Filters and Ultrafilters as Approximate Solutions in the Attainability Problems with Constraints of Asymptotic Character
Alexander Chentsov
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DOI: 10.4236/iim.2010.29062   PDF    HTML     6,398 Downloads   9,268 Views  

Abstract

Abstract problems about attainability in topological spaces are considered. Some nonsequential version of the Warga approximate solutions is investigated: we use filters and ultrafilters of measurable spaces. Attrac- tion sets are constructed. AMS (MOS) subject classification. 46A, 49 K 40.

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A. Chentsov, "Filters and Ultrafilters as Approximate Solutions in the Attainability Problems with Constraints of Asymptotic Character," Intelligent Information Management, Vol. 2 No. 9, 2010, pp. 526-551. doi: 10.4236/iim.2010.29062.

Abstract problems about attainability in topological spaces are considered. Some nonsequential version of the Warga approximate solutions is investigated: we use filters and ultrafilters of measurable spaces. Attraction sets are constructed. AMS (MOS) subject classification. 46A, 49 K 40.

1. Introduction

This investigation is devoted to questions connected with attainability under constraints; these constraints can be perturbed. Under these perturbations, jumps of the attained quality can arise. If perturbation is reduced to a weakening of the initial standard constraints, then we obtain some payoff in a result. Therefore, behavior limiting with respect to the validity of constraints can be very interesting. But, the investigation of possibilities of the abovementioned behavior is difficult. The corresponding “straight” methods are connected with constructions of asymptotic analysis. Very fruitful approach is connected with the extension of the corresponding problem. For example, in theory of control can be used different variants of generalized controls formalizable in the corresponding class of measures very often. In this connection, we note the known investigations of J. Warga (see [1]). We recall the notions of precise, generalized, and approximate controls (see [1]). In connection with this approach, we recall the investigations of R.V. Gamkrelidze [2]. For problems of impulse control, we note the original approach of N.N. Krasovskii (see [3]) connected with the employment of distributions. If is useful to recall some asymptotic constructions in mathematical programming (see [4,5]). We note remarks in [4,5] connected with the possible employment of nonsequential approximate (in the Warga terminology) solutions-nets.

The above-mentioned (and many other) investigations concern extremal problems. But, very important analogs are known for different quality problems. We recall the fundamental theorem about an alternative in differential games established by N.N. Krasovskii and A.I. Subbotin [6]. In the corresponding constructions, elements of extensions are used very active. Moreover, approximate motions were used. The concrete connection of generalized and approximate elements of the corresponding constructions was realized by the rule of the extremal displacement of N.N. Krasovskii.

In general, the problem of the combination of generalized and approximate elements in problems with constraints is very important. Namely, generalized elements (in particular, generalized controls) can be used for the representation of objects arising by the limit passage in the class of approximate elements (approximate solutions). These limit objects can be consider as attraction elements. Very often these elements suppose a sequential realizetion (see [1]). But, in other cases, attraction elements should be defined by more general procedures.

So, we can consider variants of generalized representtation of asymptotic objects. This approach is developed by J. Warga in theory of control.

Similar problems can arise in distinct sections of mathematics. For example, adherent points of the filter base in topological space can be considered as attraction elements. Of course, here nonsequential variants of the limit passage are required very often.

In the following, the attainability problem with constraints of asymptotic character is considered.

Fix two nonempty sets and H, and an operator h from into H. Elements of are considered as solutions (sometimes controls) and elements of H play the role of estimates. We consider h as the aim mapping. If we have the set of admissible (in traditional sense) solutions, then play the role of an attainability domain in the estimate space. But, we can use another constraint: instead of a nonempty family of subsets of E is given. In this case, we can use sequences in with a special property in the capacity of approximate solutions. Namely, we require that the sequence has the following property: for any the inclusion takes place from a certain index (i.e. for where is a fixed index depending on). For such solutions we obtain the sequences in H. If H is equipped with a topology t, then we can consider the limits of such sequences as attraction elements (AE) in (H,t) Of course, our AE are “sequential”: we use the limit passage in the class of sequences. This approach can be very limiting. The last statement is connected both with our family and with topology The corresponding examples are known: see [7,8]. In many cases, the more general variants of the limit passage are required. Of course, we can consider nets in and, as a corollary, the corresponding nets in H. In addition, the basic requirement of admissibility it should be preserved: for any the inclusion is valid starting from a certain index. With the employment of such nets, we can realize new AE; this effect takes place in many examples.

But, the representation of the “totality” of above-mentioned (-admissible) nets as a set is connected with difficulties. Really, any net in the set is defined by a mapping from a nonempty directed set (DS) into the concrete choice of is arbitrary (is a nonempty set). Therefore we have the very large “totality” of nets with the point of view of traditional Zermelo axiomatics. But, this situation can be corrected by the employment of filters of it is possible to introduce the set of all -admissible filters of the set In addition, the -admissibility of a filter is defined by the requirement So, we can consider nonsequential approximate solutions (analogs of sequential approximate solutions of Warga) as filters of with the property Moreover, we can be restricted to the employment of only ultrafilters (maximal filters) with the above-mentioned property. In two last cases, we obtain two variants of the set of admissible nonsequential approximate solutions defined in correspondence with Zermelo axiomatics. In our investigation, such point of view is postulated. And what is more, we give the basic attention to the consideration of ultrafilters. Here, the important property of compactness arises. Namely, the corresponding space of ultrafilters is equipped with a compact topology. This permits to consider ultrafilters as generalized elements (GE) too (we keep in mind the above-mentioned classification of Warga).

The basic difficulty is connected with realizability: the existence of free ultrafilters (for which effects of an extension are realized) is established only with the employment of axiom of choice. Roughly speaking, free ultrafilters are “invisible”. This property is connected with ultrafilters of the family of all subsets of the corresponding “unit”. But, we can to consider ultrafilters of measurable spaces with algebras and semialgebras of sets. We note that some measurable spaces admitting the representation of all such ultrafilters are known (see, for example, [9,§7.6]; in addition, the unessential transformation with the employment of finitely additive (0,1)-measures is used).

2. General Notions and Designations

We use the standard set-theoretical symbolics including quantors and propositional connectives; as usually, replaces the expression “there exists and unique”, is the equality by definition. In the following, for any two objects and, is the unordered pair of x and y (see [10]). Then, is singleton containing an object x. Of course, for any objects x and the object is the ordered pair of objects x and y; here, we follow to [10]. By we denote the empty set. By a family we call a set all elements of which are sets.

By we denote the family of all subsets of a set then, are the family of all nonempty subsets of X. Of course, for any set in the form of and we have the family of all nonempty subfamilies of and respectively.

If is a set, then we denote by the family of all finite sets of then is the family of all finite subsets of

For any sets and we denote by the set of all mappings from into If and are sets, and then

(the image of under the operation f) and is the usual -restriction of . In the following, and is the real line; Of course, we use the natural order of. If, then

Transformations of families. For any nonempty family and a set we suppose that

If X and Y are sets and, then we suppose that

(2.1)

of course, in (2.1) nonempty families are defined.

If is a family, then we suppose that

(we keep in mind that is a nonempty set and, for, is a family) and

(of course, for, is a nonempty family); moreover

So, for any nonempty family, we obtain that

of course,.

