Ideal Limit Theorems and Their Equivalence in $(\ell)$-Group Setting

We prove some equivalence results between limit theorems for sequences of $(\ell)$-group-valued measures, with respect to order ideal convergence. A fundamental role is played by the tool of uniform ideal exhaustiveness of a measure sequence already introduced for the real case or more generally for the Banach space case in our recent papers, to get some results on uniform strong boundedness and uniform countable additivity. We consider both the case in which strong boundedness, countable additivity and the related concepts are formulated with respect to a common order sequence and the context in which these notions are given in a classical like setting, that is not necessarily with respect to a same $(O)$-sequence. We show that, in general, uniform ideal exhaustiveness cannot be omitted. Finally we pose some open problems.

In this paper we continue the investigation initiated in Boccuto, Das, Dimitriou, and Papanastassiou (2012); Boccuto andDimitriou (2011a, 2011b), in the context of ( )-group-valued measures and in connection with uniform ideal exhaustiveness.We consider both cases when countable additivity and strong boundedness are intended relatively to a same order sequence, and when these notions are formulated like in the classical approach, that is not necessarily with respect to a common (O)-sequence.We give some equivalence results between limit theorems in the setting of order ideal convergence.Here, absolute continuity is intended with respect to a general Fréchet-Nikodým topology.Similar equivalence results are given in Drewnowski (1972a) in topological groups.
In particular, when it is proved that the Nikodým convergence theorem implies the Brooks-Jewett theorem, countably additive restrictions of finitely additive (s)-bounded topological group-valued measures, defined on suitable σ-algebras, are considered (see also Boccuto, Dimitriou, & Papanastassiou, 2010a, 2011a for a lattice group version).However in our setting, in order to relate finitely and countably additive measures, it is not advisable to use an approach of this kind.Indeed, in topological groups, the involved convergences fulfil some suitable properties, which are not always satisfied by order convergence in ( )-groups, because in general it does not have a topological nature.So, to prove our results, we use the Stone Isomorphism technique (see also Sikorski, 1964), by means of which it is possible to construct a σ-additive extension of a finitely additive (s)-bounded measure, and to study the properties of the starting measures in relation with the corresponding ones of the considered extensions.For lattice group-valued measures, the Stone extension is examined in Boccuto (1995) when σ-additivity and (s)-boundedness are intended in a classical like sense, that is not necessarily with respect to a same order sequence, and in Boccuto andCandeloro (2002, 2004) when these notions are formulated with respect to a common (O)-sequence or regulator.Note that, in topological groups, it is possible to use not only the Drewnowski Lemma (see Drewnowski, 1972a), but also the Stone Isomorphism Technique, to construct σ-additive measures by starting from finitely additive (s)-bounded measures (see also Candeloro, 1985c;Sion, 1969Sion, , 1973)).Moreover, to prove that the Brooks-Jewett theorem implies the Nikodým theorem, when we link uniform (s)-boundedness and σ-additivity, when these concepts are intended not necessarily with respect to a same order sequence, in general for technical reasons it is not advisable to consider a direct approach, and we use the Maeda-Ogasawara-Vulikh representation theorem for Dedekind complete ( )-groups, studying the related properties of the corresponding real-valued measures.When we deal with a common (O)-sequence, it is possible to give direct proofs, and it is not always advisable to use the tool of the Maeda-Ogasawara-Vulikh representation theorem, because it yields informations in general only about convergence of suitable ( )-group-valued sequences by means of convergence of suitable real-valued sequences, and not necessarily about whether they can be obtained with respect to a single (O)-sequence in the ( )-group involved.
The paper is structured as follows.In Section 2 we present some basic notions and results on ( )-groups, order ideal convergence, submeasures, Fréchet-Nikodým topologies and ( )-group-valued measures.In Section 3, using the Stone extension of measures in connection with uniform ideal exhaustiveness, we present our main results about ideal limit theorems and their equivalence, and we show that, in general, the condition of uniform ideal exhaustiveness cannot be dropped in our context, and it is impossible to obtain versions of limit theorems analogous to the classical ones, when pointwise convergence of the measures involved is replaced by ideal pointwise convergence.Finally we pose some open problems.
Our thanks to the referee for his/her valuable comments, remarks and suggestions to improve some parts of the paper.
