Coun tably Additiv e Restrictions and Limit Theorems in l ... fileIn this pap er w e deal with * -additiv e restrictions of (l)-group-v alued Þnitely additiv e measures and with Nik

Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia, 57 (2010), 121-134

Antonio BOCCUTO - Xenofon DIMITRIOU -Nikolaos PAPANASTASSIOU

Countably Additive Restrictions and Limit Theoremsin l -Groups

Abstract.Some Drewnowski-type lemmas are given for (l)-group-valued measures. Moreover some Vitali-Hahn-Saks, Schur andNikodym convergence-type theorems in the context of (l)-groupsare proved.

Key Words: (l)-group, order convergence, #-additive measure,MOV representation theorem, Limit theorem, Schur lemma, Dre-wnowski lemma, Semivariation, Stone Isomorphism technique,Technical lemma for order sequences.

Mathematics Subject Classification (2000): 26B10, 28B05.

1. Introduction

In this paper we deal with *-additive restrictions of (l)-group-valuedfinitely additive measures and with Nikodym and Vitali-Hahn-Saks typetheorems for (l)-group-valued set functions. In [4], when convergence ofthe involved measures is the pointwise one, some limit theorems wereproved, for positive measures taking values in suitable spaces of measur-able functions and for positive measures with values in any Dedekind com-plete Riesz space, provided that the “limit measure” satisfies some “good”properties. In some other papers, for example [5], some convergence the-orems are demonstrated, in which the concepts of (s)-boundedness, *-additivity, and so on, are formulated “with respect to a same unit”, aswell as the pointwise convergence of measures. Here we deal with (s)-boundedness, *-additivity and absolute continuity intended in the ordi-

Supported by the Universities of Perugia and Athens.Received June 22, 2010, accepted October 2, 2010.

122 A. BOCCUTO - X. DIMITRIOU - N. PAPANASTASSIOU [2]

nary sense, while pointwise convergence is supposed to be “with respectto a same unit”. In this direction, some results on limit theorems wereobtained in [8] and [10]. The results here presented deal with order con-vergence and the “classical” semivariation, that is the semivariation withrespect to the *-algebra in which our measures are defined rather thanwith respect to the *-algebra generated by suitable sequences.

We use deeply the famous Maeda-Ogasawara-Vulikh representationtheorem for Archimedean (l)-groups (see [2, 18]). In [9] we prove someother versions of limit theorems using di#erent hypotheses and contexts.

2. Preliminaries

Definitions 2.1. From now on, we suppose, if it is not di#erentlymentioned, that R is a Dedekind complete (l)-group, G is any infinite setand A 0 P(G) is any *-algebra.

A Dedekind complete l-group R is said to be super Dedekind completeif, for any nonempty set A 0 R, bounded from above, there exists acountable subset A$ 0 A, such that supA = supA$.

We say that R is weakly *-distributive if for every bounded doublesequence (ai,j)i,j , such that (ai,j)j is an (o)-sequence for all i # N, onehas:

?

+%NN

= '@

i=1

ai,+(i)

>= 0.

We say that a sequence (pn)n of positive elements of R is an (o)-sequenceif pn = 0. A sequence (xn)n in R is said to be order-convergent (or (o)-convergent ) to x if there exists an (o)-sequence (pn)n with |xn ) x| , pn

for all n # N, and in this case we write (o) limn xn = x.Given a finitely additive bounded set function m : A" R, we define

the semivariation of m (on A), v(m) : A" R, as follows: v(m)(A) =>B%A,B.A |m(B)|, A # A.

A sequence of finitely additive set functions (mj)j is said to be (RO)-convergent to m0 in A if there exists an (o)-sequence (pl)l such that forevery l # N and A # A there is j0 # N with |mj(A))m0(A)| , pl for allj / j0. In this case we say that (RO) limj mj = m0. We say that (mj)j

(UO)-converges to m0 if there is an (o)-sequence (vj)j with the propertythat |mj(A))m0(A)| , vj whenever j # N and A # A.

We say that (mj)j is order uniformly equally convergent to m0 if thereare an (o)-sequence (pj)j and a natural number n0 with

8({j # N : mj(A))m0(A) (# [)pj , pj ]}) , n0 for all A # A

(here the symbol 8 denotes the cardinality of the involved set).

