On the Differentiation of Integrals in Measure Spaces Along Filters

It is known that the study of the boundary behavior of (harmonic or) holomorphic functions, to which N. Sibony has contributed with penetrating work, is linked to the differentiation of integrals. In 1936, R. de Possel observed that, in the general setting of a measure space with no metric structure, certain phenomena, relative to the differentiation of integrals, which are familiar in the Euclidean setting precisely because of the presence of a metric, are devoid of actual meaning. In the first part of this work, we introduce the concept of functional convergence class that provides a unifying framework for various limiting processes and enables us to establish a hierarchy between them, and show that, within this hierarchy, the notion of filter (introduced by H. Cartan just a year after De Possel’s contribution) occupies the position of wider scope. In the second part of this work, we show how to reformulate some of the contributions of de Possel in the language of filters.


Introduction
In 1936, R. de Possel observed that, in the general setting of a measure space with no metric structure, certain phenomena, relative to the differentiation of integrals, which are familiar in the Euclidean setting precisely because of the presence of a metric, are devoid of actual meaning. In his work, de Possel introduced an axiomatic approach based on a limiting process centered on the preliminary choice of certain sequences of sets. The notion of filter, due to Cartan [8], appeared 1 year after de Possel's work, and yields another limiting process, which, in the context of the problem of the differentiation of integrals, turns out to be preferable, for specific reasons that will be outlined momentarily.
The purpose of this paper is, first of all, to show that the language of filters yields a notion of limiting process that has wider scope, with respect to other limiting processes. On the basis of this preliminary groundwork, we show how to reinterpret in the language of filters some of the contributions given by de Possel to the problem of the differentiation of integrals in measure spaces.
In order to achieve the first goal, we introduce the concept of functional convergence class and systematize the results on convergence along filters by using a topology on the space of all filters. Moreover, we present a self-contained and fairly complete treatment of the notions centered around the relation between filters and Moore-Smith sequences, encompassing various results not all of which appear to be known, or as well known as they ought to be.

Background
Let X ≡ (X, M, ω) be a measure space, where ω is a measure defined on a σ -algebra M of subsets of X. The vector space of measurable real-valued functions defined a.e. on X, whose p th-power is integrable ( p > 0), is denoted by L p (X). The quotient of 123 L p (X) under a.e. equality is denoted by L p (X). The corresponding projection maps f ∈ L p (X) to the class of functions which are a.e. equal to f . We denote this class by f (in bold font), and say that f ∈ L p (X) is a representative of f ∈ L p (X). The spaces L ∞ (X) and L ∞ (X) are also defined in the familiar way [29, p. 244].
Similarly, the quotient of M under a.e. equality of measurable sets is denoted by M and is called the measure algebra of (X, M, ω). The corresponding projection is called the canonical projection associated to (X, M, ω), and is a homomorphism of Boolean algebras.

The Mean-Value Operator
Consider the subcollection of M defined as follows: The sets in (1.2) are called averageable, since for each f ∈ L 1 (X) and Q ∈ A(X) the mean-value of f over Q may be defined in the familiar way, as follows: Indeed, if f is the indicator function of R ∈ A(X), i.e., f = 1 R , then 1 R ∈ L 1 (X) and instead of (1 R ) ω [Q] we write ω (R|Q). Hence ω (R|Q) = ω(Q ∩ R) ω(Q) .

The Problem of the Differentiation of Integrals
The following preliminary observation will help us make a precise statement of the problem.  [29, p. 238].
Observe that f ω : A(X) → R is a bona fide function, defined on A(X), while f is an equivalence class of functions, and that the values f (x) of a representative of f may be recovered only up to a set of measure zero (called the exceptional set of f ). The problem of the differentiation of integrals may be described in the following terms: Find a limiting process that enables us to recapture (a representative of) f ∈ L 1 (X) from f ω (i.e., from the mean-values of f ).
Observe that the notion of limiting process appears in the formulation of the problem in an informal fashion. One of the goals of the present paper is to establish a formal framework for the concept of "limiting process": This will be achieved by means of the notion of functional convergence class. Another goal, subordinate to the first one, is to identify, within this framework, the notion of limiting process that has wider scope: We will show that the concept of filter has precisely this property.
A solution to the problem of the differentiation of integrals is called a Generalized Lebesgue Differentiation Theorem. Indeed, the Lebesgue differentiation theorem solves the problem of the differentiation of integrals in the case X = (R, M, ω), where ω is Lebesgue measure and M is the σ -algebra of Lebesgue-measurable subsets of R, and says that, if f ∈ L 1 (R), then, for a.e. x ∈ R, its value f (x) is approximately equal to the mean-value of f over balls which are, in a certain sense, "close to" x. The prototype result is that, for a.e. x ∈ R, where I x (r ) is the open interval in R of center x and radius r . Observe that the limiting process used in (1.7), to which the function f ω is subject, rests on the metric structure of R.

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Lebesgue himself has given deep generalizations of his one-dimensional results to higher-dimensional Euclidean space R n , where he considered mean-values f ω [B] of f over balls B ⊂ R n which are not centered at x, or even balls which do not contain the point x, provided the balls B get "close to" x in a certain manner (which may be described as being of a "nontangential" nature; see [33]). Once more, the metric structure of the ambient space is used to obtain a "limiting value" from the function f ω ∈ hom Set (A(X), R) defined in (1.3).
In order to achieve his results, Lebesgue had to solve two problems. Firstly, he had to describe what it means for a ball to be "close to" the point x. Secondly, he had to understand which manners of approach of balls to x are compatible with the intended convergence result. The first task, in the context of a metric measure space, such as R n , is indeed not a difficult one, since the metric itself, which is used to define the balls, endows the collection of all its nonempty subsets with a pseudometric: The Hausdorff pseudometric. Indeed, in this context, we may say that a sequence converges to a point x ∈ X if, for each ball B x (r ) of center x and radius r > 0, the set Q(n) is eventually contained in B x (r ). A similar approach may be adapted, at least in principle, in the context of a topological measure space (X, M, ω, ) (where ω is the measure, defined on a σ -algebra M of subsets of X, and is a topology with ⊂ M).

René de Possel's Approach
If (X, M, ω) is a measure space with no further structure, then, although it makes sense to consider mean values, as in (1.3), it does not seem possible to define, in this degree of generality, what it means for a sequence Q : N → A(X) to converge to a point x, especially if the sets Q(n) are not assumed to contain x. This difficulty was perceived already in 1936 by René de Possel, who observed that only some of the main properties of Lebesgue measure admit d'une manière évidente (in evident ways) an extension to the case of an arbitrary measure space, but others semblent perdre toute signification dès que l'espace n'est plus métrique (appear to lose their meaning as soon as the space is not metric) [9]. Among the latter, he listed the properties related to differentiation of integrals. It is useful to present the particular solution devised by de Possel in the context of the general underlying problem, which may be formulated by replacing the space A(X) with a generic set A with no further structure. If hom Set (A, R) denotes the collection of all functions from A to R, then the general underlying problem is that of finding the limiting processes to the which the elements ϕ of hom Set (A, R) may be subjected, which yield as a result a "limiting value" y ∈ R and enable us to write y = lim ϕ, (1.8) (where lim denotes the limiting process). Formally, whatever "limiting process" we may be able to devise, its end result is the selection of a collection F of pairs (y, ϕ) ∈ The limiting process associated to the choice of V in (1.10) is then a natural one: to wit, it is the convergence of ϕ to y along each sequence q in the collection, i.e., lim n→+∞ ϕ(q(n)) = y for each q ∈ V . (1.11) The application of this limiting process to the case where A = A(X) led de Possel to adopt an axiomatic approach based on the preliminary choice of a function V of the following form: (1.12) where hom Set (N, A(X)) is the collection of all A(X)-valued sequences, with the understanding that the sequences in the collection V(x) are axiomatically assumed to be "convergent" to a given point x ∈ X. In this set-up, de Possel had to solve the following problem: specify conditions on the function V in (1.12) which ensure that (1.13) for each f ∈ R, where R ⊂ L 1 (X) is a specified class of functions, and a.e. x ∈ X.

Notation from Category Theory
We find it convenient to adapt to our needs the notation from category theory employed in [17], and, whenever it is helpful, we append to an object or a morphism a subscript that specifies in which category it is located. Hence if C is a given category, we denote by hom C (A, Z ) the collection of morphisms in C from A to Z . For example, hom Set (A, Z ) [resp. hom BA (A, Z )] denotes the collection of functions from a set A to a set Z (resp. the collection of Boolean algebra homomorphisms between Boolean algebras A and Z ). Moreover, this subscript device will be used as a shorthand for the so-called forgetful functors. For example, if A is a topological space, then A Set denotes the underlying set. However, we will depart from strict observance of these notational devices whenever they lead to unnecessary notational clutter. For example, we find it useful to write, with a slight abuse of notation, hom Set (A, Z ) instead of hom Set (A Set , Z Set ), whenever A and Z are objects in some concrete category [recall that an object A ≡ (A Set , S A ) in a concrete category is a set A Set , called the underlying set, endowed with additional structure S A ]. In the same vein, whenever the 123 precise meaning can be gathered from context, the same symbol will denote an object in a concrete category or its underlying set.

Foundational Results
The limiting process adopted by de Possel is but one of many that have been conceived.

Functional Convergence Classes
The first contribution of the present paper is the introduction of a set of axioms which describe the properties which a relation F between R and hom Set (A, R) should satisfy in order to be the outcome of some "reasonable" limiting process which acts, so to say, in the "background." Indeed, one would hardly expect that every relation F between R and hom Set (A, R) as in (1.14) will be of interest.
A relation F between R and hom Set (A, R) is called a functional convergence class if it has some specific, natural properties, encoded in certain axioms, that will be described momentarily. As far as we know, the notion of functional convergence class is new, although it is inspired by the notion of convergence class [18, p. 73], which has, however, a different character.
The output of a limiting process for real-valued functions is a subset i.e., a collection of pairs (y, ϕ) ∈ R × hom Set (A, R), where (y, ϕ) ∈ F precisely if y = lim ϕ according to the limiting process acting on the background and encoded in F. The aim of the abstract notion of functional convergence class is precisely to recapture the natural properties that are expected from F.