An ( )-group is said to be Dedekind complete iff every nonempty subset A ⊂ R, bounded from above, has lattice supremum in R, denoted by ∨A.
A Dedekind complete ( )-group is super Dedekind complete iff every nonempty set A ⊂ R, bounded from above, has a countable subset A ⊂ R, with ∨A = ∨A.
A sequence (σ p ) p in R is an (O)-sequence iff it is decreasing and ∧ p σ p = 0, where the symbol ∧ denotes the lattice infimum.
A bounded double sequence (a t,l ) t,l in R is called a (D)-sequence or regulator iff (a t,l ) l is an (O)-sequence for every t ∈ N.
for every (D)-sequence (a t,l ) t,l .
Note that weak σ-distributivity is a necessary and sufficient condition in order that, for any abstract nonempty set G and any algebra A of subsets of G, every σ-additive R-valued measure defined on A admits a σ-additive extension, defined on the σ-algebra Σ(A) generated by A (see Wright, 1971).
A sequence (x n ) n in R is said to be order-convergent (or (O)-convergent) to x ∈ R iff there is an (O)-sequence (σ p ) p such that for each p ∈ N there is n p ∈ N with |x n − x| ≤ σ p for every n ≥ n p .In this case we write (O) lim n x n = x (with respect to (σ p ) p ).
Theorem 2.1 Given a Dedekind complete ( )-group R, there exists a compact Hausdorff extremely disconnected topological space Ω, such that R can be lattice isomorphically embedded as a subgroup of λ∈Λ is any family such that a λ ∈ R for all λ, and a = ∨ λ a λ ∈ R (where the supremum is taken with respect to R), then a = ∨ λ a λ with respect to C ∞ (Ω), and the set {ω ∈ Ω: In this paper we deal with the order convergence for sequences in the ( )-group setting.Another kind of convergence is widely studied in this context, the (D)-convergence (see also Boccuto, 2003;Boccuto, Riečan, & Vrábelová, 2009;Riečan & Neubrunn, 1997).Note that, in any Dedekind complete ( )-group R, every (O)convergent sequence (D)-converges to the same limit, while the converse is true if and only if R is weakly σdistributive.
For technical reasons, there are some situations in which (O)-convergence is easier to handle than (D)-convergence, and other contexts in which it is preferable to consider (D)-convergence.In particular, we will often use the tool of replacing a series of (D)-sequences with a single regulator (Fremlin Lemma), and in this setting it is advisable to deal with regulators.
Lemma 2.2 (Fremlin Lemma, see also Fremlin, 1975, Lemma 1C;Riečan & Neubrunn, 1997, Theorem 3.2.3)Let R be any Dedekind complete ( )-group and (a (n)  t,l ) t,l , n ∈ N, be a sequence of regulators in R. Then for every u ∈ R, a t,ϕ(t) for every q ∈ N and ϕ ∈ N N .
The following result links order and (D)-sequences and will be useful to study some properties of lattice groupvalued measures.
Theorem 2.3 (see also Boccuto, 2003, Theorems 3.1 and 3.4) Given any Dedekind complete ( )-group R and any (O)-sequence (σ l ) l in R, the double sequence defined by a t,l := σ l , t, l ∈ N, is a regulator, with the property that for every ϕ ∈ N N , if l = ϕ(1), then Conversely, if R is super Dedekind complete and weakly σ-distributive, then for every (D)-sequence (a t,l ) t,l in R there is an (O)-sequence (β p ) p such that for each p ∈ N there exists

Ideal Convergence
We now recall the main properties of ideal convergence in the ( )-group setting.
A class of sets Given an ideal I of N, we call dual filter of I the family of sets An admissible ideal I of N is called a P-ideal iff for any sequence (A j ) j in I there are a sequence (B j ) j in P(N), such that the symmetric difference A j ΔB j is finite for all j ∈ N and An admissible ideal I of N is said to be maximal iff, for every subset A ⊂ N, we get that either Some examples of P-ideals of N are the ideal I fin of all finite subsets of N and the ideal I δ of all subsets of N having null asymptotic density (see also Kostyrko, Šalát, & Wilczyński, 2000/2001;Farah, 2000).Observe that I δ is not maximal.Indeed, if E is the set of all even integers, then we get E I δ and N \ E I δ .However it is known that, if we assume the continuum hypothesis, then there are several maximal P-ideals of N (see also Henriksen, 1959, (1,7)).
