Nikodým-Type Theorems for Lattice Group-Valued Measures with Respect to Filter Convergence

Abstract

We present some convergence and boundedness theorems with respect to filter convergence for lattice group-valued measures. We give a direct proof, based on the sliding hump argument. Furthermore we pose some open problems.

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Boccuto, A. and Dimitriou, X. (2014) Nikodým-Type Theorems for Lattice Group-Valued Measures with Respect to Filter Convergence. Advances in Pure Mathematics, 4, 213-221. doi: 10.4236/apm.2014.45028.

1. Introduction

In the literature there have been several recent studies about limit theorems with respect to filter/ideal convergence for measures, taking values in abstract spaces, whose a particular case is the statistical convergence, related with asymptotic density of subsets of the set of natural numbers. Though in general it is not possible to give versions of limit theorems completely analogous to the corresponding classical ones in the filter/ideal setting (see also [1] , Example 3.4), there are different kinds of results on these topics, whose an overview can be found in [2] and its bibliography, together with a historical survey on several types of kinds of such theorems and related topics since the beginning of the last century. Some classical like limit theorems for measures and integrals in the context of lattice groups or similar structures can be found, for instance, in [3] -[7] . In particular, in [1] [8] , some versions of basic matrix theorems are given, extending results of [9] [10] , which were proved in the normed space context and with respect to statistical convergence. In [11] -[17] , some Schur, boundedness, decomposition and/or convergence theorems are given in the lattice group setting with respect to a suitable class of filters, extending some results of [18] , while for positive measures it is possible to have some similar results even for a larger class of filters (see also [19] ). Analogous results have been established also in the setting of topological group-valued measures in [20] [21] . In [22] -[24] , some limit theorems are proved by means of the tool of uniform filter/ideal exhaustiveness. Moreover, in [25] , some results about equivalence between BrooksJewett, Vitali-Hahn-Saks, Nikodým convergence and Dieudonné-type theorems are presented, using the Stone Isomorphism technique and extending results of [26] , given in the classical case for topological group-valued measures. In this paper we use sliding hump-type techniques, similar to those used in the topological group-setting first by D. Candeloro and G. Letta in 1985 in [27] -[30] for proving limit and boundedness theorems for families of finitely additive group-valued measures defined on suitable Boolean algebras, and successively in [4] [5] [16] [17] [20] [21] in the setting of lattice groups and filter convergence. We prove some new further versions of Nikodým convergence, boundedness and Brooks-Jewett-type theorems for lattice group-valued measures, defined on a σ-algebra of an abstract nonempty set. The results and the proofs are direct and, differently than in [14] [16] , without using Schur-type theorems proved for measures defined on. Finally, we pose some open problems.

2. Preliminaries

A lattice group (shortly, -group) is said to be Dedekind complete iff every nonempty subset of, bounded from above, admits supremum in. A Dedekind complete lattice group is said to be super Dedekind complete iff for every nonempty set, bounded from above, there exists a countable subset, such that.

Let be a Dedekind complete -group. A sequence of positive elements of is called an -sequence iff it is decreasing and. A sequence in is said to be order convergent (or -convergent) to iff there exists an -sequence in such that for every there is a positive integer with for all, and in this case we will write. A bounded double sequence in is called a -sequence or a regulator iff for each the sequence is an -sequence. A sequence in is said to be -convergent to (and we write) iff there exists a -sequence in, such that to every

there is with for every. An -group is weakly -distributive iff for every -sequence.

Observe that, if is the -algebra of all Borel subsets of and is the Lebesgue measure, then the space of all -measurable real-valued functions on (with identification up to -null sets) is super Dedekind complete and weakly -distributive (see also [5] , [6] , Example 2.17).

We now recall the following theorem, which links - and -sequences in lattice groups.

Theorem 2.1 ([25] , Theorem 2.3, see also [31] , Theorems 3.1 and 3.4) Given any Dedekind complete -group and any -sequence in, the double sequence defined by, , is a regulator, such that for every, if, then Conversely, if is super Dedekind complete and weakly -distributive, then for every -sequence in there are an -sequence in and a sequence in, with for each k.

Let be any nonempty set and be a -algebra. A bounded finitely additive measure

is said to be -additive (resp. -bounded) on iff (resp.) for each disjoint sequence in, where,

, is the semivariation of on. Moreover, if is a compact Hausdorff topological space and is its Borel -algebra, we say that a positive finitely additive measure is regular iff for every there exists a -sequence, depending on, such that for all there are a compact set and an open set with and. We recall the following.

Theorem 2.2 ([32] , Theorem 2.2) Assume that is a Dedekind complete and weakly -distributive Riesz space, and let be a regular measure defined on the Borel -algebra of a compact Hausdorff space. Then is -additive.

The following result (Fremlin lemma, see [33] , Lemma 1C) allows us to replace a countable family or a “series” of -sequences with a single regulator.