Special families. Let I be a set. Then, we suppose that

(2.2)

elements of (2.2) are called -systems with “zero” and “unit”. Moreover,

(2.3)

elements of (2.3) are lattices of subsets of I (with “zero”). Finally,

(2.4)

Of course, in (2.4) lattices of sets with “zero” and “unit” are introduced. We note that

Of course,

(2.5)

is the set of all topologies of I. If then the pair is a topological space (TS);

(2.6)

in (2.6) we have families dual with respect to topologies. It is obvious that

(2.7)

We suppose that is the mapping for which

(2.8)

From (2.5) – (2.8), we obtain the following properties:

(2.9)

We note that in addition,

Of course, in (2.9), we have (in particular) the natural duality used in general topology. Let

(the set of all compact topologies of I) Now, we introduce in consideration algebras of sets. Namely,

(2.10)

In connection with (2.10), we note that

If then is a measurable space with an algebra of sets.

If and then by we denote the set of all mappings

for each of which:

1)

2)

Then

(2.11)

is the set of all semialgebras of subsets of I. Of course,

see (2.10). If we have a semialgebra of subsets of I, then algebra generated by the initial semialgebra is realized very simply: for any

has the properties: 1) 2)

Now, we introduce some notions important for constructions of general topology. Namely, we consider topological bases of two types:

(2.12)

(2.13)

Of course,

In connection with (2.12), we suppose that

,

;

Moreover, the following obvious property is valid:

We note the natural connection of open and closed bases:

(2.14)

Along with (2.14), we note the following important property:

(2.15)

From (2.9) and (2.15), we obtain the obvious statement:

(2.16)

So, closed bases can be used (see (2.16)) for topologies constructing. We note the following obvious property (here we use (2.14) and (2.16)):

(2.17)

Of course, in (2.17), we use the usual duality property connected with (2.14) – (2.16).

Some additions. In the following, we suppose that

(2.18)

if then TS is called -space. We use (2.18) under investigation of properties of topologies on ultrafilter spaces.

Finally, we suppose that. So, we introduce “continuous” lattices.

3. Nets and Filters as Approximate Solutions under Constraints of Asymptotic Character

In this section, we fix a nonempty set considered (in particular) as the space of usual solutions. We consider families as constraints of asymptotic character. Of course, in this case, we use asymptotic version of solutions. The simplest variant is realized by the employment of sequences in: in the set the set of -admissible sequences (see Section 1) is selected. It is logical to generalize this approach: we keep in mind the employment of nets. Later, we introduce some definitions connected with the Moore-Smith convergence. But, before we consider the filter convergence.

We denote by (by the set of all families (families) for which

Then, is the set of all filter bases on. By we denote the set of all filters on:

(3.1)

Using (3.1), we introduce the set of all ultrafilters on:

(3.2)

In connection with (3.1) and (3.2), see in particular [11, ch. I]. In addition,

(3.3)

By (3.3) we define the filter on generated by a base of

If then by (by we denote the set of all filters (ultrafilters ) such that. Then, for any filter , we have and what is more is the intersection of all ultrafilters see [11].

If a family is considered as the constra int of asymptotic character, then ultrafilters are considered as (nonsequential) approximate solutions; of course, filters can be considered in this capacity also. But, ultrafilters have better properties; therefore, now we are restricted to employment of ultrafilters as approximate solutions.

The filter of neighborhoods. If and, then

and of course, in correspondence with (3.3). We were introduce the filter of neighborhoods of in the sense of [11,ch.I]. In the following,

So, we introduce the closure operation in a TS. Moreover, we suppose that

(3.4)

The filter convergence. We follow to [11]. Suppose that

(3.5)

In addition, see (3.1). Therefore, we can use (3.5) in the case of where we note that Then, by (3.5)

(3.6)

Of course, it is possible to use the variant of (3.6) corresponding to the case where

Nets and the Moore-Smith convergence. On the basis of (3.6), we can to introduce the standard MooreSmith convergence of nets. We call a net in the set arbitrary triplet where is a nonempty DS and If is a net in the set then

(3.7)

we obtain the filter of associated with Now, for any topology a net in the set and we suppose that

(3.8)

From (3.6) and (3.7), we obtain that (3.8) is the “usual” Moore-Smith convergence (see [12]). Of course, any sequence generates the net where is the usual order of.

If, then a net in is called -admissible if In this case, can be considered as a constraint of asymptotic character and plays the role of nonsequential (generally speaking) approximate solution.

In conclusion, we note that

(3.9)

In (3.9), trivial ultrafilters are defined.

4. Attraction Sets

In this section, we construct nonsequential (generally speaking) attraction sets (AS) using different variants of the representation of approximate solutions. Since nets are similar to sequences very essential, we begin our consideration with the representation (of AS) using nets.

For brevity, in this section, we fix following two nonempty sets: and In addition, under and

(4.1)

of course, in (4.1), we can use a filter or ultrafilter instead of. In addition, the important property takes place: if and, then

(4.2)

So, by (4.2) image of an ultrafilter base is an ultrafilter base. Of course, the image of an ultrafilter is an ultrafilter base also.

Introduce AS: if and, then by we denote the set of all for each of which there exists a net in the set X such that

(4.3)

we consider as AS. In this definition, we use nets. But, for any filter there exists a net in the set X for which (see [13]).

Proposition 4.1. For any and

(4.4)

Proof. Fix and Suppose that and are the sets on the left and right sides of (4.4) respectively. Let. Then and, for a net in X, the relation (4.3) is valid under Then, by (4.3)

(4.5)

Moreover, by (3.8) and (4.3) So, by (3.6)

(4.6)

Let Then by (3.7) and (4.6), for some the following property is valid:

(4.7)

In addition, and by (3.7) and (4.5)

As a corollary, But, by (4.7) By (3.3) Since the choice of was arbitrary, the inclusion is established. By (3.5)

(4.8)

By (4.5) and (4.8) The inclusion is established.

Let Then, for we have a filter such that

(4.9)

Choose a net in X for which. By (4.1) and, as a corollary, by (3.5) and (4.9)

(4.10)

Then by (3.3) and (4.10) we obtain that. Using (2.1) we have the property: Choose arbitrary then, for some the inclusion is valid. By (3.7) and the choice of for some the following property is realized:

By the choice of we obtain that

Then, So, the important inclusion

is valid. Then (see (3.6)). By (3.8)

(4.11)

Moreover, by the choice of and the inclusion

is valid. From (4.11), we have the inclusion So, and, as a corollary,

Proposition 4.2. For any and  

(4.12)

Proof. We denote respectively by an the sets on the left and right sides of (4.12). Since, we have the obvious inclusion (see Proposition 4.1). Let Then by Proposition 4.1 for some Then and We recall (see Section 3) that. Choose arbitrary Then and Therefore, Moreover, by (2.1) and, as a corollary,

(4.13)

(we recall that by (4.1) and ). By the choice of we have the inclusion

(see (3.5)). Then by (4.13) and, as a corollary (see (3.5)),

Then, The inclusion is established.

Recall that, for any family and We note the following obvious.

Proposition 4.3. For any, the equality is valid.