Some other examples of P-ideals are the Erdős-Ulam ideals associated with a function f : N → R + , consisting on all subsets A ⊂ N for which Δ k be a partition of N into infinite sets, and The ideal I 0 is not a P-ideal (see also Kostyrko, Šalát, & Wilczyński, 2000/2001, Example 3.1 (g)).
For further properties of ideals, see also Farah (2000) and the bibliography therein.
Let I be an admissible ideal of N. A sequence (x n ) n in R (OI)-converges to x ∈ R iff there exists an (O)-sequence (σ p ) p in R with {n ∈ N: |x n − x| σ p } ∈ I for all p ∈ N, and in this case we write (OI) lim The following result relates (OI)-convergence with the classical (O)-convergence (see also Boccuto, Dimitriou, & Papanastassiou, 2012a, Proposition 2.11).
Proposition 2.4 Let R be any Dedekind complete ( )-group and I be any admissible ideal of N. If there is B ∈ I with (O) lim n∈N\B x n = x with respect to an (O)-sequence (σ p ) p in R, then (OI) lim n x n = x with respect to (σ p ) p .

Proof. By hypothesis there is
with respect to a suitable (O)-sequence (σ p ) p in R.
Choose arbitrarily p ∈ N. By (1) there exists The converse of Proposition is in general not true, and holds if and only if I is a P-ideal (see also Boccuto & Dimitriou, 2011b;Boccuto, Dimitriou, Papanastassiou, & Wilczyński, 2013).
The following property of P-ideals will be useful in the sequel.
Proposition 2.5 (Boccuto & Dimitriou, 2011b, Proposition 3.2) Let (x n, j ) n, j be a double sequence in R, I be a P-ideal of N, and suppose that (OI) lim n x n, j = x j for every j ∈ N with respect to a common (O)-sequence (σ p ) p .
Then there is a set B 0 ∈ F with (O) lim n∈B 0 x n, j = x j for all j ∈ N, with respect to the same (O)-sequence (σ p ) p .

Set Functions and FN-Topologies
We now recall some notions and properties of submeasures, ( )-group-valued measures and Fréchet-Nikodým topologies.From now on, let Σ be a σ-algebra of subsets of an abstract infinite set G.
A topology τ on Σ is called a Fréchet-Nikodým topology iff the functions (A, B) → AΔB and (A, B) → A ∩ B from Σ × Σ (endowed with the product topology) to Σ are continuous, and for each τ-neighborhood V of ∅ in Σ there is a τ-neighborhood U of ∅ in Σ with the property that, if E ∈ Σ is contained in some suitable element of U, then E ∈ V (see also Drewnowski, 1972b, §1).
Observe that a topology τ on Σ is a Fréchet-Nikodým topology if and only if there exists a family of submeasures Z := {η i : i ∈ Λ}, with the property that a base of τ-neighborhoods of ∅ in Σ is given by (see also Boccuto & Dimitriou, 2011b;Drewnowski, 1972aDrewnowski, , 1972b;;Weber, 2002).
We recall the basic properties of measures with values in a Dedekind complete ( )-group R.
Given a finitely additive measure m: Σ → R, we denote by positive part, negative part and semivariation of m the quantities m respectively, where A ∈ Σ.
We say that the finitely additive measures A finitely additive measure m:

The finitely additive measures m
A finitely additive measure m: Σ → R is said to be σ-additive on Σ iff for every disjoint sequence Let τ be a Fréchet-Nikodým topology on Σ.A finitely additive measure m: Σ → R is said to be τ-continuous on Σ, iff for each decreasing sequence The finitely additive measures Let G, H ⊂ Σ denote two lattices, satisfying the following property: (R0) the complement with respect to G of every element of H belongs to G and G is closed under countable unions.
A finitely additive measure m: The finitely additive measures m n : Σ → R, n ∈ N, are uniformly regular on Σ iff for every E ∈ Σ and r ∈ N there exist Analogously as above it is possible to formulate the notions of global and global uniform (s)-boundedness, σadditivity, τ-continuity and regularity, by requiring that the involved (O)-limits exist with respect to a common (O)-sequence.Note that, in general, these concepts are not identical.Indeed there exist bounded finitely additive lattice group-valued measures, which are not globally (s)-bounded (see Boccuto & Candeloro, 2002, Example 2.17), while every bounded finitely additive ( )-group-valued measure is (s)-bounded too (see Boccuto, Dimitriou, & Papanastassiou, 2010a, Theorem 3.1).
