* (resp., *-forward*) converges to iff there is a family , such that for each and there is with (resp., ) for any . Analogously as above it is possible to formulate the notions of -convergence and -limit.

*Remark 2. *Observe that, in our context, we will consider filters without dealing explicitly with nets, and this is not a restriction. A* net* on is a function , where is a* directed* set, namely, a partially ordered set such that for any there exists with , . Given a directed set , it is possible to associate the filter generated by the family . Note that is a* filter base* of ; that is, for every there is an element with . The filter generated by a filter base is the family . Conversely, given a filter base , it is possible to associate a directed partial order on , by setting if and only if , (see also [18, 19]).

*Example 3. *We now present some kinds of abstract space in which our approach can be applied, including both symmetric and asymmetric distance functions (for a literature, see also [3, 5, 10–13]).

(a) Let be a Dedekind complete lattice group, , and let , , be the* absolute value* of . It is possible to define different kinds of convergences, as follows (see also [16]).

Let be endowed with the usual order, (*-convergence*); let be with the usual order, , where an *-sequence* is a decreasing sequence in whose infimum is equal to (*order convergence* of *-convergence*); let be directed with the pointwise order, , where a *-sequence* or* regulator* is a bounded double sequence in such that is an -sequence for each (*-convergence*). The -convergence was presented in [25] to give direct proofs of extension theorems for vector lattice-valued functionals and replaces the -technique in dealing with suprema and infima of lattice group- or vector lattice-valued families. For technical reasons, sometimes the -convergence is easier to handle than -convergence, and in particular it is very useful when one replaces a sequence of regulators with a single -sequence (for a literature about these topics, see also [16, 23, 26, 27]).

It is not difficult to check that , , are -systems, satisfying .

(b) We can extend the examples given in (a) to the case in which is a* cone metric space* (with respect to ); that is, is a nonempty set and is a Dedekind complete lattice group endowed with a* distance function *, satisfying the following axioms: (i) and if and only if .(ii) (*symmetric property*).(iii) (*triangular property*) for every , , .(See also [6, 28].) When a cone metric space is a semigroup, we say that is a* cone metric semigroup*, a cone metric semigroup in which is said to be a* metric semigroup*. Note that the set of fuzzy numbers is a metric semigroup, but not a group (see also [20]). If satisfies the first and the third of the above axioms, but not the symmetric property, then we say that is an* asymmetric distance function* and that is a* cone asymmetric metric space* or* cone quasimetric space* (see also [3, 5, 10]). For example, let be a nonempty set, , and let be a fixed positive real number and let be a fixed element of with . For each and , setand let . It is not difficult to see that is an asymmetric distance function (see also [3, 10]).

(c) When is a lattice group and , it is advisable to deal not only with continuity, but also with upper or lower semicontinuity (see also [4]). In this setting we take , , , , as in (a), and ; ; .

(d) Let be a Hausdorff topological group with neutral element satisfying the first axiom of countability, , , , and . It is not difficult to see that is a -system (see also [16, 29]).

(e) Let be a filter of . When we consider -convergence and is a cone quasimetric space, a family of elements of is said to *-backward converge to * iff there is , , with for all . When we deal with -sequences, we say that -*backward converges to * iff there exists an -sequence in with for every . When we consider -sequences, we say that the net *-backward converges to * iff there exists a regulator in with When and , we have the classical -, -, and -(backward, forward) convergence (see also [3, 16]). If is a Hausdorff topological group and , then we say that a net , , in , -*backward converges to * iff for each neighborhood of . Similarly as above it is possible to formulate the corresponding notions of -, -, and -(forward) convergences and limits.

(f) When is a Dedekind complete lattice group, and are two families in and is the -system associated with -convergence (resp., -convergence, -convergence); we say that (resp., , ) iff . Analogously it is possible to formulate the corresponding concepts of backward and forward convergences (see also [3, 5, 10]). In particular, when endowed with the usual convergence, since it coincides with - -, and -convergence, we will denote by - and -(backward, forward) convergence the usual filter (backward, forward) convergence and the ordinary pointwise filter (backward, forward) convergence. When is a Hausdorff topological group, , are as in (d), and we get that the -convergence is equivalent to the pointwise -convergence, and hence we write for every , or .