Lemma 2.3 Let be any Dedekind complete -group and, , be a sequence of regulators in. Then for every, there exists a -sequence in with

for every and.

We now give some basic properties of filters, which will be useful in the sequel. Let be a countable set and be a filter of. A subset of is -stationary iff it has nonempty intersection with every element of. We denote by the family of all -stationary subsets of.

A filter of is said to be diagonal iff for every sequence in and for each there exists a set, such that the set is finite for all (see also [15] [16] [18] ). Given an infinite set, a blocking of is a countable partition of into nonempty finite subsets. A filter of is said to be block-respecting iff for every and for each blocking of there is a set, with for all, where denotes the number of elements of the set into brackets. A particular class of filters, which are block-respecting and diagonal at the same time, is that of the category respecting filters. A filter of is said to be category respecting iff for every compact metric space and for every family of closed subsets of, if whenever in and, then there is a set such that the interior of is nonempty (see also [18] , Theorem 4.3).

Let be a disjoint partition of into infinite subsets. For each sequence of finite subsets and every, set The filter generated by the sets of type is a non-diagonal and block-respecting filter. Furthermore, note that the filter of all subsets of having asymptotic density one is a diagonal and not block-respecting filter (see also [18] ).

If, then the trace of on is the family. It is not difficult to see that is a filter of (see also [21] ).

Remark 2.4 Observe that, if is a block-respecting filter of, then is a block-respecting filter of for every (see also [20] , Proposition 2.1, [21] , Proposition 2.3).

We now recall some main properties of filter convergence in the lattice group setting (see also [1] [16] ).

Let be a filter of. A sequence in -converges to iff there is a -sequence with the property that for each.

Let be any arbitrary nonempty set. A family is said to be -convergent to a family with respect to iff there is a regulator such that for each and we get

Given, set. For, , put, (times). Let, , be such that for every. A set is said to be --bounded by, iff, and -eventually bounded by iff it is --bounded by (see also [1] [15] [16] [34] ).

3. The Main Results

We begin with recalling the following Lemma 3.1 ([16] , Lemma 2.3) Let be a diagonal filter of, be a sequence in with with respect to an -sequence. Then for every there exists such that and with respect to the same -sequence.

We now prove the main result, by means of sliding hump-type techniques.

Theorem 3.2 Let be a Dedekind complete -group, be a block-respecting filter of, , , be a sequence of equibounded -additive measures, be a disjoint sequence in, with

(i) for any, and

(ii) with respect to. Then3.2.1) for every strictly increasing sequence in we get

(1)

3.2.2) if is also diagonal and is super Dedekind complete and weakly -distributive, then the only condition (ii) is sufficient to get (1).

Proof: For each, set. Let: such an element does exist in, thanks to equiboundedness of the’s. For each let be a -sequence related with -additivity of and the sequence. For every and there is, with

(2)

By the Fremlin Lemma 2.3 there is a -sequence with

(3)

for each and. From (2) and (3) it follows that for every and there exists

with

Let be a regulator, satisfying the condition of -convergence as in (ii).

Since for every, then for each there exists a regulator such that for every there is with for all. Since the’s are equibounded, arguing analogously as above, by the Fremlin Lemma 2.3 we find a regulator such that for each and there exists with.

Again by Lemma 2.3, there are two -sequences, , with

(4)

(5)

for every and. For every, set

(6)

We prove that the -sequence satisfies the condition of -convergence in 3.21). Otherwise there is with the property that

We get that: otherwise, there would be with

, that is and hence, a contradiction.

Let. By -additivity of, there is a cofinite subset, with

, and

where. By (i) there is an integer with whenever and

. By -additivity of, , there is a cofinite subset, withand for every, where. Proceeding analogously as abovewe find an integer with whenever and.

By induction, it is possible to find: a strictly decreasing sequence of cofinite subsets of, a strictly decreasing sequence in and two strictly increasing sequences, in such that, for every,

(7)

(8)

Since is block-respecting, there is, , with for every. As, then either or. Without loss of generality, let (see also [15] [16] [18] ). Put. We have:

(9)

Since and

(10)

from (7) and (10) we get

(11)

Moreover, since for every, from (8) we obtain

(12)

If, then from (9), (11) and (12) we have

But we know that and so we get a contradiction.

Thus for all, and hence

Since, by (ii), , we obtain that, which is absurd. This proves 3.2.1).