Proof. Recall that Therefore,; on the other hand, from (3.1), we obtain that Then, for an ultrafilter

and, as a corollary, So, since the choice of was arbitrary, and, as a corollary,

Corollary 4.1. If and, then

The corresponding proof is realized by the immediate combination of Propositions 4.2 and 4.3. We note that, by definitions of Section 2

(4.14)

In connection with (4.14), we note the following general property. Namely,

(4.15)

Then, by (4.14), (4.15), and Corollary 4.1

(4.16)

In connection with (4.16), we note that

(4.17)

Remark 4.1. By analogy with Proposition 4.3 we have that

Really, fix Then Therefore, Let. Then, and But, from (3.1), we have the equality where by the choice of. So, and, as a corollary, The inclusion is established. So,

Returning to (4.17), we note that by Proposition 4.2

(4.18)

Remark 4.2. We have that, for the case it is possible that

Indeed, consider the case is the usual -topology of real line, and

Then, and But, by (4.15)

It is obvious the following.

Proposition 4.4. If and then

Proof. The corresponding proof follows from known statements of general topology (see [11]). But, we consider this proof for a completeness of the account. In our case, we have (4.15). In addition,

(4.19)

is nonempty family of sets closed in the compact topological space (TS) Moreover, (we use known properties of the closure operation and the image operation). Since we obtain that In addition, Therefore, by [9] we have the following property: if and

then As a corollary, is the non empty centered system of closed sets in a compact TS. Then, the intersection of all sets of is not empty. By (4.19)

Using (4.15), we obtain the required statement about the nonemptyness of attraction set.

Corollary 4.2. If and then

Proof. Let Choose arbitrary topology By (4.14) Moreover, Therefore, Then, and by Proposition 4.4

Using Corollary 4.1, we obtain that

In the following, we use the continuity notion. In this connection, suppose that

(4.20)

So, continuous functions are defined. In the following, we use bijections, open and closed mappings, and homeomorphisms. Let

(4.21)

In (4.21), the set of all bijections from onto is defined. If and then

(4.22)

(4.23)

In (4.22) (in (4.23)), we consider open (closed) mappings. In addition,

(4.24)

So, in (4.24), the set of homeomorphisms is defined.

5. Some Properties of Ultrafilters of Measurable Spaces

In this section, we fix a nonempty set. We consider the very general measurable space where is fixed also. According to necessity, we will be supplement the corresponding suppositions with respect to. We suppose that is the set of all families such that

Elements of the set are filters of. In addition,

(5.1)

is the set of all ultrafilters of Recall that (see [16])

(5.2)

In the following, (5.2) plays the very important role.

We introduce the mapping by the following rule:

(5.3)

We note that and by (5.2) In addition, we recall that (see Section 2)

(5.4)

by (5.4) the pair () is a nonempty multiplicative space. We note some simplest general properties. We obtain that

We note that

In addition, for the inclusion takes place. Therefore,

(5.5)

With the employment of (5.5), we obtain that, for any and

(5.6)

Now, we return to the space Suppose that

(5.7)

(the set of filter bases of); and

We note the obvious property: In addition,

Using (5.5) and the obvious inclusion under and we obtain, that We note that, under and the filter

has the following properties

(5.8)

Of course, We can use this property in (5.8): for any and the filter has the properties

(5.9)

In connection with (5.9), we recall the very general property: if and then Using the maximality property, we obtain that

And what is more,

Of course, the above-mentioned properties are valid for

(5.10)

The following reasoning is similar to the construction of [13,§3.6] connected with Wallman extension; in addition, later until the end of this section, we suppose that (5.10) is valid (so, we fix a lattice with “zero” and “unit”).

So, if and then (under condition (5.10))

(5.11)

The property (5.11) is basic. As a corollary,

(5.12)

We note that by (5.11) the following property is valid:

As a corollary, we obtain the property

(5.13)

(so, under (5.10), the statement (5.4) is amplified). In (5.13), we have the lattice of subsets of This important fact used below.

6. Topological Properties, 1

As in the previous section, now we fix a nonempty set and a family We note the following obvious property:

From definitions of the previous section, the following known property follows:

(6.1)

Moreover, we note that

(6.2)

Moreover, we note that. Therefore, by (5.4)

(6.3)

As a corollary, we obtain (see Section 2) that

(6.4)

We recall the very known definition of Hausdorff topology; namely, we introduce the set of such topologies: if is a set, then

For any set M we suppose that

If then TS is called a compactum. Then, the obvious statement follows from the ultrafilter properties (see (5.3), (6.1)):

(6.5)

So, by (6.5) is a Hausdorff TS. Of course, we can use the previous statements of this section in the case of obtaining the Hausdorff topology (6.5). But, in the above-mentioned case, another construction of TS is very interesting. This construction is similar to Wallman extension (see [13,§3.6]). Moreover, in this connection, we note the fundamental investigation [14], where topological representations in the class of ideals are considered. We give the basic attention to the filter consideration in connection with construction of Section 3 concerning with the realization of AS. In this connection, we note that and the sets and are defined. From (3.1) and definitions of Section 5, we have the equality Moreover, from (3.2) and the above-mentioned definitions of Section 5, the equality

(6.6)

follows. By these properties (see (6.6)) the constructions of Section 3 obtain interpretation in terms of filters and ultrafilters of measurable spaces.

Now, we note one simple property; in addition, we use the inclusion chain So, by (3.3)

In particular, we have the following property:

(6.7)

We note one general simple property; namely, in general case of

(6.8)

Remark 6.1. We note that (6.8) is a variant of Proposition 2.4.1 of monograph [16]. Consider the corresponding proof. Fix Then by (6.7)

(6.9)

From (6.9), we obtain (see Section 3) that Let

Then, and In addition (see Section 5), Let Then, and, in particular, By (6.9) and, as a corollary, Then, So, the inclusion is established; we obtain that

(6.10)

From (5.1) and (6.10), we have the equality So,

Since the choice of was arbitrary, the property (6.8) is established.

7. Topological Properties, 2

In this and following sections, we fix a nonempty set and a lattice We consider the question about constructing a compact -space with “unit” This space is similar to Wallman extension for a -space. But, we not use axioms of topology and operate lattice constructions (here, a natural analogy with constructions of [14] takes place). Later we use the following simple statement.

Proposition 7.1.

Proof. We use (5.13). In particular, As a corollary,

(7.1)

Moreover, (see (5.4)). So, is a family with “zero” and “unit”. Moreover, by (5.13)

Therefore, by (2.13) the required statement is realized.

By (2.15) and Proposition 7.1 we have the following construction:

(7.2)

Proposition 7.2. The following compactness property is valid:

(7.3)

Proof. For brevity, we suppose that

(7.4)

and Of course, by (2.9) Moreover, under the family

has the following obvious property

(7.5)

We have the equality So, is the family of all subsets of closed in the TS

(7.6)

Let be arbitrary nonempty centered subfamily of (for any and the intersection of all sets is not empty). If then the family

(7.7)

has the property: Of course,

is centered. Indeed, choose and Let be a procession with the property:

Then, in particular, In addition, by (7.7)

Of course,

Since the intersection of all sets is not empty (we use the centrality of), we choose an ultrafilter

Then, under By axioms of a filter (see Section 5) we obtain that

Since is closed with respect to finite intersections, we obtain that

(7.8)

Moreover, (7.8) is supplemented by the following obvious property; namely,

From (5.7), we obtain that As a corollary,

in addition, by (7.8) Finally, we use (5.2). Let be an ultrafilter for which Then, So,

(7.9)

Let Then, and the equality

(7.10)

is valid (see (7.5)). Choose arbitrary Then, and Using (5.4), we choose for which Then

By (7.7) and, in particular,. By (7.9) and, as a corollary, see (5.3). So, Since the choice of was arbitrary, we obtain that By (7.10) So, we have the property:

Then, the intersection of all sets of is not empty. Since the choice of was arbitrary, it is established that any nonempty centered family of closed (in TS (7.6)) sets has the nonempty intersection. So, TS (7.6) is compact (see [11-13]).