The Filter of Neighborhoods of a Point in a Topological Space
Let A be a topological space. If x ∈ A, a neighborhood of x in A is a subset of A which contains an open set containing x. The set of all neighborhoods of x in A is denoted by (1.16) For example, N R (π ) is the collection We define (with a slight abuse of language) The meaning of (1.18) is that constant functions ought to converge to the constant. The meaning of (1.19) is that it is meant to exclude that every function converges to each value y ∈ R. Definition 1.2 A functional convergence relation F for real-valued functions defined on A is: then (y, β) ∈ F. Hereditary if, whenever y ∈ R, (y, ϕ) ∈ F, and (y, β) ∈ F, if γ ∈ hom Set (A, R) and there exists U ∈ N R (y) such that then it follows that (y, γ ) ∈ F.
The following observations should help the reader to assess the meaning of the axioms that describe the notion of functional convergence class.
(a) These axioms identify a class of subsets of R × hom Set (A, R).
(b) As we shall see, each F in this class arises from a certain "limiting process," expressed in purely formal terms by (1.8).
(c) The link between the "limiting process" (acting on the background) and F is given by (1.9). 123

Examples of Functional Convergence Classes
The following examples will give a first bird's eye view of the content of this paper and help to clarify the picture. More precisely, we will show that each of the following data entails a limiting process that yields a functional convergence class.
Example (i) The first example of a functional convergence class is the one induced by the choice of a nonempty collection of A-valued sequences. In Theorem 3.33 we show that, if V ⊂ hom Set (N, A) is such a collection and we define F V by [where lim n→+∞ ϕ(q(n)) = y is the familiar notion of convergence for the sequence ϕ • q : N → R] then F V is a functional convergence class.

Example (ii)
The second example of a functional convergence class is the one induced by the choice of a direction on A. In Theorem 3.24 we show that if R is a direction on A (i.e., R is a preorder on A such that for each j, k ∈ A, there exists an element l ∈ A such that jRl and kRl, as explained in Sect. 3.3) and we define F R by [where lim (A,R) ϕ = y denotes Moore-Smith convergence of ϕ : A → R along the direction R, defined in Sect. 3.4], then F R is a functional convergence class.

Example (iii)
The third example of a functional convergence class is the one induced by the choice of an A-valued Moore-Smith sequence. Theorem 3.33 implies that if q is such a sequence (hence q is a function q : D → A defined on a directed set, i.e., a set D which is endowed with a direction R) and we define F q by [where lim (D,R) ϕ • q = y denotes Moore-Smith convergence of ϕ • q : D → R along R], then F q is a functional convergence class.

Example (iv)
The fourth example of a functional convergence class is the one induced by the choice of a nonempty collection of A-valued Moore-Smith sequences (where different Moore-Smith sequences in the collection are possibly defined on different directed sets). In Theorem 3.33 we show that if V is such a collection and we define F V by [where, for each q ∈ V , (D q , R q ) is the domain of q, and lim (D q ,R q ) ϕ • q = y denotes Moore-Smith convergence of ϕ • q : D q → R along R q ] then F V is a functional convergence class. The fifth example of a functional convergence class is the one induced by the choice of a filter on A.

The Notion of Filter
The notion of filter, due to Henri Cartan, is a tool that helps clarify topological phenomena, and acts as a substitute, in case there is no topology; moreover, it is a precious tool in several mathematical areas.
The key observations leading to the notion of filter are the following. Firstly, observe that if A is a topological space and x ∈ A then the set N A (x), seen as a collection of subset of A, has the following essential properties: Secondly, the familiar -δ description of the existence of a limiting value lim z→x ϕ(z), where ϕ belongs to hom Set (A, R) shows that this notion only depends on the values of ϕ on (set-theoretically) small sets in N A (x). In view of the following definition, due to Cartan [8], N A (x) is called the neighborhood filter associated to A at x. Observe that (F1) is equivalent to the conjunction of the following two axioms: There is no filter on the empty set.
The collection of all ultrafilters on a set A is denoted by U U U (A).

The Category of Filtered Sets
The set A Set is called the total space of the filtered set A. A filter-homomorphism f : A → A between the filtered set A and the filtered set A is a function f : A Set → A Set between the underlying sets such that

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Filtered sets form the objects of a category, denoted FSet, where morphisms are filter-homomorphisms. We will return momentarily to the notion of filterhomomorphism, in order to achieve a better understanding of its meaning.

Localization
We will see that the seemingly simple hypothesis that a certain set belongs to a given filter has great import, and we use the expression the filter Z is localized in K , where K ⊂ A, as synonym for the set K belongs to the filter Z ∈ F F F (A). (1.23) Observe that and wloc(Z) satisfies (F2).
Hence if a filter is localized in K then it is weakly localized in K . The converse implication does not hold, unless the given filter is an ultrafilter, as we will see in Lemma 4.26. Indeed, we will see that a filter is an ultrafilter if and only if equality holds in (1.24).

Example 1.8
If A is a topological space and x ∈ A then (A, N A (x)) is a filtered set.

Example 1.9 The collection
is a filter on N, called the Fréchet filter on N.

Example 1.11
If (X, M, ω) is a complete probability space then the collection M F ω of measurable sets of full measure in X is a filter on X.

Limiting Values Along a Filter
Observe that the familiar -δ description of the existence of a limiting value lim z→x ϕ(z), in the case of a real-valued function ϕ defined on a topological space A, may be immediately adapted to the case where ϕ is defined on the underlying set of a filtered set A.

Definition 1.12
If A is a filtered set, Y is a topological space, ϕ ∈ hom Set (A, Y), and y ∈ Y, we say that y is the limiting value of ϕ along the filter Z, and write The meaning of the condition that ϕ : ) is a filter-homomorphism.

The Functional Convergence Class Induced by a Filter
Example (v) The fifth example of a functional convergence class is the one induced by the choice of a filter on A. In Theorem 3.2 we show that if Z is a filter on A and we define c A (Z) by (where lim Z ϕ = y denotes convergence of ϕ : A → R along the filter Z, defined in Sect. 1.2.5), then c A (Z) is a functional convergence class.

A Hierarchy of Limiting Processes
The second contribution of this paper is the clarification of the hierarchical relations between the limiting processes described in Examples (i)-(v). More precisely, we will prove the following results.

Applications to the Problem of the Differentiation of Integrals (I)
We are now ready to give a second bird's eye view of the content of this paper, where we present a reformulation of de Possel's approach in terms of filters. This reformulation is inspired by the following three implications of the results described in Sect. There is no gain in generality in the limiting process described in Example (iv), with respect to the one in Example (iii).
(C) The limiting process produced by filters, described in Example (v), has wider scope than the one produced by collections of sequences, described in Example (i).
The following set-up is based on these implications.

The Set-Up Based on Filters
Since the phenomena of interest in the present work are invariant under rescaling, the results we obtain for complete probability spaces also hold for complete measure spaces endowed with a finite measure (see Sect. 2). Hence, unless otherwise stated, we assume that (X, M, ω) is a complete probability space. Denote by F F F (A(X)) the collection of all filters on A(X).

Definition 1.14 A family of differentiation filters (based on X) is a function
which associates to each x ∈ X a filter G(x) ∈ F F F (A(X)). Definition 1. 15 We say that a family of differentiation filters (1.28) differentiates a f ω exists for a.e. x ∈ X and yields a representative of f . If R ⊂ L 1 (X), we say that G differentiates R if G differentiates f for each f ∈ R.

On the Differentiation of the Class of All Measurable Sets (I)
Perhaps the simplest class of integrable functions is given by the following one, associated to the σ -algebra of measurable sets: If R ∈ M has measure zero, then every family of differentiation filters (1.28) differentiates 1 R . Hence it suffices to restrict attention to {1 R : R ∈ A(X)}.

Definition 1.16
If G is a family of differentiation filters, as in (1.28), we say that G differentiates all measurable sets if, for each R ∈ A(X), G differentiates 1 R . Hence a lifting θ : M → M of (X, M, ω) amounts to the choice of a representative of the measure class π(Q), for each Q ∈ M, which preserves the Boolean structure of M, and hence establishes a Boolean isomorphism between the measure algebra of (X, M, ω) and some subalgebra of M.
The problem of the differentiation of the class of all integrable functions is clarified by the following result, whose proof may be obtained by adapting the techniques used in [20].

Theorem 1.18
If (X, M, ω) is a complete probability space, then a necessary and sufficient condition for the existence of a family of differentiation filters G : X → F F F (A(X)), which differentiates all integrable functions, is the existence of a lifting of (X, M, ω).
The following result, coupled with Theorem 1.18, shows that there exists a family of differentiation filters G : X → F F F (A(X)) which differentiates all integrable functions. Theorem 1.19 has a "curious history," as Fremlin puts it, which is recounted in [15, pp. 162-174], where a proof is given. The proof of Theorem 1.19 must necessarily involve the Axiom of Choice [6].

Measurability Issues (I)
In dealing with a general family of differentiation filters G : X → F F F (A(X)), we are faced with certain measurability issues, as we will see in more detail in Sect. 13. We will treat these difficulties using the same devices which de Possel used in his work.
For the collection of all subsets of a set A we use the standard notation

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In the study of filters the empty set is a nuisance, and in order to simplify many statements which otherwise would be too involved, we introduce the following notation for the collection of nonempty subsets of a given set: is a probability space, the outer measure induced by ω is defined by where, if Q ∈ P(X), then The following result is well known.

Lemma 1.21
For each Q ∈ P(X) there exists a set R ∈ M such that Q ⊂ R and ω * (Q) = ω(R).

Definition 1.22
If Q ∈ P(X) and R has the property described in Lemma 1.21, we say that R is a measurable representative of Q, and write

A Criterion for the Differentiation of Integrable Functions
Assume that G : X → F F F (A(X)) is a family of differentiation filters, f ∈ L 1 (X), α ∈ R, and Q ∈ A * (X).

Definition 1.24
We say that G is adapted to f on Q above α (resp. below α) if Definition 1. 25 We say that the mean-value of f over Q lies above α (resp. below α) Definition 1. 26 We say that G : X → F F F (A(X)) and f ∈ L 1 (X) are compatible if (a) for all Q ∈ A * (X) and for all α ∈ R, if G is adapted to f on Q above α, then the mean-value of f on Q lies above α, (b) for all Q ∈ A * (X) and for all α ∈ R, if G is adapted to f on Q below α, then the mean-value of f on Q lies below α.
Proof The proof is given in Sect. 13.

On the Differentiation of the Class of All Measurable Sets (II)
The following theorem is akin to a result due to Busemann and Feller in the context of the so-called differentiation bases [7].
) is a family of differentiation filters, then the following conditions are equivalent: Proof The proof is based on Theorem 1.27 and on an appropriate adaptation of a covering result due to de Possel. Details are omitted.

Notation
The The identity function I X : X → X is defined by It is a compact topological space which contains R as an open subset.

Sets, Collections, and Families
Since filters are elements of P • (P • (A)), in order to avoid confusion between the different levels in the hierarchy of powersets, we find it useful to reserve the term set (of points) to a generic element of P(A), and call collection (of sets) a generic element of P(P(A)); an element of P(P(P(A))) is called a family (of collections). We only deal with sets A for which x, r ∈ A ⇒ x / ∈ r .