[3] COUNTABLY ADDITIVE RESTRICTIONS 123

The sequence (mj)j is globally order uniformly equally convergent tom0 if there is an (o)-sequence (pl)l such that to every l # N there corre-sponds an integer j0 with

8({j # N : mj(A))m0(A) (# [)pl, pl]}) , j0 for all A # A.

A finitely additive set function m : A"R is said to be (s)-bounded if for ev-ery disjoint sequence (Hn)n inA we have (o) limn v(m)(Hn)=0. The mapsmj : A"R, j#N, are uniformly (s)-bounded if (o) limn[>j v(mj)(Hn)] = 0whenever (Hn)n is a sequence of pairwise disjoint elements of A.

A finitely additive map m : A " R is said to be *-additive if forevery disjoint sequence (Hn)n in A we get: (o) limn v(m)(

$'l=n Hl) = 0.

The set functions mj : A" R, j # N, are uniformly *-additive if for eachdisjoint sequence (Hn)n in A, (o) limn[>j v(mj)(

$'l=n Hl)] = 0.

We say that the sequence (mj)j is uniformly equally *-additive if thereexist an (o)-sequence (pn)n and an integer n0 with

(1) 8({n # N : v(mj) (''k=n Hk) (# [)pn, pn]}) , n0 for all j # N;

we say that (mj)j is weakly uniformly equally *-additive if there exists an(o)-sequence (pl)l with the property that to every l # N there correspondsan integer n0 such that

(2) 8({n # N : v(mj) (''k=n Hk) (# [)pl, pl]}) , n0 for all j # N.

We now introduce our concept of absolute continuity. Let 9 : A" [0,+1]be a *-additive measure: a map m : A " R is said to be 9-absolutelycontinuous, if for each decreasing sequence (Hn)n in A, with limn 9(Hn) =0, we get (o) limn v(m)(Hn) = 0. The maps mj : A " R, j # N, areuniformly 9-absolutely continuous if (o) limn[>j v(mj)(Hn)] = 0 whenever(Hn)n is a decreasing sequence in A such that limn 9(Hn) = 0.

Note that, if R is a topological group, the definition of absolute con-tinuity here given coincides with the classical one (see [15]), but in generalthis is not true in Riesz spaces (see also [4]).

The following two technical lemmas for order sequences were provedin [6].

Lemma 2.2. Let R be a super Dedekind complete and weakly *-distributive l-group, and (rn)n be any (o)-sequence in R. Then, for everyU # R, U / 0, there exists an element $ # NN such that the mapping

N ?" U @'7

n=N

r,(n)

is an (o)-sequence.


Lemma 2.3. Let R be a super Dedekind complete and weakly *-distributive l-group and {(r(k)

n )n : k # N} be an equibounded countablefamily of (o)-sequences. Then there exists an (o)-sequence (bj)j with thefollowing property: for every j, k # N there is n = n(j, k) # N such thatr(k)n , bj.

We now recall the famous Maeda-Ogasawara-Vulikh representationtheorem in its version for (l)-groups (see [2]), which will be useful in thesequel.

Theorem 2.4. Given a Dedekind complete (l)-group R, there ex-ists a compact extremely disconnected topological space !, unique up tohomeomorphisms, such that R can be embedded as a solid subgroup ofC'(!) = {f # "R" : f is continuous, and {$ : |f($)| = +1} is nowheredense in !}. Moreover, if (a-)-%# is any family such that a- # R for all:, and a = >- a- # R (where the supremum is taken with respect to R),then a = >- a- with respect to C'(!), and the set {$ # ! : (>- a-)($) (=sup- [a-($)]} is meager in !.

[We denote by sup and > the pointwise and the lattice supremumrespectively, and by inf and @ the corresponding infima.]

3. The main results

We begin with the following

Theorem 3.1. Every bounded measure m : A" R is (s)-bounded.

Proof. Let ! be as in 2.4. By Theorem 2.4 there is a nowhere denseset N0 0 ! such that the real-valued measures m(·)($), $ # ! \N0, arebounded. By virtue of the classical results, they turn to be (s)-boundedon A.

Fix now any arbitrary disjoint sequence (Hn)n in A. Then by Theo-rem 2.4 we get, in the complement of a meager set, depending on (Hn)n:

[(o) limn

v(m)(Hn)]($) = [@n(>s)nv(m)(Hs))]($)

= [@n(>s)n(>A%A,A.Hs |m(A)|))]($) = infn

(sups)n

( supA%A,A.Hs

|m(A)($)|))

= infn

(sups)n

v(m(·)($))(Hs)) = limn

v(m(·)($))(Hn) = 0.