Using Proposition 7.2, by we denote the topology (7.3); so,

(7.11)

We have the nonempty compact TS

(7.12)

Proposition 7.3. If then

The corresponding proof follows from (6.2); of course, we use (5.4) also. From (2.18), (7.11), and Proposition 7.3, we obtain the following property:

(7.13)

So, by (7.13) we obtain that (7.12) is a nonempty compact -space.

In conclusion of the given section, we note several properties. First, we recall that

(7.14)

In addition, from (7.11), the obvious representation follows:

(7.15)

With the employment of (7.15) the following statement is established.

Proposition 7.4. If then the family

is a local base of TS (7.12) at:

The proof is obvious. So, by (3.4) and Proposition 7.4

We note that, from definitions, the following property is valid:

(7.16)

8. The Density Properties

In this section, we continue the investigation of TS (7. 12). Of course, we preserve the suppositions of Section 7 with respect to E and. But, in this section, we postulate that So, in this section

(8.1)

unless otherwise stipulated. So, and. Therefore, with regard (3.9) and (8.1), we obtain that

(8.2)

Of course, for any the inclusion is valid.

Proposition 8. 1.

Proof. Let and We use Propposition 7.4. Namely, we choose a set for which

(8.3)

Since by axioms of a filter (see Section 5), we obtain that in addition, and by (2.4) and (8.1) So, Choose arbitrary point and consider the ultrafilter

(8.4)

see (8.2). In addition, As a corollary, by definitions of Section 5

(8.5)

But, by the choice of e. Therefore, by (8. 5) From (5.3) we have the property As a corollary, by (8.4)

(8.6)

From (8.3) and (8.6), we obtain that By (8.4)

(8.7)

Since the choice of was arbitrary,

Since the choice of was arbitrary, the inclusion

is established. The inverse inclusion is obvious (see (7. 11)).

So, we obtain that trivial ultrafilters (8.2) realize an everywhere dense set in the TS (7.12).

Returning to (7.11), we note one obvious property connected with (7.16). Namely, by (2.14) and Proposition 7.1, in general case of

and, in particular,

then, for

And what is more by (2.17), (7.11), and Proposition 7.1, in general case of

(8.8)

so, by (8.8) is a base of topology (7.11). We recall that by (2.8) and (5.4), for general case of

(8.9)

Connection with Wallman extension. Let Then, and by (2.18) Using (2.7), we obtain that with the employment of the above-mentioned closedness of singletons, by the corresponding definition of Section 2 we obtain that

(8.10)

Until the end of the present section, we suppose that

(8.11)

So, in our case, is the lattice of closed sets in T1- space. Then, (7.12) is the corresponding Wallman compact space (see [13]). On the other hand, by (8.10) and (8.11) we obtain that this variant of corresponds to general statements of our section (for example, see (8.2) and Proposition 8.1). In this connection, we consider the mapping

(8.12)

we denote the mapping (8.12) by. So, and

Consider some simple properties. First, we note that is injective:

(8.13)

Indeed, for and with the property by (3.9) we have that and, as a corollary, so,

Of course, is a bijection from E onto the set

(8.14)

If and then As a corollary, we obtain that

(8.15)

Remark 8.1. Of course, in (8.15), we use the representation (8.12). Fix Let Then and By (5.3) and, as a corollary, So,

(8.16)

If then see (8.12). Therefore, by (5.3) and, as a corollary, So, Therefore (see (8.16)) and coincide.

From (5.4) and (8.15), we obtain that

(8.17)

Proposition 8.2.

Proof. We use the construction dual with respect to (4.20). Let Then, by (8.8) Therefore, for some

As a result, we obtain that

(8.18)

where see (8.17). By (2.6), (2.9), (8.11), and (8.18) we have the property:

and

(8.19)

Since the choice of F was arbitrary, from (8.19) we obtain the required continuity property (see [16, (2.5.2)]).

Corollary 8.1.

Proof. Recall that In addition, by (8.14)

Let and realizes the equality By Proposition 8.2

(8.20)

In addition, (indeed,). Let Then, and But, too. Then, So, Therefore, Since the choice of was arbitrary, the inclusion

is established. So, By (8.20) Since the choice of G was arbitrary, the inclusion

is established.

Recall that (see (4.21)).

Proposition 8.3.

Proof. Let Then and By (8.11) In addition, by (5.3)

(8.21)

Of course, by (5.4) Then

As a corollary, Therefore,

(8.22)

Now, we compare and (8.22). Let Then, for some

(8.23)

Of course, By (3.9) (indeed,). By (8.23) and, as a corollary, see (8.21). We obtain that

(8.24)

Since we have the inclusion Using (8.22) and (8.24), we obtain that The inclusion

(8.25)

is established. Choose arbitrary then, by (8. 22), for some the equality is valid. So,

(8.26)

Moreover, So, By (8.21) Since by (8.26) From (3.9), the property follows. Then, Therefore, as a corollary, The inclusion is established. Using (8.25), we obtain that By (8.22)

Since the choice of G was arbitrary, by Corollary 8.1 and (4.22) we have the inclusion

By (4.24), (8.13), and Proposition 8.3 we obtain that

(8.27)

So, we construct the concrete homeomorphic inclusion of -space in the compact -space (in this connection, we recall that by Proposition 8.1

moreover, see (7.13)). So, we have the “usual” Wallman extension.

9. Ultrafilters of Measurable Space

In this section, we fix a nonempty set and an algebra of subsets of So, in this section, is a measurable space with an algebra of sets: Of course, we can to use constructions of Section 5; indeed, in particular, we have the inclusion see (2.10). As a corollary, by (2.4) So, we use the sets and of Section 5; we use properties of these sets also. We note the known representation (see [15]):

(9.1)

Now, we use (9.1) for investigation of TS (7.12) in the case First, we note the obvious corollary of (9.1):

(9.2)

Remark 9.1. Let is fixed. Choose arbitrary Then, by (7.14) By (9.1) where by axioms of an algebra of sets. So, by (5.3) The inclusion

(9.3)

is established. Let Then, by (5.3) and By axioms of a filter

So, and As a corollary, So, the inclusion

is established. Using (9.3), we obtain the required coincidence and

Returning to (9.2) in general case, we note the following obvious Proposition 9.1.

Proof. Let Using (5.4), we choose such that Then and by (9.2)

(9.4)

From (5.4), we have the obvious inclusion By (9.4)

Therefore, we obtain the following property:

The inclusion is established. Choose arbitrary

(9.5)

Using (2.8), we choose such that Let be the set for which see (5.4). Then, by (9.2)

(9.6)

where Since by (5.4) from (9.6), we obtain that

Since the choice of (9.5) was arbitrary, the inclusion

is established. So, we obtain the required equality.