The following technical lemma will be useful in the sequel to link global uniform (s)-boundedness of ( )-groupvalued measures with the other related global properties.A similar result (see Boccuto & Dimitriou, 2011b, Lemma 3.3) holds also for (s)-bounded measures, not necessarily with respect to a same (O)-sequence, but for technical reasons in this case we use the Maeda-Ogasawara-Vulikh theorem and assume uniform (s)-boundedness of the measures m n (•)(ω), n ∈ N, for ω belonging to a complement of a meager subset of Ω, where Ω is as in Theorem 2.1.
Lemma 2.6 Let G ⊂ Σ be a lattice, closed under countable unions, (σ p ) p be an (O)-sequence in R and m n : Σ → R, n ∈ N, be a sequence of finitely additive measures, globally uniformly (s)-bounded on Σ. Fix W ∈ Σ and a decreasing sequence with respect to the same (O)-sequence (σ p ) p , then Proof.Put W := {A ∈ Σ: A ∩ W = ∅}.For every A ∈ W and n, q ∈ N, we have As A ∩ H q ⊂ H q−1 \ W for any q ∈ N, from (2) and (3), for each n ∈ N we get If the thesis of the lemma is not true, then there exists p ∈ N such that for every r ∈ N there are n, k ∈ N with k > r and A ∈ Σ with A ⊂ H k \ W, |m n (A)| σ p , and thus, thanks to (4), for q large enough.
At the first step, we find a set A 1 ∈ Σ and three integers 2), in correspondence with n = 1, 2, . . ., n 1 there exists h 1 > q 1 with At the second step, there are From ( 5) and ( 6) it follows that k 2 > k 1 .
By induction, we find a sequence (A k ) k in Σ and three strictly increasing sequences in N, (k r ) r , (n r ) r , (q r ) r , with q r > k r > q r−1 for all r ≥ 2; q r > n r , A r ⊂ H k r \W, |m n r (A r \ H q r )| σ p for all r ∈ N.But this is impossible, because the sets A r \ H q r , r ∈ N, are pairwise disjoint, and the measures m n , n ∈ N, are globally uniformly (s)-bounded on Σ with respect to (σ p ) p . 2 The proof of the following result is similar to that of Lemma 2.6 (see also Boccuto & Candeloro, 2010

The Main Results
We begin with recalling the concept of uniform ideal exhaustiveness for measures (see also Athanassiadou, Dimitriou, Papachristodoulos, & Papanastassiou, 2012;Boccuto, Das, Dimitriou, & Papanastassiou, 2012;Boccuto & Dimitriou, 2011a, 2011b;Boccuto, Dimitriou, Papanastassiou, & Wilczyński, 2011, 2012), which plays a very important role in the versions of limit theorems with respect to ideal order convergence, and we deal with some properties of the Stone extension of a finitely additive measure, in connection with uniform ideal exhaustiveness and ideal pointwise convergence with respect to a common (O)-sequence.
In what follows, we suppose that R is a Dedekind complete ( )-group, I is a P-ideal of N and λ: Σ → [0, +∞] is a finitely additive measure, such that Σ is separable with respect to the Fréchet-Nikodým topology generated by λ (shortly, λ-separable).Let B := {F j : j ∈ N} be a countable λ-dense subset of Σ.
A sequence of finitely additive measures m n : Σ → R, n ∈ N, is λ-uniformly I-exhaustive on Σ iff there is an (O)-sequence (σ p ) p in R such that for every p ∈ N there are a positive real number δ and a set D ∈ I, with and for each n ∈ N \ D.
A sequence m n : Σ → R, n ≥ 0, of finitely additive measures, together with λ, satisfies property ( * ) with respect to R and I iff it is λ-uniformly I-exhaustive on Σ and (OI) lim n m n (E) = m 0 (E) for every E ∈ Σ with respect to a common (O)-sequence.
The following lemma will be useful to prove our results about equivalence of limit theorems.