Thus we obtain the assertion.


We now prove the following results about *-additive restrictions offinitely additive measures.

Theorem 3.2. Let ! be as in Theorem 2.4 and m : A " R be afinitely additive bounded measure.

Then there is a meager set N0 ! such that for every disjoint sequence(Hn)n there is a subsequence (Hnk)k with limk [v(m)('j)kHnj )](y) =infk [v(m)('j)kHnj )](y) = 0 for each y # ! \ N . In particular, m andm(·)(y), y # ! \N , turn out to be *-additive along the sequence (Hnk)k.

Proof. First of all we note that, by Theorem 3.1, the set functionm turns to be (s)-bounded, and there is a meager set N0 such that themeasures m(·)(y), y # ! \N0, are (s)-bounded too.

Let R A r := supA%A |m(A)|. For each n # N, let Tn := {$ # ! :r($) , n}. Since ! is compact and r is continuous, then Tn is compactfor all n # N. By Theorem 2.4, the set N := ! \ [''n=1 Tn] is meager in!. Without loss of generality, we can assume that N < N0. By the Bairecategory theorem, T := ! \N is dense in !.

Now, let {P 1k : k # N} be a (fixed) infinite partition of N, whose

elements are infinite sets; put H1k := 'j%P 1

kHj for every k # N and -1

j :=>k)j v(m)(H1

k). By (s)-boundedness of m, we get: -1j = 0 in R as j tends

to +1.Since T1 is compact, in correspondence with every $ # T1 there

exist j($) # N such that -1j(,)($) < 1 and an open neighborhood U,

with -1j(,)(y) < 1 whenever y # U,. Since the family {U, : $ # T1}

covers T1, we can find a finite subcovering {U,1 , . . . , U,q}, with associ-ated indexes j($1), . . . , j($q). Let j$1 := maxq

l=1 j($l): then -1j!1

(y) < 1whenever y # T1. Indeed to each fixed y # T1 there corresponds (atleast) an element $l, with l # {1, . . . , q}, such that y # U,l , and hence0 , -1

j!1(y) , -1

j(,l)(y) < 1.

Set now H1 := H1j!1

: then 0 , v(m(·)(y))(H1) , v(m)(H1)(y) < 1for each y # T1, since the pointwise suprema are less or equal than thecorresponding lattice suprema.

At the second step, define P $1 := P 1j!1

, and denote by {P 2k : k # N}

a (chosen) infinite partition of P $1 , whose elements are infinite sets. Foreach k # N, set H2

k := 'j%P 2k

Hj . Observe that the sets H2k , k # N, form

a disjoint partition of H1.Put -2

j := >k)j v(m)(H2k) for every j # N. Again by (s)-boundedness

of m, we have -2j = 0 in R as j " +1. Proceeding analogously as in the

previous step, let us consider the compact set T2 and the elements -2j : it is

possible to check the existence of an element j$2 # N such that -2j!2

(y) < 1/2for all y # T2. Set P $2 := P 2

j!2and H2 := 'j%P!

2Hj : then H2 0 H1, and

0 , v(m(·)(y))(H2) , v(m)(H2)(y) < 1/2 for any y # T2.


By induction, we construct a decreasing sequence (P $n)n of infinitesubsets of N such that, if Hn := 'j%P!

nHj , n # N, then (Hn)n is decreas-

ing, and 0 , v(m(·)(y))(Hn) , v(m)(Hn)(y) < 1/n whenever y # Tn,n # N. Set now pn := min{P $n , n # N}: without loss of generality,we can assume pn < pn+1 for all n # N. We claim that the sequence(Hpn)n is the requested one. Indeed, for any n # N, 'j)n Hpj 0 Hn and[v(m)('j)n Hpj )](y)<1/n for each y#Tn. If we put -n :=v(m)('j)n Hpj),n # N, and - := @n -n, then 0 , -(y) , -j(y) < 1/j for all y # Tj . SinceTj 0 Tj+1 for all j # N, then 0 , -(y) < 1/k for all k / j and y # Tj , andhence -(y) = 0 for each y # Tj , j # N. By arbitrariness of j, we obtain:-(y) = 0 for all y # ! \N and so, by a density argument, we get - = 0.This proves that

(3) limn

v(m)''j)n Hpj

(= 0,

and a fortiori

(4) limn

v(m(·)(y))''j)n Hpj

(= 0,

since the pointwise suprema are always less or equal than the correspond-ing “lattice” ones.