Direct Image and Inverse Image Notation
We also find it useful, for the sake of clarity, to adopt the following notation from [22, p. 154], and write, if f ∈ hom Set (A, Z ), B ∈ P(A), and C ∈ P(Z ), In particular, f * : P(A) → P(Z ) and, by the same token, ( f * ) * : P(P(A)) → P(P(Z )). The restriction of f * : P(A) → P(Z ) to P • (A) will also be denoted by f * (with a slight abuse of language). Hence

Measure-Theoretic Notation
A measure space (X, M, ω) is a nonempty set X endowed with a σ -algebra M ⊂ P(X) of subsets and a set-function (called a measure) ω : M → [0, +∞] which is countably additive and whose value at ∅ is zero [29, p. 217]. The measure ω is said to be finite if ω(Q) ∈ [0, +∞) for all Q ∈ M. A probability space is a measure space (X, M, ω) with ω(X) = 1.

Null Sets and Derived Notions
A null set in a measure space (X, M, ω) is a set Q ∈ M such that ω(Q) = 0. The measure space (X, M, ω) is complete if each subset of a null set is also a null set. In a complete probability space (X, M, ω), the σ -ideal of null subsets is the collection The collection N is called a σ -ideal because it has the following properties: (i) it contains the empty set; (ii) if Q ∈ N and R ⊂ Q then R ∈ N ; (iii) it is closed under countable unions [14, p. 16].
It is useful to introduce the binary relations "⊂ ω " and ω = between subsets of a measure space, which are obtained from the inclusion relations "⊂" and "=" by replacing the empty set with null sets. If Q, R ⊂ X, we say that Q is a.e. contained in R, and write Q ⊂ ω R if Q \ R ∈ N : This means that almost all of Q is a subset of R. We say that the sets Q, R are almost everywhere equal, and write Q ω = R if Q ⊂ ω R and R ⊂ ω Q. Observe that ω = is an equivalence relation on M and that Q ω = R if and is a null set. We say that R is a.e. disjoint from Q if Q ∩ R ω = ∅, i.e., if Q ∩ R is a null set. A set Q ⊂ X has full measure if Q is a null set. A property holds a.e. (almost everywhere) if it holds on a set of full measure. A set Q ⊂ R has full measure in R if Q ∪ R has full measure.
In a complete probability space (X, M, ω), the collection of measurable sets of full measure is defined as follows:

Functional Convergence Classes
The goal of this section is to provide some of the proofs of results concerning the abstract notion of functional convergence class, introduced in Sect. 1.2 and show why a priori it is preferable to rephrase the work of R. de Possel in terms of filters rather than in terms of the choice of collections of sequences given in (1.12), as in the original approach by R. de Possel. We will also show that an approach based on the notion of filters appears to be preferable also with respect to a variant of (1.12) where instead of sequences one uses Moore-Smith sequences. Indeed, the lack of a uniqueness (in the representation of a given functional convergence class in terms of convergence along a Moore-Smith sequence) gives rise to ambiguities in the notion of exceptional set.

Functional Convergence Classes
We now show that the notion of limiting value along a filter on A yields a functional convergence class on A.
then U ∈ N R (y) and (ϕ) * (U ) = ∅, thus lim Z ϕ = y does not hold, hence (y, ϕ) / ∈ c A (Z). We have proved that c A (Z) is a functional convergence relation.
Assume that y ∈ R, lim Z ϕ = y, lim Z β = y, and γ ∈ hom Set (A, R). Suppose that γ has the property that, for some and since ϕ * (V ∩ U ) and ϕ * (V ∩ U ) both belong to Z, and Z is a filter, it follows that In the following section, we will show that the map c A in (3.1) is one-to-one and onto.

Definition 3.3 If A is a nonempty set and F ∈ FCC(A), then define by
Firstly, observe that s A (F) is not empty, since it contains A, because F(0) contains the constant function identically equal to 0.
Secondly, we show that ∅ / ∈ s A (F). We proceed by contradiction and assume that there exists V ∈ N R (0) and ϕ ∈ F(0) such that (ϕ) * (V ) = ∅. Let β ∈ hom Set (A, R) and observe that (1.20) holds for y = 0. Since F is local, it follows that β ∈ F(0). Hence we have proved that hom Set (A, R) = F(0). Since F is translation invariant, it follows that F = R × hom Set (A, R), a contradiction.
We now show that if b ∈ s A (F) and c b then c ∈ s A (F).
Then β * (U ) is either equal to c or it is equal to A. In either case, β * (U ) contains b = ϕ * (V ). Since F is local, it follows that (0, β) ∈ F. Observe that β * (V ) = c. Hence c ∈ s A (F), and the proof of (3.4) is complete. Now, assume that U ∈ N R (0). We claim that Finally, we prove that Hence (3.6) follows from (3.4) and (3.5). The proof that s A (F) is a filter is complete.
implies that lim Z ϕ = 0, hence it implies that ϕ * (U ) ∈ Z, and thus b ∈ Z. Hence we have proved that Z ⊃ s A (c A (Z)), and the proof is complete.
We claim that Proof It suffices to apply Lemmas 3.5 and 3.6.

Moore-Smith Sequences
In 1915 and 1922 Eliakim Hastings Moore and Herman Lyle Smith attempted to subsume different limiting processes under the same notion [23,24]. They were motivated by the following heuristic principle: The existence of analogies between central features of various theories implies the existence of a more fundamental general theory embracing the special theories as particular instances and unifying them as to those central features.
In their work, they created the notion of Moore-Smith sequence (see below), and, in so doing, they introduced the notion of a direction. As is customary, if R is a binary relation on a set S, i.e., a subset of S × S, we write jRk instead of ( j, k) ∈ R. Definition 3.8 A preorder R on a nonempty set S is a reflexive and transitive binary relation on S, i.e., a subset of S × S with the following properties: (R) jR j for each j ∈ S (reflexivity); (T) if jRk and kRl then jRl (transitivity).
Definition 3.9 A partial order R on a nonempty set S is a preorder R on S which also satisfies the following condition: (A) if jRk and kR j then j = k (antisymmetry).
A poset A is a set A Set endowed with a partial order R A .

Definition 3.12 A subset T ⊂ A is called final in A if it contains some tail, and the collection of all final sets in A is
We may write Fin[R] instead of Fin[A] in case we need to emphasize the role of the direction R.
The notion of tail, and the associated notion of final set, display their full power only if some other assumptions are made on the preorder.

Definition 3.13
A direction on a set S is a preorder R on S such that, for each j, k ∈ S, there exists an element l ∈ S such that jRl and kRl. We define

We will see that A is a directed set if and only if Fin[A] is a filter on A.
123 Example 3.14 N is a directed set under the natural order: The following result shows that reverse inclusion in a filter is a direction.

Example 3.15 If Z ∈ F F F (A), then reverse inclusion between sets (⊃) is a direction on Z. In particular, if is a topology on A and x ∈ A, then (N (x), ⊃) is a directed set.
Directed sets serve as domains of definition of Moore-Smith sequences. It is useful to emphasize the role of the codomain, as in the following definition.
The elements of S (Y) are called Y-valued Moore-Smith sequences. The directed set which appears in (3.12) is called (with slight abuse of language) the direction of w.
Observe that i.e., each element of Y may be seen as a constant sequence, and each sequence is a Moore-Smith sequence.
Proof The proof follows at once from the fact that the composition of functions is associative whenever defined, and I Y • w = w. For background, see [22, p. 501].
Observe that a direction is not necessarily a partial order, since antisymmetry may fail. An antisymmetric direction on a set A is a direction R on A for which if jRk and kR j then j = k.

Definition 3.18
If w is a Y-valued Moore-Smith sequence, is a topology on Y, A is the direction of w, and y ∈ Y, we say that y is the limiting value of. w along A, and write lim w = y

Example 3.19
On R the preorder ≤ [resp. the preorder ≥] yield the familiar notions lim r →+∞ w(r) [resp. lim r →−∞ w(r)] for a Y-valued Moore-Smith sequence w : If we need to emphasize more explicitly the preorder R or the direction A, we write instead of (3.14).
The following elementary remark is useful in topological spaces where points are not necessarily separated. We will see that the set F F F (A) is endowed with a topology of this kind. Indeed, we will see that F F F (A) is compact but not Hausdorff, while U U U (A) is compact and Hausdorff.

Lemma 3.21
If z, w ∈ Y and is a topology on Y, then the following conditions are equivalent:

Remark 3.22
Of course the statement is interesting only if z = w. Observe that {w} is the closure in the given topology, and that (2) rests on the fact that, according to (3.13), we may identify w with the constant sequence w identically equal to w, and indeed (2) says that z is the limiting value of this sequence.
, since z ∈ {w}, hence z = lim w, and (2) follows from the identification of w with w in (3.13). Lemma 3.21 is a special case of the following, more general, result, due to Garrett Birkhoff [3]. Observe that if W ⊂ Y then each W -valued Moore-Smith sequence may be seen as a Y-valued Moore-Smith sequence: is a topological space, W ⊂ Y, and z ∈ Y then the following conditions are equivalent.
Hence O ∩ W is not empty and (1) holds.

Remark on notation
In order to facilitate the distinction between the setting of filters and the setting of Moore-Smith sequences, and also to gain a better appreciation of the connection between the two viewpoints, we use bold Sans Serif font to denote limiting notions pertaining to filters, such as lim, cluster, liminf, limsup, and Typewriter font to denote notions pertaining to Moore-Smith sequences, such as lim and ClusterSet. Indeed, it seems to us that if we used the same notation for the different notions then the connection between the two viewpoints would be obscured by the uniform notation.

The Functional Convergence Class Associated to a Direction
The following result says that convergence along a direction, described in Definition 3.18, is a limiting process that yields a functional convergence class, just as the limiting process of convergence along a filter does. However, we will see that the limiting process of convergence along a filter on A, described in Definition 1.12, has wider scope and higher synthetic power than the limiting process of convergence along a direction on A, described in Definition 3.18.
is a functional convergence class on A.
Proof The proof will be given in Sect. 3.6.

A Comparison of the Two Notions
In 1938, Herman Lyle Smith considered the following notion of limiting value, due to Arnaud Denjoy [10, p. 165], [32], [13, p. 158]. We say that ϕ has approximate limiting value equal to y at x 0 if the following condition holds.
has density equal to 1 at x 0 . H. L. Smith observed that the limiting notion in (Example MS 4) may be readily subsumed under Definition 1.12 but that it cannot be covered by Definition 3.18 "without a somewhat artificial transformation." Observe that the collection of final sets in a directed set is a filter. For example, the filter generated by the natural order on N is the Fréchet filter (see Example 1.9). Indeed, we now show that every directed set A yields a filtered set A FSet , whose underlying set is the underlying set of A and whose filter is the one generated by the given direction on A. In other words, we define a map (3.16)

Lemma 3.25 If A is a directed set, then the collection Fin[A] of final sets in
A is a filter on A, called the filter generated by (the tails of) R.
Lemma 3.25 enables us to subsume Definition 3.18 under Definition 1.12.