Furthermore, from αα) and λ-uniform I fin -exhaustiveness of the m n 's, n ∈ M 0 , it follows that for each p ∈ N there are δ > 0 and Z p ∈ I fin with 8) and ( 9) it follows that the measures m n (ω), n ∈ M 0 , are λ-uniformly I fin -exhaustive and lim

This proves ααα). 2
In order to prove the equivalence between our ideal limit theorems, we will relate (globally) (s)-bounded and (globally) σ-additive measures.Indeed, in general, many problems involving finitely additive measures can be solved by finding suitable σ-additive measures, related to them, and then studying their properties.One of the main tools in this setting is the Stone extension, by means of which it is possible to construct a globally σ-additive measure, defined on a larger σ-algebra than the original one (see also Boccuto & Candeloro, 2002;Boccuto & Candeloro, 2004;Sikorski, 1964).
Let R be a super Dedekind complete and weakly σ-distributive ( )-group, λ: Σ → [0, +∞], m n : Σ → R, n ∈ N, be finitely additive measures, Q be the Stone space associated with Σ, that is a totally disconnected Hausdorff compact space, such that the algebra Q of its clopen subsets is algebraically isomorphic to Σ.If we denote by ψ: Σ → Q such isomorphism, then it is possible to "transfer" the measures λ and m n , n ≥ 0, to Q, by putting (λ Observe that, by the particular structure of Q, every monotone sequence (H k ) k of sets of Q is eventually constant.This implies that the measures λ • ψ −1 and m n • ψ −1 are globally σ-additive.By Boccuto and Candeloro (2004, Theorem 4.4), these measures admit (unique) globally σ-additive extensions λ, m n respectively, to the σ-algebra Σ(Q) generated by Q.These extensions are called the Stone extensions of λ and m n respectively.We now prove that the Stone extensions "inherit" property ( * ).
Theorem 3.2 Let λ: Σ → R, m n : Σ → R, n ≥ 0, be finitely additive measures, which together with λ satisfy property ( * ) with respect to I and R.
Proof.We first claim that Σ(Q) is λ-separable.Fix arbitrarily ε > 0. By Boccuto and Candeloro (2002, Theorem 4.4), for each This proves the claim, and we get also that E is a countable λ-dense subset of Σ(Q).
We now prove that the m n 's are λ-uniformly I-exhaustive.According to λ-uniform I-exhaustiveness of the m n 's on Σ, let (σ p ) p be an (O)-sequence related with it, choose arbitrarily p ∈ N and pick δ > 0, D ∈ I in correspondence with p.
By proceeding analogously as in Boccuto and Candeloro (2004, Theorems 4.4 and 4.5), it is possible to see that for every n ∈ N there is a (D)-sequence (a (n)  t,l ) t,l in R, such that for every ϕ ∈ N N and A 1 , Since, by construction, the m n 's are equibounded (see also Boccuto & Candeloro, 2004), then by Lemma 2.2, in correspondence with u := for every q ∈ N and ϕ ∈ N N , and hence Since R is super Dedekind complete and weakly σ-distributive, by Theorem 2.3 we find an (O)-sequence (v p ) p , such that for every p ∈ N there exists Thus we obtain that for each p, n ∈ N and Moreover we get: Thus condition ( 7) is satisfied, and then we have Let w p : = σ p + 2 v p , p ∈ N. Note that (w p ) p is an (O)-sequence, and we have obtained that for each p ∈ N there exist δ > 0 and D ∈ I such that for every pair Thus we have proved that, if the measures m n : Σ → R are λ-uniformly I-exhaustive on Σ, then the measures m n : The last step is to prove that with respect to a common (O)-sequence.Since the ψ(F j )'s, j ∈ N, form a countable λ-dense subset of Σ(Q) and satisfy ( 11), then by Lemma 3.1, α) applied to the sequence m n : From this and Proposition 2.4 we obtain (11).This concludes the proof of the theorem. 2 We now recall the following Brooks-Jewett-type theorem in the ( )-group context with (O)-convergence (with respect to a same (O)-sequence or not).
Furthermore, if the m n 's are globally (s)-bounded and R is super Dedekind complete and weakly σ-distributive, then they are also globally uniformly (s)-bounded.
We now are ready to prove the following limit theorems for ( )-group-valued measures with respect to ideal convergence and their equivalence (see also Boccuto & Dimitriou, 2011b, Theorem 3.10).