The *-additivity of the measures m and m(·)(y), y # ! \ N , on the*-algebra generated by the sets Hpn , n # N, is an easy consequence of (3)and (4), thanks to monotonicity of v(m) and v(m(·)(y)), y # ! \N .

The following result is an (l)-group-version of the well-known Drewno-wski Lemma (see [15, 16]).

Theorem 3.3. Let R, G, A, ! be as in Theorem 3.2, and (mn :A" R)n be a sequence of bounded measures. Then there exists a meagerset N 0 ! such that for every disjoint sequence (Hs)s in A there is asubsequence (Hsk)k of (Hs)s with limk v(mn)

''j)k Hsj

((y) = 0 for each

y # ! \ N and n # N. In particular, the measures mn and mn(·)(y),n # N, y # ! \N , are *-additive along the *-algebra generated by (Hsk)k.

Proof. By virtue of Theorem 3.2, we get that to every n # N therecorresponds a meager set Nn 0 !, satisfying the thesis of Theorem 3.2.Set N := ''n=1 Nn: we claim that N is the requested set.

By Theorem 3.2, there exists an infinite subset P1 0 N withlimh v(m1) ('j%P1,j)h Hj) (y) = 0 for all y # ! \ N . Moreover there isan infinite subset P2 0 P1 such that limh v(m2) ('j%P2,j)h Hj) (y) = 0for all y # ! \ N . Let p1 := min P1: without loss of generality, we cansuppose that P2 does not contain p1.

By induction, it is possible to construct a decreasing sequence ofinfinite subsets Pn 0 N, n # N, with lim

hv(mn) ('j%Pn,j)h Hj) (y) = 0


for each y # ! \ N and n # N, and pn < pn+1 for all n # N, wherepn := min Pn. Let now P := {pn : n # N}. For every fixed n # N andy # ! \N , we get:

0 = limh

v(mn) ('j%Pn,j)h Hj) (y) = limh)pn

v(mn) ('j%Pn,j)h Hj) (y)

= lim suph)pn

v(mn) ('j%Pn,j)h Hj) (y) / lim suph)pn

v(mn) ('j%P,j)h Hj) (y)

=lim suph

v(mn) ('j%P,j)h Hj) (y) / lim infh

v(mn) ('j%P,j)h Hj) (y)/0.

From this and by a density argument it follows that

(5) limh

v(mn) ('j%P,j)h Hj) = 0,

and a fortiori

(6) limh

v(mn(·)(y)) ('j%P,j)h Hj) = 0

for every n # N and y # ! \ N , since the pointwise suprema are less orequal than the corresponding “lattice” ones. The *-additivity of the setfunctions mn and mn(·)(y), n # N, y # ! \N , on the *-algebra generatedby the sets Hpn , n # N, follows easily from (5) and (6).

This lemma will be useful to prove our Nikodym and Vitali-Hahn-Saks-type theorems.

Lemma 3.4. With the same assumptions as above, suppose that thereis a meager set N 0 ! such that the real-valued measures mj(·)($), j # N,are uniformly (s)-bounded on A for all $ (# N . Fix W # A, and assumethat (Hn)n is a decreasing sequence in A, such that W 0 Hn for all n # N.If

(7) limn

[ supA%A,A.Hn\W

|mj(A)($)|] = 0 for all j # N

for $ belonging to the complement of a meager set NW 0 !, then

(8) limn

(supj

[ supA%A,A.Hn\W

|mj(A)($)|]) = 0

whenever $ # ! \ (N 'NW ).

Proof. Fix arbitrarily $ # ! \ (N ' NW ), and let W := {A # A :A 3W = %}. For every A #W and j, q # N we get:

(9) mj(A)($))mj(A 3Hcq )($) = mj(A 3Hq)($).