Lemma 3.26
If w is a Y-valued Moore-Smith sequence, is a topology on Y, A is the direction of w, and y ∈ Y, then the following conditions are equivalent Proof Definition 3.18 says precisely that w :

Proof of Theorem 3.24 Lemma 3.26 says that
and we know from Theorem 3.2 that the right-hand side of (3.17) is a functional convergence class, since Fin[A] is a filter on A.
Hence (3.15) and Theorem 3.24 yield a map Moreover, the following result also follows immediately from Theorem 3.24.

Theorem 3.27 If
A is a nonempty set, then the following diagram is commutative where the map on the top is the one given in (3.16) and the diagonal map is the one given in (3.18).

123
Proof The result follows immediately from Lemma 3.26.
Recall that Theorem 3.7 says that c A in (3.19) is 1-1 and onto, and this means that every functional convergence class on A is associated to a unique filter. One may wonder whether the diagonal map in (3.19) is also onto, i.e., whether every functional convergence class is associated to a direction on A. As far as we know, the following result is new.

Theorem 3.28 If
A is equal to the unit disc in C, then the diagonal map in (3.19) is not onto. Indeed, the functional convergence class associated to nontangential convergence is not associated to any direction.
Proof We now show that Theorem 3.28 may be reduced to Theorem 3.29, to be stated momentarily. Recall that if D def = {z : z ∈ C, |z| < 1} is the unit disc in C, then there exists a filter S ∈ F F F (D), called the nontangential filter on D ending at 1 (see [11,12,33], for background), such that the following result holds. A precise definition of the nontangential filter S will be given in Sect. 5. Theorem 3.28 follows at once from the following result.

Theorem 3.29
The nontangential filter S on D is not equal to the filter of tails of any direction on D.
The proof of Theorem 3.29 will be given in Sect. 5. Remark 3. 30 We do not know whether an intrinsic characterization of the image of the map (3.16) is known, i.e., whether it is possible to give an intrinsic characterization of those filters which are generated by a direction.

Remark 3.31 Lemma 3.25 shows that every directed set A yields a filtered set A
, whose underlying set is the underlying set of A and whose filter is the filter of tails of the given direction on A. We will look at DSet as a full subcategory of FSet, i.e., we will declare that DSet-homomorphisms from A to A , where A and A are directed sets, are precisely the FSet-homomorphisms from A FSet to A FSet . However, we will not base the notion of Moore-Smith subsequence on this identification, since it would lead to "irregularities" [1, p. 285] and make the subject somewhat "contentious," as Saitulaa Naranong puts it in Translating between Nets and Filters (2010) (unpublished). We will return to this theme in Sect. 12.
Directed sets form a proper subclass of the objects of the category FSet of filtered sets, since (Lemma 3.26) the relevant data in a directed set is the filter of tails of the given direction, and (Theorem 3.29) The map (3.18) is not necessarily onto.
In his work, R. de Possel used sequences of measurable sets. One may be tempted to employ instead Moore-Smith sequences, and we will do so in Sect. 3.7, but we will see that filters appear to be more flexible and direct. This conclusion may appear to be counterintuitive, since convergence phenomena are based on an idea of movement, and Moore-Smith sequences appear to be especially suited to represent them, because a "dynamic" is encoded in the directed set which acts as their basis, while filters are seemingly "static" objects, in the sense that there is no apparent "sense of direction" in them. In Sect. 9, we will show that this impression is erroneous, since the collection of all filters on a given nonempty set is endowed with a natural topology, which is especially suited to be used in the study of convergence phenomena. Hence, the advantage of filters is that there is no need to rest on the additional structure of a directed set, since a "sense of direction" is encoded in their intrinsic structure.

The Functional Convergence Class Associated to a Family of Moore-Smith Sequences
We now introduce another method for constructing functional convergence classes.

Theorem 3.33 If A is a nonempty set and a nonempty set V ⊂ S (A) is given, then
, is a functional convergence class.
Proof The proof will be given in Sect. 7.
, every y ∈ Y, and each function ϕ : A → Y, the following conditions are equivalent: Observe that Z is a directed set, by Example 3.15, and hence S ⊂ S (A). Let (Y, ) be a topological space, ϕ : A → Y, and y ∈ Y, and assume that (1) holds. If is arbitrary, it follows that lim ϕ • t = y, and since t ∈ S is arbitrary, it follows that (2) holds. If (1) does not hold, then there exists does not hold.

Theorem 3.35 If A is a nonempty set, then every functional convergence class on
Proof This result follows at once from Theorems 3.7 and 3.34.
We have thus seen that a functional convergence class on A may be represented in terms of a unique filter, or in terms of a family V , as in Theorem 3.33. The advantage of the representation in terms of filters is precisely given by uniqueness. Indeed, the lack of uniqueness would cause some difficulties in the determination of the exceptional set for a.e. convergence. Hence the approach based on the notion of filter has higher synthetic power and flexibility.
Theorem 3.34 shows that the functional convergence class c A (Z) of a given filter Z on A may be described as F V , in terms of a set V of A-valued Moore-Smith sequences. One may wonder whether it is possible to choose as V a set of A-valued sequences, and whether it is possible to choose as V as set consisting of just one Moore-Smith sequence. We will see that the answer to the first question is in the negative, and that the answer to the second question is in the positive.

Theorem 3.36 There exists a nonempty set A and a functional convergence class
Proof Let A = N and let Z ∈ F F F (N) be an ultrafilter on N which contains the Fréchet filter.

Theorem 3.37 For every nonempty set A and each functional convergence class F on A there exists an
Proof The proof will be given in Sect. 8.

Preliminary Results on Filters
The goal of this section is to give a self-contained presentation of the basic results on filters.

Basic Lattice-Theoretic Properties of F F F (A)
A preliminary examination of some lattice-theoretic properties of F F F (A) will be useful, as we will see, in order to gain a better understanding of the topological implications of the notion of filter. Recall that if C is a family of filters, i.e., if C ⊂ F F F (A), then (Cartan [8]) The intersection C of any nonempty family C of filters on a set is not empty and is a filter.
Observe that P(X) is a poset, hence F F F (A) is a poset under set inclusion (Example 3.10). Observe that antisymmetry of R implies that, in a poset, a meet of x and y, if it exists, is unique.
The notion of greatest lower bound of a subset S of a poset X is defined in a natural way, to wit: If it exists, it is an element l ∈ X such that (i) l is a lower bound of S (i.e., lRa for each a ∈ S), and (ii) if b is a lower bound of S, then bRl; If it exists, it is unique. If it exists, the greatest lower bound of a subset S is denoted by s∈S s.  Antisymmetry implies that, in a poset, a join of x and y, if it exists, is unique. In Sect. 9.2 we will see that the existence of the join of two filters is more delicate.

The Operator A ↑
In order to present a basic technique for the construction of filters and exhibit more examples of filters, we introduce an operator A ↑ associated to every nonempty set. This operator is actually implicit in the definition of the notion of filter, so it is not surprising that it serves as a useful tool to construct new ones.
The basic building block for the operator A ↑ is contained in the following observation, which shows that F F F (A) contains a copy of P • (A). Indeed, consider the following diagram where the function ı : , and the functions A • and A ↑ will be defined momentarily.
is a filter which contains b as an element: it is the smallest filter on A which contains b as an element, and is called the principal filter generated by b on A.
The map A ↑ is designed to make the diagram (4.1) commutative (i.e., to extend A • to P • (P • (A))) and to commute with the union of collections.
for each indexed family of collections {Z α } α∈I , where Z α ∈ P • (P • (A)) and I is a nonempty set of indexes. Proof We define the map as follows: We now show that A ↑ commutes with the union of collections. Indeed, the statement that b ∈ A ↑ α∈I Z α means that there exists α ∈ I and c ∈ Z α such that b ⊃ c, and this means precisely that b ∈ α∈I A ↑ [Z α ]. In order to show uniqueness, it suffices to observe that W = b∈W {b} = b∈W ı(b) for each W ∈ P • (P • (A)).
. The conclusion follows from Lemma 4.9.

Bases and Subbases
Observe that W ∈ P • (P • (A)) satisfies (F2) in the axioms of a filter (Sect.
is not a filter, since a filter cannot contain disjoint sets.

Generating Bases for a Filter
Lemma 4.17 If Z ∈ F F F (A), W ⊂ Z, and the following condition holds: then W is a filter base on A and A ↑ [W] = Z.
On the other hand, (4.7) and Lemma 4.10 imply that  Observe that (4.7) says that for each b ∈ Z there exists c ∈ W such that c ⊂ b.
If any of these equivalent conditions holds, we say that W and Y are equivalent.

Filter Subbases
Observe that if W ∈ P • (P • (A)) then the collection satisfies Condition (2) in Lemma 4.11 but it is not necessarily true that W ∩ ∈ P • (P • (A)), since it may happen that ∅ ∈ W ∩ . However, we have the following result, stated in terms of (4.8).
Lemma 4.21 (Cartan [8]) If W ∈ P • (P • (A)) then a necessary and sufficient condition for the existence of a filter on A which contains W is that ∅ / ∈ W ∩ . If ∅ / ∈ W ∩ , then W ∩ is a filter base on A, and the filter A ↑ W ∩ is said to be generated by the subbase W.
Condition (2) in Lemma 4.11 holds for W ∩ by its very construction, and A ↑ W ∩ is a filter which contains W.

Definition 4.22
If W ∈ P • (P • (A)) and ∅ / ∈ W ∩ , then we then say that W is a filter subbase on A, and A ↑ W ∩ is the filter generated by the subbase W on A: It is the broadest filter which contains W.

Ultrafilters and Compactness
If Z ∈ F F F (A), then Z ⊂ P • (A), and, in particular, if b ∈ Z, then b ⊂ A. Thus (4.9) Hence F F F (A) inherits from P • (P • (A)) the partial order given by inclusion. The notion of ultrafilter, due to H. Cartan, introduced in Definition 1.5, is useful in several areas: topology, functional analysis, mathematical logic, among many others.
Observe that a filter on A is an ultrafilter if it is a maximal element of F F F (A) under inclusion.

Corollary 4.27 If
Observe that, if T ∈ P(P(A)) is an open cover of K ⊂ A, then for each

Lemma 4.29 If T ∈ P(P(A)) is an open cover of K ⊂ A, then the collection
is a filter if and only if T has no finite subcover of K .
Proof Observe that ∅ ∈ W T K if and only if T has a finite subcover of K , hence it suffices to observe that (i) The following characterization of compactness is useful.
Lemma 4.30 (Cartan [8]) Assume that (A, ) is a topological space and that K ⊂ A. Then the following conditions are equivalent: Proof If K is not compact, then there exists T ⊂ which is an open cover of K with no finite subcover. Then W T K in (4.10) is a filter. Observe that K ∈ W T K . Theorem 4.25 We claim that Z does not cover K , and hence there exists If U ∈ and x ∈ U then U / ∈ Z , and thus U / ∈ Z, hence U ∈ Z, by Lemma 4.26. Hence N (x) ⊂ Z.
We now prove the claim. If Z covers K , then a finite subcover of Z covers K , and since Z is closed under finite unions, K is contained in one of the sets in Z , hence K ∈ Z, which is impossible since K ∈ Z.