3.2.2) Put, , and let be a -sequence, satisfying -convergence in condition (ii). Of course, for every -stationary set, the regulator satisfies (ii) also with respect to -convergence. Since is super Dedekind complete and weakly -distributive, by Theorem 2.1 there is an -sequence, satisfying condition (ii), when -convergence is replaced with -or -convergence. For every there is, , with, , with respect to (see also Lemma 3.1). From this and Theorem 2.1 it follows that there is a regulator, such that for every there exists, , with

(13)

with respect to. Let now, , be regulators associated to -additivity of the’s, be as in the proof of 3.2.1), be as in (3) and, , be as in (4), (5), (6) respectively. We prove that the regulator satisfies 3.2.2). Otherwise, by proceeding analogously as in the proof of 3.2.1), we find and with for each. In correspondence with, there is, , satisfying (13). Note that the sequence, , does not converge to 0 (see also [18] ). Since and is block-respecting, then, by Remark 2.4, is block-respecting too. By 3.2.1) used with, , and, it follows that, getting a contradiction. This proves 3.2.2). A result analogous to Theorem 3.2 holds in the setting of finitely additive measures.

Theorem 3.3 Let be a Dedekind complete -group, be as in Theorem 3.2, be a blockrespecting filter of, , , be an equibounded sequence of finitely additive measures, and assume that

(i) for any;

(ii) with respect to.

Then for every strictly increasing sequence in we get

(14)

If is also diagonal and is super Dedekind complete and weakly -distributive, then the only condition (ii) is enough to get (14).

Indeed, it will be enough to apply Theorem 3.2 to the measures, defined by

Analogously as Theorem 3.2 it is possible to prove a Nikodým boundedness-type theorem in the context of -groups and filter convergence, extending [34] , Theorem 4.6 (see also [15] Lemma 3.4).

Theorem 3.4 Let be any Dedekind complete -group, , , , be a block-respecting filter of, , , be a sequence of finitely additive measures, and assume that

3.4.1) for every disjoint sequence in and there is a cofinite set with for each.

Let be a disjoint sequence in and be an increasing sequence of positive elements of. For each, set and. Moreover suppose that:

(i) the set is -eventually bounded by for each;

(ii) the set is --bounded by for each.

Then we get:

3.4.2) for every strictly increasing sequence in, the set is -- bounded by;

3.4.3) if is also diagonal, then the only condition (ii) is enough in order that is --bounded by.

Proof: For every, let. If the thesis of the theorem is not true, then . Set. By 3.4.1) there is a cofinite set, with and for each. By (i) there is with for each and. By induction, there are a strictly decreasing sequence of subsets of and two strictly increasing sequences, of positive integers such that, for each

,; for every and;

for any and.

As is block-respecting, proceeding analogously as in the proof of Theorem 3.2, we find a set , , with for every. For any we have:

(15)

, , , and

(16)

Put. If, then from (15) and (16) we get

and. This contradicts. Thus for every, and hence.

From this, arguing as in 3.2.1), we obtain a contradiction, and this proves 3.4.2). From 3.4.2), proceeding analogously as in the proof of Theorem 3.2, we get 3.4.3). Open problems: (a) Find some versions of limit theorems with respect to some other classes of filters and/or algebras satisfying suitable properties.

(b) Find some convergence and boundedness theorems with respect to other kinds of -boundedness, - additivity or boundedness.

3. Conclusions

The Schur, Nikodým convergence and boundedness, Vitali-Hahn-Saks and Dieudonné limits have been widely investigated in the literature since the beginning of the last century, and there are several extensions of them along different directions, concerning for example the set of definition of the involved set functions, their properties and the structure of their range. The novelties in our context are both the structure of the space in which the considered measures ta ke values and the types of convergence: indeed we deal with filter/ideal convergence in lattice groups introduced in [1] . Moreover we use a new technique in the filter setting, inspired by those in [27] -[30] and [18] , and similar to that in [20] , which allows us to prove some Nikodým-type convergence and boundedness theorems with respect to filter convergence for measures defined on a -algebra of parts of an abstract nonempty set with a direct approach, without using earlier Schur-type results for measures defined on. In this context the main properties of diagonal and block-respecting filters are used, which allow us to apply the sliding hump argument to filter convergence. So it is possible to pose the question of finding some other classes of filters for which similar theorems hold or for which they are not valid, for measures, or even not necessarily finitely additive set functions, taking values in different types of abstract structures, like topological or lattice groups or metric semigroups, and defined in algebras, which satisfy similar properties and are not necessarily -algebras. Note that, in general, filter convergence is not inherited by subsequences. In this context, another problem that one can pose is to find similar results on convergence and boundedness theorems for non-additive set functions, or results like some kinds of uniform -boundedness of -additivity when the limits of the involved sequences are intended in the filter sense, and/or with respect some other kinds of convergence in the lattice group-context, like order convergence (see also [2] ).

Acknowledgements

Our thanks to the anonymous referee for his/her remarks which improved the exposition of the paper.

2010 A. M. S. Classifications

Primary: 26E50, 28A12, 28B10, 40A35, 54A20. Secondary: 06E15, 46G12, 54H11, 54H12.

NOTES

*Corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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