(g) Observe that, in general, a family can be backward (resp., forward) convergent to more than one element. For example, if is a Dedekind complete lattice group, is a nonempty set, is any filter of , for every , , for every , and is* any* element of with (resp., ), then it is not difficult to see that -backward (resp., -forward) converges to .

(h) In general, backward and forward convergence are not equivalent. For example, similarly as in (1), let be a nonempty set, let be endowed with the usual order, let be a filter of containing all half lines with , pick , and let , be those functions which associate with every element of the real constants , , respectively. For any , and , setand put . It is not difficult to check that is an asymmetric distance function (see also [3, 10]). For each , set and . Note that , , , and . From this it is not difficult to deduce that the family -forward converges to and -backward converges to , while does not -backward converge to and does not -forward converge to .

However, if is any nonempty set, is any filter of , is as in (1), and , then it is not difficult to see that whenever , . From this it follows that a family in is -backward convergent if and only if it is -forward convergent. We claim that, in this case, the involved limit coincides. Indeed, if -backward converges to and -forward converges to with respect to , then there exist , , such that for every there are with for every , whenever . Note that . If is any fixed element of , then from the triangular property of we deduce that Thus, by arbitrariness of , we get , and hence , getting the claim.

#### 3. The Main Results

In this section we give the fundamental results of the paper in our unified setting, which includes lattice groups, cone metric spaces, metric groups and topological groups, symmetric and asymmetric distances, continuity and semicontinuity of the limit, and families of functions and of measures. We first present the notion of weak filter backward and forward exhaustiveness in our abstract context, which extends the corresponding ones given in the literature and the classical concept of equicontinuity (see also [4, 8, 16, 30]).

*Definition 4. *(a) Let be a nonempty set; fix and let be a filter of . One says that the family is* weakly **-backward* (resp.,* forward*)* exhaustive at * iff there exists a family such that for each there is a set such that for every there is a set with (resp., ) for any . The family is said to be* weakly **-exhaustive at * iff it is both weakly -backward and weakly -forward exhaustive at .

(b) Let , , be a family of filters of . One says that is* weakly **- (backward, forward) exhaustive on * iff it is weakly - (backward, forward) exhaustive at every with respect to a single family , independent of .

*Example 5. *We now show that, in general, weak -backward and forward exhaustiveness do not coincide. Let , , be equipped with the usual convergence; that is, let be endowed with the usual order, and . Let us define by It is not difficult to see that is an asymmetric distance function (see also [10, Example 5.3]). Let be any filter of , and, for every , let be the filter of all neighborhoods of with respect to the topology generated by . Set , , . We claim that the family is weakly -forward exhaustive at . Indeed, in correspondence with , take , and set for any , where denotes the ball of center and radius with respect to . For every and we get , getting the claim.

Now, in correspondence with every and , let and take . Note that . Choose arbitrarily . It is not hard to see that for every . Hence, the family is not weakly -backward exhaustive at . Furthermore note that, analogously as in (3), it is not difficult to check that -forward (resp., backward) convergence does not imply -backward (resp., forward) convergence with respect to .

The following result deals with characterizations and properties of the limit family and extends [3, Theorem 3.1], [4, Theorems 2.5, 2.6], and [6, Theorem 3.1] to the abstract context.

Theorem 6. *Assume that -converges to , fix , and let be a filter of . Then the following are equivalent:*(i)* is weakly -backward (resp., forward) exhaustive at .*(ii)*-backward (resp., forward) converges to as .*

*Proof. *We give the proof only in the “backward” case, since the other case is analogous.

Let be a family related to -backward exhaustiveness of at . By hypothesis, for each , there exists a set , associated with weak -backward exhaustiveness. Pick arbitrarily . There is a set with for any . Moreover, thanks to -convergence, there is a family such that for every there exists with and whenever . From this and it follows that , getting (ii).

By hypothesis, there exists a family such that for each there is a set withChoose . By -convergence of to , there is a family such that for every there is a set withfor each . From (6), (7), and we get that for every there is such that for each there exists with whenever . Thus the family is weakly -backward exhaustive at . This ends the proof.