Theorem 3.4 (Brooks-Jewett (BJ)) Let R be a Dedekind complete ( )-group, λ: Σ → [0, +∞] be a finitely additive measure, such that Σ is λ-separable, I be a P-ideal of N, m 0 : Σ → R, m n : Σ → R, n ∈ N, be equibounded finitely additive measures, which together with λ satisfy property ( * ) with respect to R and I.Then, I) there exists a set M 0 ∈ F (I), such that the measures m n , n ∈ M 0 , are uniformly (s)-bounded on Σ.

II)
If the m n 's are globally (s)-bounded and R is super Dedekind complete and weakly σ-distributive, then M 0 can be chosen in such a way that the measures m n , n ∈ M 0 , are globally uniformly (s)-bounded on Σ.

II)
If each m n is globally σ-additive and R is super Dedekind complete and weakly σ-distributive, then M 0 can be chosen in order that the measures m n , n ∈ M 0 , are globally uniformly σ-additive.

I)
If each m n is regular, then a set M 0 ∈ F (I) can be found, with the property that the measures m n , n ∈ M 0 , are uniformly (s)-bounded and uniformly regular on Σ.

II)
If each m n is globally (s)-bounded and globally regular, and R is super Dedekind complete and weakly σ-distributive, then M 0 can be chosen in such a way that the measures m n , n ∈ M 0 , are globally uniformly (s)-bounded and globally uniformly regular.
To prove Theorem 3.4 (BJ), observe that there exist M 0 ∈ F (I) and N 0 ⊂ Ω, satisfying the thesis of Lemma 3.1.The assertion of (BJ) follows by applying Theorem 3.3 to the sequence m n , n ∈ M 0 , and to N 0 .
We begin with the implication (BJ) II) =⇒ (VHS) II).Let m n : Σ → R, n ∈ N, be a sequence of globally (s)bounded and globally τ-continuous finitely additive equibounded measures, satisfying together with λ property ( * ) with respect to R and I.By Lemma 3.1, αα), there is M 0 ∈ F (I) such that the measures m n , n ∈ M 0 , and m 0 satisfy property ( * ) with respect to R and I fin .By (BJ) II) used with I = I fin , there is a set M 0 ⊂ M 0 , such that M 0 \ M 0 is finite and with the property that the measures m n , n ∈ M 0 , are globally uniformly (s)-bounded, that is there is an (O)-sequence (σ p ) p with (O) lim with respect to (ζ p ) p , and so we get global uniform τ-continuity of the m n 's, n ∈ M 0 .Thus, (BJ) II) implies (VHS) II).
We We now prove that the m n 's are globally regular.Fix arbitrarily n ∈ N. Obviously, condition (R1) is fulfilled for each set E ∈ Q.We now claim that the class of all sets satisfying (R1) is a σ-algebra.From this it will follow that every set E ∈ Σ(Q) fulfils (R1), and hence that m n is globally regular.Without loss of generality, assume that m n is positive (indeed, in the general case, it will be enough to consider m n + and m n − ).It is readily seen that, if E ∈ Σ(Q) fulfils (R1), then Q \ E does.Let E k , k ∈ N, be a disjoint sequence in Σ(Q), satisfying (R1) and (σ p ) p be a related (O)-sequence.For each t, l ∈ N, put a t,l = σ l .Note that (a t,l ) t,l is a (D)-sequence, such that for every ϕ ∈ N N there is p ∈ N, p = ϕ(1), with σ p ≤ ∞ t=1 a t,ϕ(t) , so that the (O)-limit in the condition (R1) of global regularity with respect to (σ p ) p is a (D)-limit with respect to the (D)-sequence (a t,l ) t,l .For every k ∈ N there are two sequences (G (k r ) r and (F (k)  r ) r in G * and H * respectively, with r for all r ∈ N, and such that for every By Lemma 2.2 there is a (D)-sequence (b t,l ) t,l , with By global σ-additivity of m n there is a (D)-sequence (c t,l ) t,l , such that for every ϕ ∈ N N there is a natural number k 0 with We get: By Theorem 2.3 we find an (O)-sequence (π p ) p in R with the property that for every p ∈ N there is Thus the set E satisfies condition (R1).This proves the claim.