From (6.4), (8.8), and Proposition 9.1, the simple (but useful) statement follows.

Proposition 9.2.

So, for measurable spaces with algebras of sets, the topological representations of Sections 6 and 7, 8 realize the same topology. By (6.5), (7.13), and Proposition 9.2

(9.7)

So, we obtain a nonempty compactum. Recall that (see (7.11), Proposition 9.2)

(9.8)

is the family of all sets closed in the sense of topology (9.7). We note the following obvious property (see [15, ch.I])

(9.9)

Remark 9.2. We recall (5.4). Let Using (5.4), we choose such that Then, and by (9.2)

(9.10)

By (5.4) and (9.10) So, we establish that

(9.11)

From (2.10), (5.4), and (9.11), the property (9.9) follows.

Proposition 9.3.

Proof. Recall that by statements of Section 2 and (9.8) the inclusion

(9.12)

From (6.4), the inclusion follows too. So, by (9.12)

(9.13)

Let Since is open, then by (6.4) we obtain that, for some family

(9.14)

the following equality is realized:

(9.15)

If then by (9.15) and, as a corollary, where So, by (5.4) we obtain the implication

(9.16)

Let Then, Since is a closed subset of a compactum, we have the compactess property of; then, by (9.14), for some

(9.17)

In particular, We note that is closed with respect to finite unions (indeed, by (9.9) is an algebra of sets). Therefore, by (9.17) in the case So,

(9.18)

Using (9.16) and (9.18), we obtain that in any possible cases. Since the choice of was arbitrary, the inclusion

(9.19)

in established. From (9.13) and (9.18), the required statement follows.

So, is the family of all open-closed sets in the nonempty compactum

(9.20)

In connection with the above-mentioned property of nonempty compactum (9.20), we recall [15, ch. I]. With the employment of (9.1), the following obvious property is established: in our case of measurable space with an algebra of sets

(9.21)

Remark 9.3. For a completeness, we consider the scheme of the proof of (9.21). For this, we note that by (3.9) and the corresponding definition of Section 5

(9.22)

In particular, by (9.22) Fix and suppose that

of course, In addition, Then,

(9.23)

Of course, by (3.9), for, we have the following obvious implications:

Then, by (9.23) Since the choice of was arbitrary, by (9.1) So, (9.21) is established.

Using (9.21), we introduce the mapping

(9.24)

Of course, in (9.24) we have analog of the mapping (8.12). But, in the given case, we realize the immersion of points of the initial set in the ultrafilter space under other conditions. We will use the specific character of measurable space with an algebra of sets. Now, we note the obvious property:

(9.25)

In (9.25), the statement of the premise has the following sense: algebra is distinguishing for points of.

If then by analogy with Section 4 we suppose that

(9.26)

of course, and moreover the following property is valid:

(9.27)

Returning to (9.25), we note that

(9.28)

In (9.28), we can use as constraints of asymptotic character. Of course, (see Section 5). Then, by (3.3)

(9.29)

By analogy with (9.29) we note that and These properties permit realize an asymptotic analogs of solutions of the set (9.28). In this capacity, we can use elements of the sets and where is used as “asymptotic constraints”. Of course, bounds our possibilities: we can use only subfamilies of.

Proposition 9.4.

Proof. Fix Let Then So, and Choose arbitrary Then, by (9.24)

(9.30)

By the choice of we have the inclusion Since we obtain that Then, by (9.30) Since by (5.3)

(9.31)

By (9.30) and (9.31) we obtain the following property

Since the choice of was arbitrary, we have (see (8.3)) the statement

(9.32)

Choose arbitrary Then, for some the inclusion is valid. Therefore, and By (6.4), there exists such that

(9.33)

From (9.32), the property is valid. By (9.33) we obtain that

(indeed,). Since the choice of was arbitrary,

Then, So, the inclusion

is established. The opposite inclusion is obvious.

We note that Proposition 9.4 is similar to Proposition 8.1. But, in the given section, the condition

(9.34)

is supposed not; in Section 8 (in particular, in Proposition 8.1), the condition similar to (9.34) is essential. So, Proposition 9.4 has the independent meaning.

10. Attraction Sets Under the Restriction in the Form of Algebra of Sets

In the following, we fix a nonempty set E, a TS where and a mapping Elements are considered as usual solutions and elements play the role of some estimates. The natural variant of an obtaining of is realized in the form where But, we admit the possibility of the limit realization of This is natural in questions of asymptotic analysis. In the last case, it is natural to use “asymptotic constraints” in the form of a nonempty subfamilies of Then, we obtain constructions of Section 4 under and But, we admit yet one possibility: along with “usual” AS, we use the sets

(10.1)

Of course, we use remarks of the conclusion of the previous section.

Proposition 10.1. If and then

(10.2)

Proof. We use reasoning analogous to the proof of Proposition 4.2. We denote by the set on the right side of (10.2). Since (see Section 9), by (10.1)

(10.3)

Let Then, by (10.1) and, for some

(10.4)

Recall that (see Section 9). Therefore, by (4.1) Then, (10.4) denotes that

(10.5)

(see (3.5)). In addition, by the choice of we have the inclusion see (9.26). By (9.27), for some the inclusion is valid. Then,

As a corollary, by(3.3) and (10.5)

where (see Section 9). Then, by (3.5)

(10.6)

By definition of we obtain that Since the choice of was arbitrary, the inclusion

(10.7)

is established. Using (10.3) and (10.7), we obtain the required equality

(10.8)

From the definition of and (10.8), we obtain (10.2).

Recall that and therefore

By definitions of Section 3, (6.6), and (9.26) we obtain that

(10.9)

From Propositions 4.2 and 10.1, we have (see (10.9)) the property:

So, our new construction is coordinated with AS of Section 4. Moreover, under we can consider AS for

Proposition 10.2. If and then

(10.10)

Proof. We use (6.8). Choose Then, and, for some the convergence

(10.11)

is valid. Then, and see (9.26). By (6.8) for some the equality is valid. Then, As a corollary, Now, we return to (10.11). In addition, Therefore, and by (3.3)

From (3.5) and (10.11), we have the obvious inclusion

(10.12)

In addition, and see (4. 1). Since the inclusion is valid. As a corollary, by (3.3)

Using (10.12), we obtain the basic inclusion

(10.13)

From (3.5) and (10.13), we obtain the following convergence

(10.14)

So, has the property (10.14). Then, by Proposition 4.2

Since the choice of was arbitrary, the required inclusion (10.10) is established.

So, by (10.1) and (10.2) some “partial” AS are defined. Of course, the case for which (10.10) is converted in a equality is very interesting. For investigation of this case, we consider auxiliary constructions. In the following, in this section, we fix So, is a measurable space with an algebra of sets. In this case, we can supplement the property (6.8). Namely,

(10.15)

Remark 10.1. We omit the sufficiently simple proof (10.15). Now, we are restricted to brief remarks. Namely, by ultrafilter we can realize a finitely additive (0,1)-measure on the family supposing that under and under In connection with such possibility, we use [9,(7.6.17)] (moreover, see [9,(7.6.7)]). The natural narrowing v of on our algebra is finitely additive (0,1)-measure on (of course,). Therefore, for some by [9,(7.6.17)] is defined by the rule

(10.16)

On the other hand, the family realizes by the obvious rule:

(10.17)

From (10.16) and (10.17), the required equality follows. Then, by the choice of we have the inclusion

Using (6.8) and (10.15), we obtain that

(10.18)

By (10.18) we establish the natural connection of and Now, we consider some other auxiliary properties.