*Remark 7. *Observe that Theorem 6 holds also if -convergence is replaced by -forward convergence, under the hypothesis that forward convergence implies backward convergence (see also [10]). In general this last condition is essential. Indeed, let be endowed with the usual order, let be a filter of containing all half lines with , let be equipped with the usual distance, let , , be the filter of all neighborhoods of , let be endowed with the usual convergence, , and let be defined by It is not difficult to check that is an asymmetric distance function. For every and , set . Observe that and for every and . It is not difficult to see that the family -forward converges to , where , , but does not -backward converge. Moreover, since , for every , , the family is neither -backward nor -forward convergent to as . Furthermore, we getfor every and . From (9) it is not difficult to deduce that the family is both weakly -forward and weakly -backward exhaustive at (see also [3, Example 3.7], [10, Example 5.10]).

We now give some kinds of convergences for families, which are some necessary and sufficient conditions for exchange of limits, which extend to our context some results proved in [1, 2, 4, 6, 9] about necessary and sufficient conditions for continuity of the pointwise limit of continuous functions. We extend to our setting the concepts of Arzelà, Alexandroff, and strong uniform convergence given in [1, 15, 31, 32].

*Definition 8. *(a) Fix , and let be a filter of . One says that *-forward strongly uniformly converges to ** at * (shortly, ) iff there exists a family such that for each there is such that for every there is a set with whenever .

(b) One says that is *-forward Arzelà convergent to ** at * (in brief, ) iff there exists a family such that for every and there are a finite set and a set , such that for each there is with .

(c) If , , is a family of filters of , then one says that a* finitely uniform cover* of is a family of subsets of such that , and for every there are a set and a finite subset of , such that for each there exists with .

(d) The family is said to *-forward strongly uniformly* (resp., *-forward Arzelà*)* converge to ** on * iff it -strongly uniformly (resp., -Arzelà) converges to at for every with respect to a single family , independent of .

(e) One says that is *-forward Alexandroff convergent to ** on * (shortly, on ) iff there exists a family such that for each and there are a nonempty set and a finitely uniform cover of with for any and .

Note that, analogously as above, it is possible to formulate the corresponding concepts of (backward) filter strong uniform, Arzelà, and Alexandroff convergence.

The next result extends [2, Theorem 3.9], [4, Theorems 2.9, 2.11 and Corollary 2.10], and [9, Proposition 3.5].

Theorem 9. *Let be fixed, let be a filter of , and suppose that *(3.6.1)*;*(3.6.2)*the family -converges to .**Then the following are equivalent: *(i)*-backward converges to as .*(ii)* at .*(iii)* at .*

*Proof. * Let , , and be three families associated with (i), (3.6.1), and (3.6.2), respectively, and take arbitrarily . By (3.6.2), there is with for all . By (3.6.1) and (i), for each there is with and for any . For such ’s, taking into account , we have , getting (ii).

Let be a family, according to -strong uniform convergence. Choose arbitrarily and , and let be associated with -strong uniform convergence. Pick any finite set : since is a filter, does exist. For every , let be related to -strong uniform convergence, and set . Note that . By construction, for each and , we get . Thus, we obtain (iii).

Let , , and be families related to (iii), (3.6.1), and (3.6.2), respectively. By (3.6.2), there is a set withChoose arbitrarily . By (iii), in correspondence with and , there exist a finite set and a set such that for every there is withThanks to (3.6.1), we find a set , without loss of generality , withfor each . From (10), (11), (12), and it follows that for every , getting (i).

*Remark 10. *(a) In general, Theorem 9 does not hold, when the involved “forward” convergences are replaced by the corresponding “backward” ones. Indeed, for example, let , be any filter of , let be endowed with the usual metric, , and let be the filter of all neighborhoods of contained in , , , , . Put , , . We get for every , and Note that for each and we get Hence, at . On the other hand, for every and for each neighborhood of contained in there is a real number , close enough to , with , and hence Thus, at . The family -forward, but not backward, converges to as : indeed for every we have for each , but . Note that the function , , is upper semicontinuous, but not lower semicontinuous, at .

(b) Observe that Theorem 9 does not hold, where in (3.6.1) the involved convergence is replaced by the corresponding backward or forward convergence (see also [2, Example 3.3]).