Functorial Properties of Direct Images and Application to Limiting Values
Recall from Lemma 4.16 that if Z ∈ F F F (A) and f ∈ hom Set (A, Y), then ( f * ) * (Z) is a filter base on Y.

Proposition 4.33
Assume that A and Y are filtered spaces, and f ∈ hom Set (A, Y).

Then f is a filter-homomorphism if and only if
and Lemma 4.32 says that this is the same as asking that (A, Z) is a filtered set, (Y, ) is a topological space, f : A → Y is a function, and y ∈ Y, then the following conditions are equivalent:

Corollary 4.34 If
Proof The result follows at once from Proposition 4.33 and Definition 1.12. Given a function f : A → Y we have thus defined the map . Hence we have almost completely proved the following result.

Extension of Filters from a Subset and Restriction to a Subset
If Ω A and ı : Ω → A is the standard injection, defined by ı(x) def = x, then the associated map is injective, as we will see in Lemma 4.38, but this does not mean that if Z ∈ F F F (Ω) then Z ∈ F F F (A). The following result clarifies this point.

Lemma 4.37
If Ω A and Z ∈ F F F (Ω) then Z / ∈ F F F (A).
Proof Observe that Z only contains subsets of Ω, and Z ∈ F F F (A) ⇒ A ∈ Z, which is impossible.
A precise description of the map (4.13) will now be given.

Lemma 4.38
If Ω A and Z ∈ F F F (Ω) then Z is a filter base on A, and Z ⊂ ı (Z). Proof The fact that Z is a filter base on A follows at once from Z ∈ F F F (Ω).

Definition 4.39
If Ω A and Z is a filter on Ω then the filter in (4.14) is a filter on A called the extension of Z from Ω to A.

Proposition 4.40 If W ∈ F F F (A)
and Ω ⊂ A, then the following conditions are equivalent: The following collection is a filter on Ω The equivalence between (2) and (3) then follows at once from Lemma 4.21. We now show that (1) and (2) are equivalent. If (2) does not hold then there exists d ∈ W such that b ∩ d = ∅, and this means that d ⊂ b, hence b ∈ W, i.e., (1) does not hold. If (1) does not hold then b ∈ W, hence b ∩ b = ∅, hence (2) does not hold. Observe that (4) implies (2) at once, since no set in a filter can be empty. We now show that (2) implies (4).

Definition 4.41
If Ω A, and W is a filter on A which is weakly localized in Ω, then the filter in (4.16) is a filter on Ω called the restriction of Z from A to Ω.

Theorem 4.42
If Ω A and W ∈ F F F (A) then the following conditions are equivalent: Proof Assume that (1) holds, and observe that Ω = Ω ∪ ∅ and Ω ∈ Z. Hence (4.14) implies that Ω ∈ W. Assume that (2) holds. Then W is weakly localized in Ω, and Proposition 4.40 implies that the collection in (4.16) is a filter on Ω. We claim that ı (Z) = W. Let d ∈ ı (Z). Then there exists b ∈ W and c ∈ P(A \Ω) such that Hence (4.14) implies that d ∈ ı (Z). Hence we have proved that ı (Z) ⊃ W, and the proof is concluded.

Separable Filters
. Hence G is a generating basis for Z.

Proof of Theorem 3.29
If we specialize (3.11) to D we obtain the following: See [11,12,33], for background. It is convenient to replace the open Euclidean triangles which appear in ( * ) with the more symmetrical nontangential approach regions in D at 1, as follows. For > 0, let be the open disc in C of center 1 and radius 1.
Remark If we used strict inequality in (5.2) (instead of the nonstrict inequality which appears inside the curly brackets) and if we were to omit the intersection with D [1], then we would obtain the same filter, and hence the same notion of convergence, but the proof would become a bit more involved. Indeed, observe that Γ α contains the following set, which will be useful in the proof: The nontangential filter S may now be defined as follows. Choose, once and for all, a bijective function Our goal is to prove the following statement.
(♠) There exists no direction R on D such that the filter of tails of R is equal to the nontangential filter S.
Indeed, we will prove that S does not belong to the image of the map (5.1). In other words, we will prove that the following set is empty: The following useful criterion follows at once from Lemma 3.26. (1) lim S w = z, (2) lim R w = z.

123
The nontangential filter is related to the nontangential approach regions described in (5.2) as follows.
( * * ) For each w : D → R and each z ∈ [−∞, +∞], lim S w = z if and only if lim Γ j z→x w(z) = z for each j ≥ 1.

Remark
The fact that ( * * ) holds implies that the choice made in (5.4) does not change the resulting filter. See [11].
If R is a direction on D and x ∈ D, the R-tail in D from x ∈ D is defined just as in (3.9), with an emphasis on the direction rather than on the directed set: We will also need the following notion. Proof Since Q → 1, it is possible to define a sequence r j > 0 j≥0 in such a way that 2 = r 0 > r 1 > · · · > r j > r j+1 for each j ≥ 0, lim j→+∞ r j = 0, and  Proof Assume that R ∈ dir(D), x ∈ D, α > 1, Q ⊂ Γ α , Q → 1, and Q ∩tail R (x) = ∅. Define w : D → R as a function that vanishes identically on D \ Q and which on Q is equal to the function described in Lemma 5.6. Then lim S w does not exist, by ( * * ), but lim R w exists. Indeed, lim R w = 0, since for each > 0 the values of w on the R-tail from x all lie in (− , ). Lemma 5.4 then implies that R / ∈ Fin * (S).

Lemma 5.8 If R ∈ Fin * (S), then Fin[R] is separable.
Proof Let q n def = 1 − 1 n , for each n ∈ N with n ≥ 1 and let Q def = {q n : n ∈ N, n ≥ 1}. Then Q → 1 and Q ⊂ Γ α for each α > 1. Let C def = {b ∈ P • (D) : b = tail R (q n ) for some n ∈ N}. Observe that C is countable and C ⊂ Fin[R]. Lemma 5.7 implies that for each x ∈ D there exists n ∈ N such that q n ∈ tail R (x), i.e., tail R (x) ⊃ tail R (q n ), and this means that Proof Let r 1 , r 2 , . . . , r k , . . . be a sequence of elements of hom Set (N, Q + ) such that G = {Q(r k ) : k ∈ N}. We claim that there exists s ∈ hom Set (N, Q + ) such that for each k ∈ N, Q(r k ) \ Q(s) = ∅. (5.7) Hence Q(s) ⊃ Q(r k ) for each k ∈ N, and since Q(s) ∈ S, this means that S ⊂ D ↑ [G]. We will construct s ∈ hom Set (N, Q + ) and a sequence z 1 , z 2 , . . . , z k , . . . of points in D in such a way that, for each k ∈ N, and , in order to ensure that (5.9) holds it is necessary that, for each n ∈ N, The values c j will be specified momentarily. Define and, for each k ≥ 1, and then define c k+1 Define c 1 = |1 − z 1 |. Observe that (5.12) implies that, for each k ∈ N, Hence (5.8) holds for each k ≥ 1.
The proof of (5.10) for a generic value of k is similar, and is achieved by first showing that it holds if n ∈ I 1 ∪ I 2 ∪ · · · I k , and then by showing that it holds for n ∈ I k ∪ I k+1 ∪ · · · Indeed, in the first case, observe that α(n) < α(n k ). Thus (5.14) implies that z k / ∈ Γ α(n) , hence (5.10) holds. In the second case, (5.11) and (5.14) imply that s(n) ≤ c k+1 < |1 − z k | hence z k / ∈ D[s(n)], and (5.10) holds also for these values of n.

Proposition 5.10
The nontangential filter on D ending at 1 is not separable.
Proof Recall that the collection B, defined in (5.6), is a generating basis for S. Let us assume that S is separable. Then Lemma 4.44 implies that there exists a countable subcollection G ⊂ B which is a generating basis for S, but this is impossible by Lemma 5.9.
Proof of Theorem 3.29 Assume that the set Fin * (S) is not empty, and let R ∈ Fin * (S). Then Lemma 5.8 implies that Fin[R] is separable. Now R ∈ Fin * (S) means that Fin[R] = S, hence it follows that S is separable, in contradiction with Proposition 5.10.
topic has an interest of its own, even though its application to the main task of this paper appears to be limited, both because of a lack of a topology, and for the reasons illustrated in Sect. 3.
Since Moore-Smith sequences of points are a special case of Moore-Smith sequences of nonempty subsets of the given topological space, we find it useful to begin with the former case.

Functorial Properties of the Filter of Tails
The convergence properties of a Moore-Smith sequence are entirely determined by the associated filter of tails, which is encoded in the operator

natural transformation between two functors. Recall that, if A is a directed set, then Fin[A]
is the filter of tails of A, described in Lemma 3.26.  The following result says that the convergence properties of a Moore-Smith sequence are entirely determined by the associated filter of tails.

Theorem 6.2 If w ∈ S (Y) is a Y-valued Moore-Smith sequence, is a topology on
Y, and y ∈ Y, the following conditions are equivalent Proof It suffices to apply Lemma 3.26 and Corollary 4.34.
In the following result, due to Bruns and Schmidt [5, p. 171], we show that the map (6.1) is onto, i.e., that for each Z ∈ F F F (Y) there exists w ∈ S (Y) such that

Y-valued Moore-Smith sequence w.
Proof Given a filter Z on Y, define consider the direction R on A defined by and define Lemma 6.3 may be strengthened so as to yield the following result, due to Bruns and Schmidt [5].

Lemma 6.4 If Y is a nonempty set, then every filter Z on Y is the filter of tails of a
Y-valued Moore-Smith sequence w such that the direction of w is antisymmetric.
Proof Instead of (6.3), define endowed with the lexicographic partial order R defined by (b 1 , n 1 , Observe that R is an antisymmetric direction. Define the Moore-Smith sequence w on A by w(b, n, x) def = x for each (b, n, x) ∈ A. Then t Y [w] = Z.
We now show that the function which associates to each nonempty set A the map in (6.1) is a natural transformation between the functor in Lemma 3.17 and the one in Lemma 4.36 (see [21] for background on categorical language). Lemma 6.5 For each pair of nonempty sets Y and X and every f : Y → X, the diagram (6.4) is commutative. and , it suffices to follow these steps backwards.

Set-Valued Moore-Smith Sequences
We will examine not only Y-valued Moore-Smith sequences (where Y is a given topological space) but also P • (Y)-valued Moore-Smith sequences, and show that the latter category of Moore-Smith sequences enjoys properties that are more streamlined with respect to the Y-valued Moore-Smith sequences. We now show that the second class of Moore-Smith sequences includes the first one.

Definition 6.6 The injective function
Observe that the injective map (6.5) is obtained by composition of w : Hence we will think of S (Y) as a subset of S (P • (Y)), i.e., we will identify w : A → Y with ı Y • w.