If and then we have the following equivalence

(10.19)

Of course, we can use instead of the corresponding image of a filter base in Indeed, by (4.1) and (10.19)

(10.20)

Moreover, in connection with (10.20), we note that

(10.21)

Remark 10.2. Consider the proof of (10.21). Fix and Let Then, by (10.20)

Therefore, for any, there exists such that As a corollary,

Then, Since the choice of was arbitrary,

So, Let

(10.22)

Choose arbitrary neighborhood Then, by (10.22) Therefore, for some the inclusion is valid. In addition, and

Then, Therefore, and by (10.20) So,

The proof of (10.21) is completed.

We note that, in (10.21), we can use instead of arbitrary filter of In this connection, we recall that by constructions of Section 5, for any we obtain (in particular) that and

(10.23)

Then, from (10.21) and (10.23), we have the following property:

(10.24)

Of course, (10.24) is the particular case of (10.21); in (10.23), we have the useful addition. We note that

(10.25)

Remark 10.3. Fix and Consider the proof of (10.25). By (3.4) and (3.5) we have the following implication

(10.26)

Let Choose arbitrary Then, by (2.18), for some the inclusion is valid. Since by filter axioms (see 3.1)) So, the inclusion is established. By (3.5) we have the convergence So,

Now, with the employment of (10.26), we obtain (10.25).

We note the following obvious corollary of (10.25) (in this connection, we recall (10.21)):

(10.27)

Remark 10.4. Consider the proof of (10.27). We fix and Since (see (3.4) and definitions of Section 3), by (10.21)

(10.28)

Let the corollary of (10.28) is valid. Fix with the property

(10.29)

Let Then, by (3.4), for some the inclusion is valid, where By (10.29) and From (3.1) and (3.3), the inclusion follows. Since the choice of was arbitrary, the inclusion

is established. By (10.21) So, we obtain that

Using the last implication and (10.28), we obtain the required property (10.27).

Using (10.15), we obtain the obvious corollary of (10.27):

(10.30)

Remark 10.5. Consider the proof of (10.30), fixing and Then, by (10.15) In particular (see Section 9), Now, (10.30) follows from (10.27).

Condition 10.1.

Remark 10.6. It is possible to consider Condition 10.1 as a weakened variant of the measurability of The usual measurability of is not natural since is only algebra of sets.

Until the end of the present section, we suppose that Condition 10.1 is valid.

Proposition 10.3. If Condition 10.1 is fulfilled, then

Proof. Let Condition 10.1 be fulfilled. Fix and Then, and, for some

(10.31)

(see Proposition 4.2). Then, by (10.21) and (10.31) we have the inclusion since by (3.1). As a corollary,

(indeed, for we can choose such that therefore, by (2.1) and by (3.1)). By Condition 10.1 there exists such that In addition,

(10.32)

therefore, where by (10.15) We recall that (see Section 9) and

(10.33)

Of course, by (4.1) In addition, by (10.33)

By (10.27) Recall that Since we obtain that Therefore (see (9.26)),

By Proposition 10.1 Since the choice of was arbitrary, we have the inclusion

(10.34)

Using (10.34) and Proposition 10.2, we obtain the equality

So, we can use (see Proposition 10.1 and Condition 10.1) ultrafilters of the space as nonsequential approximate solutions in the case, when a nonempty subfamily of is used as the constraint of asymptotic character. This property is very useful in the cases of spaces for which the set is realized effectively. In addition, for a semialgebra with the property (see Section 2), we consider the passage

as an unessential transformation (see [9,§7.6] and [16,§ 2.4]; here it is appropriate to use the natural connection of ultrafilters and finitely additive (0,1)-measures). Then, after unessential transformations, the examples of [9,§ 7.6] can be used in our scheme sufficiently constructively.

11. Ultrasolutions

First, we recall some statements of [17]. In addition, we fix a nonempty set E and a TS where We consider the nonempty set Suppose that Then, we suppose that

(11.1)

So, we introduce the limit sets corresponding to ultrafilters of By analogy with Proposition 5.4 of [17] the following statement is established.

Proposition 11.1. If then

Proof. Fix Then and by (4.2)

(11.2)

(recall that Since is a compact TS, there exists such that see [9, ch. I]. Then, by (3.6) or

(11.3)

(see (11.2)). By (3.5) and (11.3) Then, by (11.1) So,

(11.4)

Let Then and By (3.5) and (11.2) In addition, by (11.2) Then, by (3.1)

By (2.1) we obtain that

Since and for we have the property:

So, The inclusion

is established. Using (11.4), we obtain that

We note the following obvious property too: if and then

(11.5)

Remark 11.1. Let the premise of (11.5) be fulfilled. Then, by (3.6)

(11.6)

Then, Indeed, suppose the contrary: Then, by (6.1), for some and the equality is valid. But, by (11.6) and Then, by (3.1) The obtained contradiction means that is impossible. So,

Proposition 11.2. If and then

Proof. The corresponding proof is the obvious combination of (11.1), (11.5), and Proposition 11.1. Indeed, by Proposition 11.1 and Let Then, and

(11.7)

Let Then, and

(11.8)

For, by (11.7) and (11.8)

So, by (3.6) and From (11.5) the equality is valid. Then, The inclusion

(11.9)

is established. But, by the choice of we have the inclusion Using (11.9), we obtain that

The uniqueness of is obvious.

From Proposition 11.2 the natural corollary follows: if then

(11.10)

In the following, we postulate that

(11.11)

Then (see (11.10) and (11.11)), we suppose that

(11.12)

is defined by the following rule: if then has the property:

(11.13)

We note that by (11.13) and Proposition 11.1

(11.14)

So, by (11.12) and (11.14) In this connection, we note the following typical situation: under condition (11.11), and Of course, by (11.11)

Indeed, any closed set in a compact TS is compact too. Recall that

(11.15)

Returning to (11.1) and (11.13) we note that

(11.16)

With the employment of (3.9), we introduce the natural immersion of E in supposing that

(11.17)

In connection with (11.17), we note the following obvious equality:

(11.18)

Remark 11.2. Consider the proof of (11.18). Fix By (3.9) we obtain that

(11.19)

So, by (2.1) and (11.19) we obtain the following property:

(11.20)

In addition, by (3.5), (3.9), and (11.16)

So,. Then, by (11.20) Using the separability of (11.11), we obtain the equality chain

Since the choice of was arbitrary, we obtain that (11.18) is fulfilled.

Proposition 11.3. If then the following equality is valid:

Proof. Let Then and by (11.16)

Then, by (3.5) Therefore, for any there exists such that (see (3.3) and (4.1)).