Thus the finitely additive measures m n , n ∈ N, are globally (s)-bounded and globally regular on Σ(Q).Arguing analogously as in the previous implication, by (D) II) used with Σ(Q) and I fin , there is a finite set M 0, * ⊂ M 0 , such that the measures m n , n ∈ M * := M 0 \ M 0, * , are globally uniformly (s)-bounded and globally uniformly regular on Σ(Q)."Coming back" to Σ, we get global uniform (s)-boundedness of the measures m n , n ∈ M * , on Σ.Since M * ∈ F (I), then it follows that (N) II) implies (BJ) II).
We start with the implication (BJ) I) =⇒ (VHS) I).Let R be a Dedekind complete ( )-group, Ω be as in Theorem 2.1 and m n : Σ → R, n ≥ 0, be a sequence of τ-continuous finitely additive equibounded measures, satisfying together with λ property ( * ) with respect to R and I.By Lemma 3.1, ααα), there are a meager set N 0 ⊂ Ω and a set M 0 ∈ F (I) with the property that for every ω ∈ Ω \ N 0 the real-valued measures m n (•)(ω), n ∈ M 0 , and m 0 (•)(ω) satisfy together with λ property ( * ) with respect to R and I fin .By (BJ) I) applied with R = R and I = I fin , for each ω ∈ Ω \ N 0 there is a set M (ω) 0 ⊂ M 0 , such that M 0 \ M (ω) 0 is finite and the real-valued measures m n (ω), n ∈ M (ω) 0 , are uniformly (s)-bounded on Σ, that is and so we obtain uniform τ-continuity of the m n 's.
The proof of (VHS) I) =⇒ (N) I) is similar to that of (VHS) II) =⇒ (N) II).
We now prove (N) I) =⇒ (BJ) I).Assume that m n : Σ → R, n ∈ N, is a sequence of equibounded finitely additive measures, satisfying property ( * ) with respect to I and R. Let Ω be as in Theorem 2.1.By Lemma 3.1, ααα), there exist a meager set N * ⊂ Ω and M 0 ∈ F (I), such that for every ω ∈ Ω \ N * the real-valued measures m n (•)(ω), n ∈ M 0 , are (s)-bounded and satisfy property ( * ) with respect to R and I fin .
Let Q be the Stone space associated with Σ, Q be the algebra of all clopen subsets of Q and Σ(Q) be the σ-algebra generated by Q.For each n ≥ 0 and ω ∈ Ω \ N * , let m n,ω : Σ(Q) → R be the Stone extensions of m n (•)(ω), and let λ: Σ(Q) → R be the Stone extension of λ.
whenever A, B ∈ I and for each A ∈ I and B ⊂ A we get B ∈ I.An ideal of N is said to be admissible iff N I and I contains all singletons.

Lemma 3. 1
Let m n : Σ → R, n ∈ N, be a λ-uniformly I-exhaustive sequence of finitely additive measures, and assume that (OI)lim n m n (F j ) =: m(F j ) for any j ∈ N with respect to a common (O)-sequence.Then, α) there exist a set M 0 ∈ F = F (I) and a finitely additive measure m 0 : Σ → R, which extends m and such that (O) lim n∈M 0 m n (E) = m 0 (E) for each E ∈ Σ and with respect to a same (O)-sequence (b p ) p ; αα) the measures m n , n ∈ M 0 , and m 0 satisfy together with λ property ( * ) with respect to R and I fin ; ααα) if Ω is as in Theorem 2.1, then there is a meager set N 0 ⊂ Ω such that for each ω ∈ Ω \ N 0 the real-valued measures m n (•)(ω), n ∈ M 0 , and m 0 (•)(ω), satisfy together with λ property ( * ) with respect to R and I fin .Proof.α) It is a consequence of Boccuto and Dimitriou (2011b) Lemma 3.9 and Proposition 2.5.αα) By λ-uniform I-exhaustiveness of the m n 's there is an (O)-sequence (σ p ) p such that for every p ∈ N there are a positive real number δ and a set A p ∈ I with |m n (E) − m n (F)| ≤ σ p for any E, F ∈ Σ with |λ(E) − λ(F)| ≤ δ and n A p .For each p ∈ N, set M p := N \ A p .Since I is a P-ideal, there exists a sequence (M p ) p of subsets of N such that M p ΔM p is finite for each p ∈ N and M := ∞ p=1 M p ∈ F .For every p ∈ N, set Z p := M \ M p .Note that Z p is finite for every p ∈ N, and so |m n (E) − m n (F)| ≤ σ p whenever E, F ∈ Σ with |λ(E) − λ(F)| ≤ δ and n ∈ M \ Z p .This proves αα).ααα) Let m 0 , M 0 , (b p ) p be as in α) and (σ p ) p be as in αα).By α) and Theorem there is a meager set N 0 ⊂ Ω such that the sequences (b p (ω)) p and (σ p (ω)) p are (O)-sequences in R for each ω ∈ Ω \ N 0 , and with the property that for every p ∈ N and E ∈ Σ there is an integer n ∈ M 0 with |m n (E)(ω) − m 0 (E)(ω)| ≤ b p (ω) for all n ≥ n, n ∈ M 0 , and ω ∈ Ω \ N 0 .