Lemma 6.7 The assignment Y → S (P • (Y)) is the object function of a functor from the category of sets to the category of sets. The associated arrow function assigns to each function f
Proof The proof follows at once from the fact that the composition of functions is associative.
The function f • : S (P • (Y)) → S (P • (X)), restricted to S (Y), recaptures the function described in Lemma 3.17. For this reason, it is denoted by the same symbol. Recall that in Sect. 6.1 the map has been defined, which associates to each w ∈ S (Y) the filter t Y [w] ∈ F F F (Y), called the filter of tails of w, and recall the natural injection (6.5)

S (Y) → S (P • (Y)).
In Lemma 6.9 we show that the dotted arrow in the following diagram may be defined so as to make it commutative: where A is the direction of w and j ∈ A Set . The collection is called the filter on Y generated by the tails of w. This terminology is justified by the following result.

Lemma 6.9 If w is a P • (Y)-valued Moore-Smith sequence, then the collection T Y [w]
defined in (6.9) is a filter on X, and the map makes the diagram (6.7) commutative.
Proof The first statement follows from the fact that the collection of tails, defined in (6.8), forms a filter base. Indeed, given j 1 , j 2 ∈ A Set , there exists j 3 ∈ A Set such that j 1 R j 3 and j 2 R j 3 , and then it follows that Tail The second statement follows at once from the identification of w ∈ S (Y) with the element of S (P • (Y)) described in Definition (6.6).
Hence, for every nonempty set Y, we have defined a map Consider the following diagram:

Lemma 6.10
For each nonempty sets Y and X and every f : Y → X, (6.11) is commutative.
Hence there exists j ∈ A Set , where A is the direction of w, such that Tail j [w] ⊂ Q.
. In order to prove the claim, let y ∈ Tail j [w] and x = f (y). Then y ∈ w(k) for some k ∈ A Set with jRk. Then Observe that Lemma 6.10 says that the assignment Y → T Y : We now show that every filter on Y is the filter generated by the tails of a generalized sequence of nonempty subsets of Y.

Definition 6.12 From Example 3.15 we obtain a map
as follows: If Z ∈ F F F (Y), then the natural injection The proof of the following result is independent of Lemma 6.3.
123 Lemma 6.13 says that a filter Z is represented by the generalized sequence s Z : Z → P • (Y). The following result follows at once from Lemmas 6.10 and 6.13.

Proof of Theorem 3.33
Assume that V = {w α } α∈I , where I is a set of indices and w α : (3.20), is equal to c A (Z), and apply Theorem 3.2.

Proof of Theorem 3.37
Let F be a functional convergence class on A. Theorem 3.7 implies that there exists a filter Z on A such that F = c A (Z). Lemma 6.3 implies that there exists a Moore-

A Natural Topology on the Collection of Filters
The goal of this section is to show that the collection F F F (A) of all filters on a given set A is naturally endowed with a topology. This result, among many others, indicates that filters are intrinsically associated to the notion of convergence.

Topological Preliminaries
Observe that, if is a topology on a nonempty set X, then (1.16) defines a function N : X → F F F (X) (9.1) called the family of (neighborhood) filters associated to . Observe that the map → N is injective, i.e., may be recaptured from N . Indeed, Q ∈ if and only if for each x ∈ Q there exists U ∈ N (x) such that x ∈ U ⊂ Q.

Definition 9.1 A function N : X → F F F (X) is called a family of filters on X based on X.
We now list some properties that a map ϕ : P(X) → P(X) may have. Definition 9.2 A map ϕ : P(X) → P(X) may have one or more of the following properties.
A map ϕ : P(X) → P(X) is called regular if it preserves the ambient space and finite intersections, and is an idempotent contraction. Observe that (pas) and (pfi) say that ϕ is a homomorphism of the multiplicative semigroup of P(X) as a Boolean algebra (see Sect. 9.3).
Observe that

Lemma 9.3 If is a topology on X, then the associated topological interior operator
Proof We omit the proof, which follows immediately from the notion of topology. We now wish to express Condition (r) in Theorem 9.7 directly in terms of N.

Proposition 9.8
For each map N : X → F F F (X), the following conditions are equivalent: (t) N is the family of neighborhood filters associated to some topology on X.
(r)N is an inflating contraction (r ) x ∈ X, Q ∈ N(x) ⇒ x ∈ Q and ∃ R ∈ N(x) such that y ∈ R ⇒ Q ∈ N(y).

Observe that the set R is necessarily contained in Q.
Proof In Theorem 9.7 we proved that (t) and (r) are equivalent, hence it suffices to show that (r) and (r ) are equivalent. Assume that (r) holds. ThenN is regular [by Lemma 9.6 and (9. 2)]. Hence if x ∈ X and Q ∈ N(x) then x ∈N(Q) has the property in (r ). Assume that (r ) ,N is a contraction. If x ∈N(Q) then Q ∈ N(x) and thus by (r ) there exists R ∈ N(x) with y ∈ R ⇒ Q ∈ N(y), i.e., y ∈ R ⇒ y ∈N(Q), i.e., R ⊂N(Q), and since R ∈ N(x) and N(x) is a filter on X it follows thatN(Q) ∈ N(x), i.e., x ∈N(N(Q)). Hence x ∈N(Q) ⇒ x ∈N(N(Q)), i.e.,N is inflating.
Hence in order to describe a topology on F F F (A), it suffices to describe a map N : is an inflating contraction. We first need some more preliminary work.

Further Lattice-Theoretic Properties of F F F (A)
We now go back to a question that was left open in Sect. 4.1, to wit: the existence in F F F (A), seen as a poset, of the l.u.b. of two given filters Z 1 , Z 2 ∈ F F F (A). Recall from Sect. 4.1 that the l.u.b. of Z 1 , Z 2 ∈ F F F (A) (called the join of Z 1 and Z 2 ), if it exists, is denoted Z 1 ∨ Z 2 and has the following two properties: (i) Z 1 ⊂ Z 1 ∨ Z 2 and We now show that the obstruction to the existence of the join of two filter lies in the existence of a filter which contains both. The following result follows at once from Lemma 4.21. However, it is instructive to provide a direct proof. Lemma 9.9 If b, c ∈ P • (A) and b ∩ c = ∅ then there is no filter on A which contains both A b and A c .
These lattice-theoretic issues are actually useful in order to gain a better understanding of the topological implications of the notion of filter, as we will see.  We read "Z W" as "Z and W are eventually disjoint." See [11] for the motivation behind this terminology. In order to clarify this terminology and its meaning, observe that (4.2) defines an injective map Hence F F F (A) contains a copy of P • (A). If A is a nonempty set and if f : A → Y is a function, then we have the following diagram, where A and Y are given in (9.7), f * is given in (2.1), and f in (4.12) The following result says that the map A → A is a natural transformation from the functor

Lemma 9.12
For each nonempty set A and Y and every f : A → Y, the diagram (9.8) is commutative.
Consider also the natural injection (9.9) given by ı A (x) def = {x}. Lemma 4.23 says that the composition of (9.9) with (9.7) yields the injection A → F F F (A) (9.10) which maps x ∈ A to the principal ultrafilter generated by x over A.

Lemma 9.13
The injective map (9.7) is order reversing. That is, if b 1 , b 2 ∈ P • (A) then The fact that if b ⊂ c then A b ⊃ A c follows at once from transitivity of inclusion. Observe that b 1 ∪ b 2 ⊂ c if and only if b 1 ⊂ c and b 2 ⊂ c, hence (9.11) follows at once. Lemma 9.14 If b 1 , b 2 ∈ P • (A) then the following conditions are equivalent: 1 and b 2 overlap (as sets).

If any of these conditions hold, then
Proof It suffices to apply Lemma 4.21 The following result also follows from Lemma 4.21.

Lemma 9.15
If Z 1 , Z 2 ∈ F F F (A) then the following conditions are equivalent: Proof If (1) holds then Z 1 ∨Z 2 is a filter which contains both Z 1 and Z 2 , hence if b 1 ∈ Z 1 and b 2 ∈ Z 2 then both b 1 and b 2 belong to Z 1 ∨ Z 2 and therefore their intersection cannot be empty. If (2) holds, then Z 1 ∪ Z 2 is a filter subbase, and Lemma 4.21 implies that there exists a filter Z which contains Z 1 ∪ Z 2 . Lemma 9.15 says that b and Z intertwine if and only if b and c overlap for each c ∈ Z. This result implies at once the following one.

Corollary 9.16
The filters Z 1 , Z 2 ∈ F F F (A) are eventually disjoint if and only if there exist sets b 1 ∈ Z 1 and b 2 ∈ Z 2 such that b 1 ∩ b 2 = ∅.

Boolean Algebras
In this section, we prove some useful results which highlight the connection between filters (ultrafilters) and the algebraic structure of a Boolean algebra.

Definition 9.17
A Boolean algebra is a ring R with unity where a 2 = a for each a ∈ R. A function f : R 1 → R 2 between Boolean algebras R 1 and R 2 is called a The collection of Boolean algebra homomorphisms from a Boolean algebra R 1 to a Boolean algebra R 2 is denoted by hom BA (R 1 , R 2 ). The collection of nonzero elements of hom BA (R, Z 2 ) [resp. hom Set (R, Z 2 )] is denoted by hom * BA (R, Z 2 ) [resp. hom * Set (R, Z 2 )].
123 Lemma 9.18 Every Boolean algebra R is commutative, and a+a = 0 holds identically for each a ∈ R.

Examples of Boolean Algebras
The simplest example of a Boolean algebra is {0, 1} = Z 2 , endowed with the usual ring operations of Z 2 ≡ Z/2. In order to avoid ambiguities, the sum of a, b ∈ Z 2 is denoted by a + 2 b. A large class of Boolean algebras may be constructed as follows. If S is a set then hom Set (S, Z 2 ) inherits the algebraic structure from Z 2 by the familiar procedure of having the operations performed "pointwise." See [22] for this general technique. Indeed, if f , g ∈ hom Set (S, Z 2 ), we define f + g and f · g as elements of Hence hom Set (S, Z 2 ) inherits from Z 2 a Boolean algebra structure.
Since a natural identification of P(A) with hom Set (A, Z 2 ) is established by the map it follows that P(A) inherits the Boolean algebra structure from hom Set (A, Z 2 ). Observe that, under this identification, the symmetric difference of two elements b 1 , b 2 of P(A) corresponds to the sum 1 b 1 + 2 1 b 2 in hom Set (A, Z 2 ), and the intersection of b 1 and b 2 corresponds to the product 1 b 1 · 1 b 2 . Proof (1) (⇒) If Z σ is a filter then σ (∅) = 0 (since a filter does not contain the empty set). Moreover, if b ∩ c / ∈ Z σ then at least one of the sets b, c does not belong to Z σ ,

The Natural Topology on F F F (A)
We are now ready to apply Theorem 9.7 and define a map which satisfies the compatibility condition described in (r) in Theorem 9.7.
and Proof If Z ∈ F F F (A) and Q ∈ N(Z) then ∃b ∈ Z such that Hence (r ) in Proposition 9.8 is satisfied.
123 Definition 9.24 If A is a nonempty set, then the topology on F F F (A) associated to the map N defined above is called the natural topology on F F F (A). (F F F (A)) is the map defined in (9.12) thenN

Corollary 9.26 A set Q ⊂ F F F (A) is open in the natural topology if and only if
Proof It suffices to apply Lemmas 9.3 and 9.25.
The following examples are meant to illustrate these ideas.
is not an ultrafilter, and this means that

Proposition 9.32 If (A, ) is a topological space and F F F (A) is endowed with the natural topology, then the function N : A → F F F (A) is continuous.
Proof Let x ∈ A and let U ∈ N F F F (A) (N (x)). Then there exists b ∈ N (x) such that Hence we have proved that if z ∈ c then N (z) ∈ U , i.e., the function N is continuous at x. Since x is arbitrary, the proof of continuity of N is complete.