Let , , , , , and be as in (a), let be endowed with the usual metric, , and let be the filter of all neighborhoods of . Set Observe that for every , so that (3.6.2) holds, and condition (i) of Theorem 9 is fulfilled. Moreover it is not difficult to see that, for each , converges backward, but not forward, to as tends to , and hence (3.6.1) is not verified. However, note that for every and for every neighborhood of there is with , and hence . Thus, condition (ii) of Theorem 9 is not satisfied.

Furthermore, if we define , , , by then Hence, (3.6.2) is satisfied, but condition (i) of Theorem 9 does not hold. Observe that, for any , converges forward, but not backward, to as tends to , and hence (3.6.1) is not satisfied. On the other hand, since for any and , we get that condition (ii) of Theorem 9 is fulfilled.

We now turn to the main theorem in our abstract setting, which extends [1, Theorems 4.7, 4.11], [2, Theorem 3.10], [4, Theorem 2.12], and [6, Corollary 3.5] to our abstract unified setting.

Theorem 11. *Let , , be a family of filters of , with the property that for every and . Suppose that (3.6.2) holds and that*(3.8.1)

*for each with respect to a single family , independent of both and .*

*Then the following are equivalent:*(i)

*-backward converges to as for every , with respect to a single family , independent of .*(ii)

*on .*(iii)

*on .*(iv)

*on .*(v)

*is weakly -backward exhaustive on .*

*Proof. * It is similar to Theorem 6.

It is similar to Theorem 9.

Let be a family associated with --convergence of to . Choose arbitrarily and . By (ii), for every , there exists a set , such that for every there is with for any . Set : note that , where . For every , let Pick arbitrarily and choose . We have whenever . Thus, . For each , set . Note that is a cover of . For every and there is with , and hence . Now, in correspondence with , choose an element and pick . Note that and . Thus, is a finitely uniform cover of , with and . Therefore, -Alexandroff converges to .

Let be a family associated with -Alexandroff convergence of to . Pick arbitrarily , , and . By (iii), there are a nonempty set and a finitely uniform cover of , with for each and . Since is a finitely uniform cover, in correspondence with , there exist a set and a finite subset , such that for every there is with . Thus , and so we obtain (iv). This ends the proof.

*Remark 12. *Observe that when the function is symmetric, Theorems 6, 9, and 11 can be viewed as necessary and sufficient conditions in order to have exchange of limits (for a related literature, see also [16, 26, 33]).

#### 4. Applications to Set Functions

In this section, as consequences of Theorems 6, 9, and 11, we will give some necessary and sufficient conditions for some kind of continuity and semicontinuity of the limit of set functions. We begin with proving a result on continuity from below the limit measure. Note that, thanks to the limit theorems existing in the literature, these conditions are often fulfilled (for a comprehensive historical survey, see [16] and its bibliography). However, we give an example in which these properties do not hold in the setting of filter convergence.

Let be any nonempty set, let be any filter of , let be any infinite set, let be -algebra of subsets of , and let be a (symmetric) cone metric semigroup, , , . It is not difficult to check that is a filter of . Moreover, let be a fixed -system associated with .

A set function is said to be *-continuous from below* (resp.,* from above*) on iff for every increasing (resp., decreasing) sequence in whose union (resp., intersection) is equal to . A consequence of Theorems 6 and 9 is the following.

Theorem 13. *Let , , be a family of set functions, -continuous from below on , with respect to a family independent of . Suppose that *(4.1.1)*, , exists in with respect to a family independent of .**Then the following are equivalent: *(i)* is -continuous from below on .*(ii)*For every increasing sequence in there is a family such that for any there is such that, for every , there is a set with for each .*(iii)*For any increasing sequence in there is a family such that for every there is such that for each there exists a positive integer with for any .*(iv)*For every increasing sequence in there is a family such that for each and there are and such that for each there exists with .*

Indeed, it is enough to take where is a fixed increasing sequence in , whose union is . Conditions (i) of Theorem 6 and (ii) and (iii) of Theorem 9 become conditions (ii), (iii), and (iv) of Theorem 13, respectively.

*Remark 14. *(a) Observe that results analogous to Theorem 13 hold when the involved set functions , , are -continuous from above or *-bounded* on , that is, if for every disjoint sequence in .