Let Then, If then, for some the inclusion is valid; moreover, and

(11.21)

where So, Since the choice of was arbitrary, we obtain that

Therefore, Since the choice of was arbitrary too, we have the inclusion Therefore,

Choose arbitrary Then and

(11.22)

Then, for we obtain the property Using (3.3), we have the following statement:

Therefore, (see Section 5), where

(11.23)

(we use (4.2)). In addition, using (6.6), (11.23), and statements of Section 5, we obtain that

Therefore, Since the choice of was arbitrary, we obtain that

So, by (3.5) Then, we have the following properties:

By (11.5) Then Since the choice of was arbitrary, we obtain that

The opposite inclusion was established previously. Therefore, and the intersection of all sets coincide.

From (4.15) and Proposition 11.3 we obtain that

(11.24)

So, ultrafilters of realize very perfect constraints of asymptotic character.

12. Ultrafilters of Measurable Space with Algebra of Sets

In this section, we fix a nonempty set E, TS and Moreover, we fix Finally, we suppose that Condition 10.1 is fullfiled. Then, we have the statement of Proposition 3 and other statements of Section 10. We suppose that (11.11) is valid also. So, we have the mapping (11.12). In addition, we have the natural uniqueness of the filter limit: (11.5) is fulfilled. Now, we supplement (11.5). Namely,

(12.1)

Remark 12.1. For the proof of (12.1), we fix and Let the premise statement of (12.1) is valid: converges to and. Then, for (see (3.3)), the inclusions

are fulfilled. Therefore, by (3.6) the following two properties are valid:

By (11.5) So, (12.1) is established.

We recall (10.23) and (10.24): if then and (see (4.1)). We use (10. 15).

Proposition 12.1.

Proof. Let and Then and by Condition 10.1, for some, the inclusion is fulfilled. In addition, by (11. 16) Since and then

(12.2)

From (12.2) the inclusion follows (namely, for any there exists such that then, by (12.2) and by axioms of a filter). In addition, Then,

and (by the choice of Since by (10.27)

By definition of

Proposition 12.2.

Proof. Fix Using (10.18), we choose such that

(12.3)

Then, by (11.12) and by (12.3) and Proposition 12.1

(12.4)

In addition, and Therefore, by (12.1) and (12.4)

From Proposition 12.2, the obvious corollary follows; namely

Now, we suppose that the mapping

(12.5)

is defined by the following rule: if then

(12.6)

From (10.15) and (12.5), the obvious property follows; namely,

Proposition 12. 3.

Proof. Fix Then, by (11.12) By (10.15) we obtain that In particular, and by (4.1) From Proposition 12.1, we have the following convergence

(12.7)

Using Proposition 12.2, (12.1), (12.6), and (12.7), we obtain that

Proposition 12.4. If and then

Proof. Using (10.18), we choose such that Then, by Proposition 11.3 we have the inclusion

(12.8)

(we use the obvious inclusion realized by the choice of). By Proposition 12.3

From (12.8), the inclusion follows.

We note that (see [11,12,13]) by (11.11) the space is regular: if then

(12.9)

Proposition 12.5. The mapping (12.5) is continuous:

(12.10)

Proof. Fix Then, by (12.5) In addition, by (12.6)

(12.11)

Of course, and (see (4.1)). As a corollary, by (3.3)

(12.12)

From (3.5), (12.11), and (12.12), we obtain the following inclusion:

(12.13)

From (2.1), (3.3), and (12.13), we obtain that

(12.14)

Fix Using (12.9), we choose such that Then, by (3.4), for some the inclusion

(12.15)

is valid. Therefore, Of course, Therefore, by (12.14), for some

(12.16)

In addition, (see (5.4)). By (6.4) In addition, by (5.3) Therefore,

(12.17)

Choose arbitrary ultrafilter Then, and see (5.3). By Proposition 12.4

(12.18)

By the closedness of and (12.16) So, from (12.18), we have the inclusion

Using (12.15), we obtain that Since the choice of was arbitrary, the inclusion

(12.19)

is established. Since the choice of was arbitrary too, from (12.17), we obtain that

So, the mapping is continuous at the point. Since the choice of was arbitrary, the required inclusion (12.10) is established (see [16], (2.5.4)).

In connection with Proposition 12.5, we recall Proposition 9.2 and known statement about the possibility of an extension of continuous functions defined on the initial space; in this connection, see, for example, Theorem 3.6.21 of monograph [13]. For this approach, constructions of Section 8 are essential. Of course, under corresponding conditions, we can use the natural connection with the Wallman extension (see (8.27) and Proposition 9.2).

In this case, Proposition 12.5 can be “replaced” (in some sense) by statements similar to the above-mentioned Theorem 3.6.21 of [13] (of course, this approach requires a correction, since we consider ultrafilters of the measurable space). But, we use the “more straight” way with point of view of asymptotic analysis: we construct the required continuous mapping by the limit passage (see Proposition 12.5). We recall (9.24). Then, by (9.24) and (12.5) the mapping

(12.20)

is defined; moreover,

Proposition 12.6. The equality is valid.

Proof. Fix Then by (11.17) In addition, by (11.18) the obvious equality follows:

(12.21)

Moreover, by (9.24) we obtain that

(12.22)

Then, by Proposition 12.3 and (12.22) we have the equality chain

So,. Since the choice of was arbitrary,

Since (9.7) is valid, from Propositions 12.5 and 12.6, we have the important corollary connected with Proposition 5.2.1 of [9]:

(12.23)

is a compactificator, for which (in the considered case)

(12.24)

in (12.24) we use Proposition 3.1 and Corollary 3.1 of [18]. In addition, Therefore, by (12. 24)

(12.25)

In (12.25), we have the important particular case. We consider this case in the following section.

13. Ultrafilters as Generalized Solutions

We suppose that and satisfy to the conditions of Section 12. We postulate (11.11). Finally, we postulate Condition 10.1. Therefore, we can use constructions of the previous section. In particular, (12.25) is fulfilled (the more general property (12.24) is fulfilled too). In connection with (12.25), the obtaining of more simple representations of AS

(13.1)

is important. For this goal, we use the natural construction of Theorem 8.1 in [17]. Namely, we have the following Proposition 13. 1. If then

Proof. Let Then, by the corresponding definition of Section 4 (see (4.3)) and, for some net in the set E,

(13.2)

Fix Then, by (13.2) Using (3.7), we choose such that

(13.3)

Of course, And what is more, Indeed, let us assume the contrary:

(13.4)

Recall that Therefore, By (9.1) and (13.4) we have the inclusion Then, by (5.4) In particular (see (6.4)),

(13.5)

Moreover, by (5.3) Using (13.5), we obtain that

(13.6)

where From (13.6) and the second statement of (13.2) we have the following property: there exists such that

(13.7)

By axioms of DS there exists for which and By (13.3) Moreover, by (13.7)

By (9.24) Therefore,

From (5.3), the inclusion follows; in particular, By (3.9) So,

We have the obvious contradiction. This contradiction means that (13.4) is impossible. So, Since the choice of was arbitrary, the inclusion is established. Then (see (9.26)), So, we obtain the inclusion

(13.8)

Choose arbitrary Then, by (9.26) and By Proposition 9.4 and [9,(3.3.7)], for some net in the set, the convergence

(13.9)

is fulfilled. Now, we use axiom of choice. Fix Then, by the choice of the inclusion is fulfilled. Of course, by (5.4)

(13.10)

in addition, by (5.3). Since by (6.4) and (13.10) we have the inclusion

(13.11)

From (13.9) and (13.11), we have the property: for some we obtain that.