Observe that if (A, ) is Hausdorff then
The following result will be better appreciated by keeping in mind Lemma 3.21.
Therefore W ∈ U . These steps are reversible, hence the other implication follows. Hence (1) is equivalent to (2). Recall from Lemma 3.21 that the meaning of (3) is that if we denote by w the constant sequence w : N → F F F (A) which is identically equal to W then lim w = Z in the topology of F F F (A). Hence for some d ∈ Z 0 and some c ∈ P(A \Ω). We claim that if Z ∈ F d (Ω) then ı (Z) ∈ U , and this will prove continuity at Z 0 , and since Z 0 is arbitrary, it will prove continuity. Let Z ∈ F d (Ω).
where b ∈ P(A). We claim that and W * (∅) = 0. (9.15) Hence Lemma 9.19 implies that We now prove the claim. Observe that W * (∅) = 1 W (F ∅ (A)) = 1 W (∅) = 0, since a filter cannot contain the empty set. In order to prove (9.14), observe that ∈ W, and then both members of (9.14) are equal to 0. A similar result follows if W * (b 2 ) = 0. ( We claim that W * ∈ Boole * (P(A), Z 2 ). Lemma 9.19 then implies that Z W , defined in (9.16), belongs to U U U (A), and (9.17) says that N F F F (A) (Z W ) ⊂ W, and the proof is concluded by Lemma 4.30. In order to prove that (9.18), and, in particular, b 1 ∩b 2 = ∅.
Let us assume that Hence there exists an ultrafilter Y on A such that Since Y is an ultrafilter on A, Lemma 4.26 implies that both sides of (9.19) are equal to 1. Since the case v = (1, 0) is symmetric, the proof is concluded if we show that (9.19) holds if v = (0, 0). In this case, Hence there exists an ultrafilter Y on A such that and since Y is an ultrafilter on A, Lemma 4.26 implies that ∈ W, and both sides of (9.19) are equal to 0. Hence U U U (A) is compact. In order to show that it is Hausdorff, let Z 1 , Z 2 ∈ U U U (A), with Z 1 = Z 2 . Then there exists b 1 ∈ Z 1 \ Z 2 and there exists b 2 ∈ W 2 . We claim that b 2 \ b 1 ∈ W. Indeed, Lemma 4.28 and b 2 ∈ Z 2 imply that either b 1 ∩ b 2 ∈ Z 2 , or b 2 \ b 1 ∈ Z 2 but the first possibility is impossible since it implies that b 1 ∈ Z 2 . Hence

Other Properties of the Natural Topology
We now show that filters have a dual character. On the one hand, a filter W on A may be seen as a "static" object, i.e., as an element of F F F (A), which is endowed, as we have seen, with a natural topology. On the other hand, we may look at W in various other ways which bring to the forelight a certain dynamic character that is encoded in the intrinsic structure of a filter. Recall from Example 3.15 that if W is a filter on a nonempty set A, then (W, ⊃) is a directed set, where ⊃ is reverse inclusion between sets.
The following commutative diagram displays the functions which appear in Proposition 9.37.

F F F (A)
Recall from Definition 6.12 that s W (and hence w W ) may be seen as generalized sequences, since W is directed by reverse set inclusion. Also recall that A has been defined in (9.7), and ı A in (9.9). The maps δ A and w W are defined by composition: Proposition 9.37 If Z, W ∈ F F F (A), then the following conditions are equivalent.
(4) lim W = Z.  (9.21) and this means that ∀b ∈ Z ∃d ∈ W such that ( Thus (1) says that ∀b ∈ Z, ∃d ∈ W such that d ⊂ b, and this condition is equivalent to (5). The diagram on the left side of (9.20) commutes, i.e., δ A = A • ı A , and thus the functoriality properties established in Lemma 4.36 imply that Hence (1) and (2) are equivalent to each other. Observe that (3) amounts to saying that ∀d ∈ Z ∃b ∈ W such that c ∈ W and c ⊂ b implies that A c ∈ F d (A). The condition A c ∈ F d (A) means that d ∈ A c , i.e., c ⊂ d. Hence (3) says that ∀d ∈ Z ∃b ∈ W such that c ∈ W and c ⊂ b implies that c ⊂ d, and this means that ∀d ∈ Z ∃b ∈ W such that b ⊂ d, which is equivalent to (5).
is a function, and y ∈ Y, then the following conditions are equivalent: Proof The result follows at once from Corollary 4.34 and Lemma 9.33.
The meaning of Proposition 9.38 is that the limiting behavior of a function f along a filter Z is completely determined by the behavior of f (Z). Lemma 9.33 enables us to reformulate Theorem 6.2 as follows.
Corollary 9.39 If w ∈ S (Y) is a Y-valued Moore-Smith sequence, is a topology on Y, and y ∈ Y, then the following conditions are equivalent: In all these results the same underlying idea emerges, to wit: It is useful to interpret everything in terms of filters and then exploit the natural topology on F F F (Y). For example, if is a topology on Y, it is customary to say that y is a point of convergence for W ∈ F F F (Y) if lim W = N (y) in the natural topology of Y. We will study other useful applications of this idea in the following sections.
, then the following conditions are equivalent: Proof The result follows at once from Lemma 9.33.

Definition 10.3 If is a topology on Y, the filter
The cluster set of W on Y is the following subset of Y: The following result says that the search for points of convergence of a filter should be restricted to the cluster set of the filter.

Application to Compactness
We are now ready to obtain a more flexible version of Lemma 4.30. (1) K is compact.
(2) For each ultrafilter on A which is localized in K there exists x ∈ K such that lim W = N (x).

(3) For each filter Z on A which is localized in K , K ∩ Cluster[Z, ] = ∅.
Proof It suffices to prove that (2) and ( The following result follows at once from Lemma 10.5 and Proposition 10.6.

Filters on the Real Line
The goal of this section is to develop appropriate machinery for the study of convergence properties of filters on R, since, in view of Corollary 4.34, these filters control the convergence properties of real-valued functions defined on a filtered set.

The Structure of the Cluster Set of Filters on the Real Line (I)
The main application of the notion of cluster set of a filter, introduced in Definition 10.3, is linked to Lemma 10.4 and Theorem 10.9, which imply that, in order to understand whether a given filter on a compact topological space converges, it suffices to control its cluster set. However, real-valued functions or sequences may very well diverge to +∞ or −∞, and indeed, if Z ∈ F F F (R), then the statement that lim Z = N R (+∞) means that N R (+∞) ⊂ Z, but then Cluster[Z, R] = ∅. In particular, in this situation, the set Cluster[Z, R] does not fully reflect the convergence properties of Z ∈ F F F (R). In order to obtain uniform results, which are useful in dealing with pointwise estimates, as we will see, we set as ambient space the extended real line R ≡ [−∞, +∞], a compact space which allows us to apply Theorem 10.9. Accordingly, we enlarge the ambient space which hosts the filters used in the notion of cluster set. In other words, we move from F F F (R) to F F F (R).
If Given Z ∈ F F F (R), in order to be able to apply Corollary 10.7, it is necessary to consider the extension of Z from R to R, described in Sect. 4.6 and denoted by Z .
Indeed, since Z ∈ F F F (R), Theorem 11.1 implies that

Cluster[Z , R] is a nonempty compact interval of [−∞, +∞].
We now go back to the difference between the neighborhood filter of +∞ which appears in (11.1) and the filter N +∞ (R) defined in (1.17). Since the starting datum is a real filter, i.e., an element Z ∈ F F F (R), it would be desirable to express the cluster set of the extension of Z directly in terms of Z. This task is achieved by the following definition, where we introduce the "extended real cluster set." Observe that all filters which appear in (11.3) are filters on R. However, (11.3) has a slightly spurious appearance, since Z is a filter on R, but the resulting cluster set lies inside R. Indeed, the advantage of Definition (11.2) is that the cluster set is expressed directly in terms of the original filter Z ∈ F F F (R) and, moreover, the simpler filter (1.17) (a filter on R) is used instead of (11.2) (a filter on R). This technical convenience has no serious side effects, as shown in the following result. Assume that x ∈ Cluster[Z , R] and x ∈ R. Then Z N R (x) and, for each > 0 and each b ∈ Z , the intersection (x − , x + )∩b is not empty. If b ∈ Z then b ∈ Z (by Lemma 4.38) and it follows that (x − , x + )∩b = ∅. Thus x ∈ clusterset(Z, R). Assume that +∞ ∈ Cluster[Z , R]. Then Z N R (+∞). Let b ∈ Z. Since Z ∈ F F F (R), it follows that b ⊂ R. Moreover, b ∈ Z (by Lemma 4.38). Let a ∈ R. Then b ∩ (a, +∞] = ∅ (since Z N R (+∞)). Since b ⊂ R, it follows that b ∩ (a, +∞) = ∅. Since b ∈ Z and a ∈ R are arbitrary, it follows that Z N R (+∞). Hence +∞ ∈ clusterset(Z, R). The proof that if −∞ ∈ Cluster[Z , R] then −∞ ∈ clusterset(Z, R) follows by symmetry. Hence we have proved that Cluster[Z , R] ⊂ clusterset(Z, R).

Proof
The result follows at once from Lemma 11.3, Theorem 10.9, and Lemma 11.4.

The Filters N R (±∞) from the Viewpoint of the Natural Topology on F F F (R)
The following result looks at these matters from the viewpoint of the natural topology of F F F (R).
for each r ∈ R. Then Proof Let U be a neighborhood of N R (+∞) in F F F (R). We may assume, without loss of generality, than U = F b (R) for b ∈ N R (+∞), and that b = (x, +∞), where x ∈ R. We claim that i.e., w(r) ∈ U . The proof of the second statement is symmetrical. Proof It suffices to apply Lemma 3.23.

