Nikodým-Type Theorems for Lattice Group-Valued Measures with Respect to Filter Convergence ()
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
, with
and 
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.