(13.12)

From (9.24) and (13.12), we have the following property:

(13.13)

By (5.3) and (13.13) we obtain that, for with the property the inclusion is valid and, as a corollary, by (3.9) So, and

Then, by (3.7) Since the choice of was arbitrary, the inclusion

(13.14)

is established. So, by (13.9) and (13.14) we obtain that the net in the set E has the following properties:

By definition of Section 4 (see (4.3)) So, the inclusion

is established. Using (13.8), we have the required equality

From (12.25) and Proposition 13.1, we have the following Theorem 13.1. If then, AS in with constraints of the asymptotic character defined by is realized by the rule

We note that, in Theorem 13.1, the set plays the role of the set of admissible generalized solutions.

14. Some Remarks

In our investigation, one approach to the representation of AS and approximate solutions is considered. This very general approach requires the employment of constructions of nonsequential asymptotic analysis. This is connected both with the necessity of validity of “asymptotic constraints” and with the general type of the convergence in TS. We fix a nonempty set of usual solutions (the solution space), the estimate space, and an operator from the solution space into the estimate space. In the estimate space, a topology is given. Then, under very different constraints, we can realize in this space both usual attainable elements and AE. But, if usual attainable elements are defined comparatively simply (in the logical relation), then AE are constructed very difficult. For last goal, extensions of the initial space are used. In addition, the corresponding spaces of GE are constructed. Ultrafilters of the initial space can be used as GE. But, the realizability problems arise: free ultrafilters are “invisible”. In addition, free ultrafilters realize limit attainable elements which nonrealizable in the usual sense. In this connection, we propose to use ultrafilters of (nonstandard) measurable space; we keep in mind spaces with an algebra of sets. But, it is possible to consider the more general constructions with the employment of ultrafilters. In our investigation, ultrafilters of lattices of sets are used. On this basis, the interesting connection with the Wallman extension in general topology arises.

It is possible that the proposed approach motivated by problems of asymptotic analysis can be useful in other constructions of contemporary mathematics.

[1] J. Warga, “Optimal Control of Differential and Functional Eguations,” Academic Press, New York, 1972.

[2] R. V. Gamkrelidze, “Foundations of Optimal Control Theory,” Izdat. Tbil. Univ., Tbilissi, Russian, 1977.

[3] N. N. Krasovskii, “The Theory of the Control of Motion,” Nauka, Moscow, Russian, 1968.

[4] R. J. Duffin, “Linear Inequalites and Related Systems,” Annals of Mathematics Studies, Vol. 38, 1956, pp. 157- 170.

[5] E. G. Gol‘stein, “Duality Theory in Mathematical Programming and Its Applications,” Nauka, Moscow, Russian, 1971.

[6] N. N. Krasovskii and A. I. Subbotin, “Game-Theoretical Control Problems,” Springer Verlag, Berlin, 1988.

[7] A. G. Chentsov, “Constructing Operations of the Limit Passage with the Emploument of Ultrafilters of Measurable Spaces,” Avtomatika i telemekhanika, Vol. 6, No. 11, 2007, pp. 208-222.

[8] A. G. Chentsov and S. I. Tarasova, “Extensions of Abstract Analog of Unstable Control Problems,” Functional Differential Equations, Vol. 16, No. 2, 2009, pp. 237-261.

[9] A. G. Chentsov, “Asymptotic Attainability,” Kluwer, Dordrecht, 1997.

[10] K. Kuratowski and A. Mostowski, “Set Theory,” NorthHolland, Amsterdam, 1967.

[11] N. Bourbaki, “General Topology,” Nauka, Moskow, Russian, 1968.

[12] J. L. Kelley, “General Topology,” Van Nostrand, Prnceton, NJ, 1957.

[13] R. Engelking, “General Topology,” PWN, Warszawa, 1977.

[14] G. Crätzer, “General Lattice Theory,” Akademie-Verlag, Berlin, 1978.

[15] J. Neveu, “Bases Math Matiques Du Calcul Des Probabilit s,” Masson, Paris, 1964.

[16] A. G. Chentsov and S. I. Morina, “Extensions and Relaxations,” Kluwer Academic Publishers, Dordrecht/ Boston/London, 2002.

[17] A. G. Chentsov, “Some Constructions of Asymptotic Analisis Connected with Stone-Čech Compactification,” Contemporary Mathematics and its Applications, Vol. 26, 2004, pp. 119-150.

[18] A. G. Chentsov, “Extensions of Abstract Problems of Attainability: Nonsequential Version. Proceedings of the Steklov Institute of Mathematics,” Suppl. 2, 2007, pp. S46-S82. Pleiades Publishing, Ltd., 2007.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] J. Warga, “Optimal Control of Differential and Functional Eguations,” Academic Press, New York, 1972.
[2] R. V. Gamkrelidze, “Foundations of Optimal Control Theory,” Izdat. Tbil. Univ., Tbilissi, Russian, 1977.
[3] N. N. Krasovskii, “The Theory of the Control of Motion,” Nauka, Moscow, Russian, 1968.
[4] R. J. Duffin, “Linear Inequalites and Related Systems,” Annals of Mathematics Studies, Vol. 38, 1956, pp. 157- 170.
[5] E. G. Gol‘stein, “Duality Theory in Mathematical Programming and Its Applications,” Nauka, Moscow, Russian, 1971.
[6] N. N. Krasovskii and A. I. Subbotin, “Game-Theoretical Control Problems,” Springer Verlag, Berlin, 1988.
[7] A. G. Chentsov, “Constructing Operations of the Limit Passage with the Emploument of Ultrafilters of Measurable Spaces,” Avtomatika i telemekhanika, Vol. 6, No. 11, 2007, pp. 208-222.
[8] A. G. Chentsov and S. I. Tarasova, “Extensions of Abstract Analog of Unstable Control Problems,” Functional Differential Equations, Vol. 16, No. 2, 2009, pp. 237-261.
[9] A. G. Chentsov, “Asymptotic Attainability,” Kluwer, Dor- drecht, 1997.
[10] K. Kuratowski and A. Mostowski, “Set Theory,” North- Holland, Amsterdam, 1967.
[11] N. Bourbaki, “General Topology,” Nauka, Moskow, Russian, 1968.
[12] J. L. Kelley, “General Topology,” Van Nostrand, Prnceton, NJ, 1957.
[13] R. Engelking, “General Topology,” PWN, Warszawa, 1977.
[14] G. Cr?tzer, “General Lattice Theory,” Akademie-Verlag, Berlin, 1978.
[15] J. Neveu, “Bases Math Matiques Du Calcul Des Probabilit s,” Masson, Paris, 1964.
[16] A. G. Chentsov and S. I. Morina, “Extensions and Rela- xations,” Kluwer Academic Publishers, Dordrecht/ Boston/London, 2002.
[17] A. G. Chentsov, “Some Constructions of Asymptotic An- alisis Connected with Stone-?ech Compactification,” Co- ntemporary Mathematics and its Applications, Vol. 26, 2004, pp. 119-150.
[18] A. G. Chentsov, “Extensions of Abstract Problems of Attainability: Nonsequential Version. Proceedings of the Steklov Institute of Mathematics,” Suppl. 2, 2007, pp. S46-S82. Pleiades Publishing, Ltd., 2007.

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