The Structure of the Cluster Set of Filters on the Real Line (II)
We now look at the cluster set of Z ∈ F F F (R) from a more concrete viewpoint, which is useful in dealing with pointwise estimates, and introduce the notion of lim sup and lim inf of a filter on R. These notions are related, as we will see, to the familiar notions of lim inf and lim sup of a real-valued sequence or function, but formally different, hence it is convenient to use a different notation. Recall from Sect. 9.2 that if Z, W are filters on A then Z W means that Z ∨ W exists in F F F (A), and Z W means Definition 11.8 If Z ∈ F F F (R) then we define with the understanding that sup ∅ = −∞ and inf ∅ = +∞.

Example 11.9
It is useful to keep in mind the following examples.
(1) Z + = (−∞, 0) if Z is the filter generated by the filter base in Example 4.13.
if Z is the filter generated by the filter base in Example 4.14.
Definition 11. 10 We say that a subset I of R is a left-interval in R if it has one of the following forms: (i) I = (−∞, a) for some a ∈ R; (ii) I = (−∞, a] for some a ∈ R; (iii) I = R; (iv) I = ∅. The notion of right-interval is defined by symmetry.

123
Lemma 11.11 If Z ∈ F F F (R) then Z + is a left-interval in R, and Z − is a right-interval in R.

Proof
The conclusion for Z + follows at once from the fact that if r ∈ R, (r , +∞) Z, and r < r , then (r , +∞) Z. The reasoning for Z − is symmetric. Proof By symmetry, it suffices to prove the first statement in each part. (1) If Z ⊂ W and r ∈ W + then (r , +∞) ∩ b = ∅ for each b ∈ W, hence the same conclusion holds for each b ∈ Z. (2) It suffices to observe that (i) the statement Z + = ∅ means that ∀ Z then, in particular, (−∞, y) ∩ b = ∅, but this is impossible, since c ⊂ [β, +∞). The proof of (2) is similar.
By symmetry, it suffices to prove the first statement, which amounts to show that (i) limsup Z ∈ clusterset(Z, R)and (ii) if limsup Z < x then x / ∈ clusterset(Z, R).
In order to prove (ii), it suffices to examine the following two cases: If (ii.a) holds and limsup Z < x, then there exists > 0 such that (x − , x + )∩Z + = ∅, and Lemma 11.15 implies that x / ∈ clusterset(Z, R). Hence (ii) holds in this case. If (ii.a) holds then Z + = ∅, and Lemma 11.12 implies that Z ⊃ N R (−∞), i.e., lim Z = N R (−∞). Then Theorem 11.5 implies that clusterset(Z, R) = {−∞} hence (ii) holds in this case as well, and the proof is complete.
(11.9) Theorem 11.18 If Z ∈ F F F (R) and y ∈ R then the following conditions are equivalent: Proof The result follows at once from Theorem 11.5 and Lemma 11.16.
We now present a different description of the limsup Z and liminf Z.
Recall that if (A, Z) is a filtered set and w : A → R is a function then w (Z) ∈ F F F (R). The following result will be useful in applications. (A, Z) is a filtered set, w : A → R is a function, and y ∈ [−∞, +∞], then the following conditions are equivalent:

Applications of the Natural Topology to Moore-Smith Sequences of Sets
In this section, we present further results pertaining to Moore-Smith sequences of nonempty subsets of a topological space, which may be obtained as an application of the results on filter presented so far.

Cofinal Subsets in a Directed Set
The notion of cluster point of a Moore-Smith sequence is based on the notion of cofinal subsets of a directed set. Recall that Fin[A] is the collection of all final sets in the directed set A and that it is a filter on A Set , by Lemma 3.25.  Observe that if lim w = y then y ∈ ClusterSet(w, ), hence the limiting values of a Moore-Smith sequence w belong to the cluster set of w. Definition 12.8 If w is a P • (Y)-valued Moore-Smith sequence and is a topology on Y, we say that w converges to y ∈ Y and write lim w = y if for each U ∈ N (y) the inner shadow w • [U ] is final in A, where A is the direction of w. If w ∈ S (Y) then this notion recaptures Definition 3.18.

The Cluster Set of Set-Valued Moore-Smith Sequences
Recall that if w ∈ S (P • (Y)) then T Y [w] ∈ F F F (Y) is the filter of tails of w, introduced in Sect. 6.2. The following result extends Corollary 9.39 to w ∈ S (P • (Y)). Lemma 12.9 If w ∈ S (P • (Y)) and is a topology on Y then the following conditions are equivalent: (1) lim w = y, Proof Let A be the direction of w. If (1) holds then for each U ∈ N (y) there exists j ∈ A such that if k ∈ A and jR A k then k ∈ w • [U ], i.e., w(k) ⊂ U , and this means that Tail j [w] ⊂ U , i.e., U ∈ T Y [w], hence (2) holds. Since all these steps are reversible, the converse implication holds as well. Proof Let y ∈ Y and let A be the direction of w. Observe that the condition that y ∈ ClusterSet(w, ) means that for each U ∈ N (y) and for each j ∈ A there exists k ∈ A such that jRk and w(k) ∩ U = ∅, i.e., such that jRk w(k) ∩ U = ∅, and since jRk w(k) = Tail j [w], this means that for each U ∈ N (y) and for each j ∈ A the intersection between Tail j [w] and U is not empty, and this is equivalent to the condition that y ∈ Cluster[T Y [w], ].

Applications to the Notion of Moore-Smith Subsequence
There is a close analogy with the situation where lim W = Z and the one where a sequence w = {w n } n∈N is a subsequence of a sequence z = {z n } n∈N . The following definition makes this analogy more precise. If t and w are Y-valued sequences and t is a subsequence of w (in the ordinary sense) then t is a Moore-Smith subsequence of w. Lemma 12.12 If w ∈ S (P • (Y)), is a topology on Y, and y ∈ ClusterSet(w, ), then then the set Proof Reflexivity and transitivity are immediate. Assume that ( j, U ) and (k, V ) are elements of A y (w). Since A is directed, there exists l ∈ A with jRl and kRl. Since y ∈ ClusterSet(w, ) and U ∩ V ∈ N (y), there exists g ∈ A with lRg and w(g)∩U ∩V = ∅. Hence (g, U ∩V ) ∈ A y (w), ( j, U )R(g, U ∩V ) and (k, V )R(g, U ∩ V ).
In the following result, we extend to the context of Moore-Smith sequences of sets a familiar fact about sequences of points. If (1) holds, apply Lemma 12.12 and obtain the directed set A y (w) described therein. Now define a P • (Y)-valued Moore-Smith sequence whose direction is A y (w) as follows: We claim that t is a Moore-Smith subsequence of w and that lim t = y.
In order to show that t is a Moore-Smith subsequence of w, i.e., and r ∈ t(g, V ), i.e., r ∈ w(g) ∩ V . Since jRk and kRg we have jRg. Moreover, r ∈ w(g, V ) ⊂ w(g). We have thus proved that Tail  We now show that y = lim t. Let U ∈ N (y). Since y ∈ ClusterSet(w, ), there exists j U ∈ A with w( j) ∩ U = ∅. Now observe that if ( j, V ) ∈ Tail ( j U ,U ) [t] then t( j, V ) = w( j) ∩ V (by definition), w( j) ∩ V = ∅ (since ( j, V ) ∈ A y (w)), and V ⊂ U (since ( j U , U )R( j, V )). Hence t( j, V ) ⊂ U . it suffices to show that ∀R ∈ A * (Q), f ω [R ] > α, (13.4) where R is a measurable representative of R.

Proof of Theorem 1.27
Recall from Lemma 13.1 that, in order to show that G differentiates f , it suffices to prove the two inequalities (13.1) for a.e. x ∈ X. Let us examine the inequality on the right. Our task is then to prove that . Let us assume that α > 0, Q ∈ A * (X), ∀x ∈ Q g(x) > α. (13.6) As we observed in Remark 13.2, the function g is not necessarily measurable. The crucial observation is that g(x) > α, i.e., limsup R G ω f (x) > α, means, according to Definition 11.8, that there exists r ∈ R such that α < r and (r , +∞) ( f ω ) (G(x)). (13.7) Observe that (13.7) means that ∀b ∈ G(x), (r , +∞) ∩ ( f ω ) * (b) = ∅ (13.8) and this means that ∀b ∈ G(x) ∃R ∈ b such that f ω [R] > r . (13.9) It follows that (13.6) implies that ∀x ∈ Q ∀b ∈ G(x) ∃R ∈ b such that f ω [R] > r (13.10) and this means that G is adapted to f on Q above α. Observe that, a fortiori, this means that, for each S ∈ A * (Q), G is adapted to f on S above α. Since G and f are compatible, it follows that the mean-value of f over R lies above α for each R ∈ A * (Q Hence we have shown that (13.2) holds, and Lemma 13.3 then implies (13.5). The other inequality in (13.1) follows along similar lines.

The Maximal Operator Associated to a Family of Filters
Stein's theorem on limits of sequences of operators shows that the role played by the boundedness properties of the maximal operator, associated to the study of problems of almost everywhere convergence, is not coincidental but essential; see [11]. It is natural to wonder whether to a given family G of filters on A(X) based on X, as in (1.28), it is possible to associate a maximal operator which would play a similar role. As we will see presently, since filters on A(X) takes us one level higher in the hierarchy of powersets, as observed at the beginning of Sect. 13, the definition of such a maximal operator also depends on the choice of a generating basis for G(x), for each x ∈ X. for each λ > 0 and each f ∈ L 1 (X), and if there exists a dense subset C ⊂ L 1 (X) such that G differentiates C, then G differentiates L 1 (X).
Proof The proof follows a standard argument, presented for example in [11,Sect. 5.2.5].

Miscellaneous Notes
The notion of filter is due to Cartan [8]. In 1909, Frigyes Riesz understood the role played by the objects that are now called ultrafilters in the study of the notions of continuum and completeness [26, p. 23], foreshadowing the use of ultrafilters in the construction of a compactification of certain topological spaces, implicitly used by Marshall Harvey Stone in 1937 and Henry Wallman in 1937 and 1938, and explicitly adopted by Samuel [30]. These ideas, as well as those of Felix Hausdorff, who formulated the abstract definition of neighborhoods [16, p. 213], were picked up by Root [27,28]. In 1938, Herman Lyle Smith also attained the notion of filter, in order to build a theory that could include cases seemingly not covered by the Moore-Smith convergence. More information can be found in [34].
The existing literature has apparently not yet reached a consensus on how the notion of a Moore-Smith subsequence of a given Moore-Smith sequence should be defined. This is a bit surprising, since the "right" definition is virtually contained in an observation made by H. Cartan in 1937, and later in 1955 in the work by Bartle [2] and more conclusively in 1972 in a work by Aarnes and Andenaes [1]. In Sect. 12.4 we have given the "right" notion of Moore-Smith subsequence of a given Moore-Smith sequence. The reason this is the most appropriate notion is fully articulated in [1].
Funding Open access funding provided by Universitá degli Studi G. D'Annunzio Chieti Pescara within the CRUI-CARE Agreement.

Conflict of Interest
There is no conflict of interest.
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