(b) Note that conditions (ii)–(iv) of Theorem 13 are just satisfied, for example, when is a Dedekind complete lattice group, , , , , and is a sequence of -additive positive -valued measures, thanks to the classical limit theorems (see also [16, 34, 35]).

The next step is to give necessary and sufficient conditions for absolute continuity of the limit measure.

Let be a finitely additive measure. We endow with the Fréchet-Nikodým topology generated by the pseudometric , . Pick now , and for each let be the filter generated by the base .

We say that is* weakly **-exhaustive at * iff there is a family (depending on ) such that for each there is such that for every with there is a set with whenever . We say that is* weakly **-exhaustive on * iff it is weakly --exhaustive at every with respect to a family independent of .

A measure is said to be *-continuous at * iff there is a family (depending on ) such that for every there is with whenever . We say that is* globally **-continuous on * with respect to iff it is --continuous at with respect to for each , relative to a family , independent of .

The next result is a consequence of Theorem 11.

Theorem 15. *Let , , be a family of measures --continuous at a fixed set (resp., globally --continuous on ) with respect to a family independent of and -convergent to a measure . Then the following are equivalent: *(i)*The limit measure is --continuous at (resp., globally --continuous on ).*(ii)*The net , , is weakly -exhaustive at (resp., on ).*(iii)*There is a family , depending on (resp., independent of ), such that for each there is such that for every there is with for each with .*(iv)*There is a family , depending on (resp., independent of ), such that for every and there are and a positive real number such that for any with there exists with .**Moreover, if ’s are globally --continuous, statements (i)–(iv) are equivalent to the following: *(v)*There is a family such that for any and there exist a nonempty set and a finitely uniform cover of with whenever and .*

*Remark 16. *(a) Observe that when , , , , are positive -additive measures, is a Dedekind complete lattice group, and , , , , we get that conditions (ii)–(v) of Theorem 15 are fulfilled, thanks to the limit theorems existing in the literature (see also [16, 34, 35]).

(b) Let be the class of all subsets of ; let be a filter containing and , . For each , let us define the Dirac measure byIt is not difficult to see that is -additive on . Moreover, is -continuous at (i.e., -absolutely continuous): indeed, if and , then , and hence . We claim that the sequence is not weakly -exhaustive at . Indeed, observe that for each there is a cofinite set with . Note that since contains , every element of is infinite; otherwise , which is impossible. Furthermore, observe that for every infinite subset , and a fortiori for any , there is a sufficiently large integer , so that . From this we deduce that the sequence is not weakly -exhaustive at . If is an ultrafilter of containing (the existence of such ultrafilters follows from the Axiom of Choice; see also [19, 36]), then for every we haveWe claim that is not -continuous at . Indeed, fix arbitrarily and let be such that . Let be any element of and set ; then . We get and , getting the claim.

Furthermore, in this case, conditions (i)–(iv) in Theorem 13 do not hold. Indeed, choose a filter of containing , and let , . Observe that, as said before, every element of is infinite. For every and for any infinite set there is , and hence we get . Thus, in this case, condition (ii) of Theorem 13 is not fulfilled. If is an ultrafilter of , then the measure defined in (22) is not -additive on . Indeed, if is any element of , then we get and .

When is a Dedekind complete lattice group, , , and , , are as in Example 3(c); we obtain some results similar to the previous ones also for semicontinuous set functions (for a related literature, see also [20] and the references therein).

In this setting, the concepts of weak backward (resp., forward) filter exhaustiveness and lower (resp., upper) semicontinuity are formulated as follows.

*Definition 17. *(a) One says that is* weakly **-backward* (resp.,* forward*)* exhaustive at * iff there is a family (depending on ) such that for each there is such that for every with there is a set with whenever .

(b) One says that is* weakly **-backward* (resp.,* forward*)* exhaustive on * iff it is weakly --backward (resp., forward) exhaustive at every with respect to a family independent of .

(c) One says that is* weakly **-exhaustive at * (resp.,* on *) iff it is weakly --backward and forward exhaustive at (resp., on ).

(d) One says that is *-lower* (resp.,* upper*)* semicontinuous at * iff there is a family (depending on ) such that for every there is with (resp., whenever . We say that is* globally **-lower* (resp.,* upper*)* semicontinuous on * iff it is --lower (resp., upper) semicontinuous at for each with respect to a family