Theorem 3.5(Vitali-Hahn-Saks (VHS)) Let R, Σ, λ, I, m n be as in Theorem 3.4, and τ be a Fréchet-Nikodým topology on Σ.I)If each m n is τ-continuous, then there exists M 0 ∈ F (I), such that the measures m n , n ∈ M 0 , are uniformly τ-continuous on Σ.II)If the m n 's are globally (s)-bounded and globally τ-continuous, and R is super Dedekind complete and weakly σ-distributive, then M 0 can be chosen to have global uniform τ-continuity of the m n 's, n ∈ M 0 .Theorem 3.6 (Nikodým (N)) I) If R, Σ, λ, I are as above and the m n 's, n ∈ N, are σ-additive, then there is M 0 ∈ F (I), such that the measures m n , n ∈ M 0 , are uniformly σ-additive on Σ.
now prove (VHS) II) =⇒ (N) II).Let τ be the Fréchet-Nikodým topology generated by the family of all order continuous submeasures defined on Σ.If (H k ) k is any decreasing sequence in Σ with τ-limk H k = ∅ and H = ∞ k=1 H k ,then we get η(H) = 0 for every order continuous submeasure η on Σ, and hence H = ∅.From this it follows that, if m n : Σ → R, n ∈ N, is a sequence of globally σ-additive measures, then they are globally τ-continuous.Since the m n 's are also globally (s)-bounded, then by (VHS) II) they are globally uniformly τ-continuous, and hence also globally uniformly σ-additive.Thus, (VHS) II) implies (N) II).We now prove (N) II) =⇒ (BJ) II).Let m n : Σ → R, n ∈ N, be a sequence of equibounded finitely additive globally (s)-bounded measures, satisfying together with λ property ( * ) with respect to I and R. By Lemma 3.1, αα), a set M 0 ∈ F (I) can be found, with the property that the measures m n , n ∈ M 0 , and m 0 satisfy property ( * ) with respect to I fin and R. If m n : Σ(Q) → R, n ∈ N, and λ: Σ(Q) → R, are the Stone extensions of m n and λ respectively, then, by Theorem 3.2, the σ-algebra Σ(Q) is λ-separable, and the m n 's, n ∈ M 0 , are σ-additive measures, satisfying together with λ property ( * ) with respect to I fin and R. By (N) II) used with Σ(Q) and I fin , we find a finite set M 0 ⊂ M 0 , such that the measures m n , n ∈ M * := M 0 \ M 0 , are globally uniformly σ-additive, and hence also globally uniformly (s)-bounded, on Σ(Q)."Coming back" to Σ, we get global uniform (s)-boundedness of the measures m n , n ∈ M * .Since M * ∈ F (I), then it follows that (N) II) implies (BJ) II).We now prove (D) II) =⇒ (BJ) II).Let G * , H * be the lattices of all open and all closed subsets of the Stone space Q which belong to Σ(Q) respectively.It is not difficult to see that G * and H * satisfy condition (R0).Let m n : Σ → R, n ≥ 0, be equibounded globally (s)-bounded finitely additive measures satisfying, together with λ, property ( * ) with respect to I and R. Arguing as in the previous implication, let us consider the global σ-additive Stone extensions m n : Σ(Q) → R and λ: Σ(Q) → [0, +∞] of m n and λ respectively.