Since A3Hq 0 Hq!1 \W for all q # N, from (7) and (9) for all j # Nwe get

(10) mj(A)($) = limq

mj(A 3Hcq )($)

uniformly with respect to A #W.If we deny the thesis of the lemma, then there exists ; > 0 with the

property that to every p # N there correspond n # N, n > p, j # N andA # A such that A 0 Hn \W , |mj(A)($)| > ;, and hence, thanks to (10),

(11) |mj(A 3Hcq )($)| > ;

for q large enough.At the first step, in correspondence with p = 1, there exist: A1 # A;

three integers n1 # N \ {1}, j1 # N and q1 > max{n1, j1}, with A1 0Hn1 \ W , |mj1(A1)($)| > ; and |mj1(A1 3 Hc

q1)($)| > ;. From (7), in

correspondence with j = 1, 2, . . . , j1 we get the existence of an integerh1 > q1 such that

(12) |mj(A)($)| , ;

whenever n / h1 and A 0 Hn \W .At the second step, there exist: A2 # A; three integers n2 > h1,

j2 # N and q2 > max{n2, j2}, with A2 0 Hn2 \W and

(13) |mj2(A2)($)| > ;; |mj2(A2 3Hcq2

)($)| > ;.

From (12) and (13) it follows that j2 > j1.Proceeding by induction, it is possible to construct a sequence (Ak)k

in A and three strictly increasing sequences in N, (nk)k, (jk)k, (qk)k, with:qk > nk > qk!1 for all k / 2; qk > jk, Ak 0 Hnk\W , |mjk(Ak3Hc

qk)($)| >

; for all k # N. But this is impossible, since the sets Ak 3Hcqk

, k # N, arepairwise disjoint elements of A, $ # !\(N'NW ), and the maps mj(·)($),j # N are uniformly (s)-bounded on A for each fixed $ # ! \ N . Thisconcludes the proof.

In [8] we proved the following Brooks-Jewett-type theorem.

Theorem 3.5. Under the assumptions as above, suppose that (mj :A" R)j is a sequence of equibounded finitely additive measures. Supposethat there is a map m0 : A " R such that the sequence (mj)j (RO)-converges to m0 on A.

Then the functions mj(·)($) are uniformly (s)-bounded on A (withrespect to j) for $ belonging to the complement of a meager subset of !.Moreover the mj’s are uniformly (s)-bounded on A.


The following versions of the Nikodym and of the Vitali-Hahn-Sakstheorem are refinements of the respective ones proved in [8].

Theorem 3.6. (Nikodym convergence theorem) Assume that (mj :A " R)j is a sequence of equibounded *-additive set functions, and sup-pose that there is a mapping m0 : A " R such that (RO) limj mj = m0.Then m0 is *-additive and the mj’s are uniformly *-additive.

Proof. By Theorem 3.5, we get the existence of a meager set N 0 !with the property that the real-valued measures mj(·)($), j # N, areuniformly (s)-bounded on A for any $ # ! \N .

Fix arbitrarily any disjoint sequence (Cl)l, and set Hn := 'l)n Cl.Since the Cl’s are pairwise disjoint, we get Hn = %.

By *-additivity of mj , j # N, and Theorem 2.4 there exists a meagerset N " 0 !, without loss of generality N " < N (depending on (Cl)l), suchthat limn [supA%A,A.Hn

|mj(A)($)|] = 0 for all $ # ! \N " and j # N. ByLemma 3.4 used with W = %, for all $ (# N " we get

limn

(supj

[ supA%A,A.Hn

|mj(A)($)|]) = infn

(supj

[ supA%A,A.Hn

|mj(A)($)|]) = 0.

From this and Theorem 2.4 we get

{@n(>j [>A%A,A.Hn |mj(A)|])}($)={(o) limn

(>j [>A%A,A.Hn |mj(A)|])}($)

=0

for any $ (# N ". Since the complement of any meager subset of ! is densein !, it follows that

0 = @n(>j [>A%A,A.Hn |mj(A)|])= @n(>j v(mj)(Hn)) = (o) lim

n(>j v(mj)(Hn)),

getting uniform *-additivity of the mj ’s.The *-additivity of the “limit measure” m0 is an easy consequence of

(RO)-convergence of (mj)j to m0 and uniform *-additivity of the mj ’s.

Theorem 3.7. (Schur theorem) Let R be a super Dedekind completeweakly *-distributive (l)-group, and (mj : P(N) " R)j be any sequence ofequibounded *-additive measures. Then (RO)-convergence of the measuresmj to some limit m0 implies (UO)-convergence to m0.


Proof. By virtue of Theorem 3.6 the measures mj are uniformly*-additive, and m0 is *-additive. From this it follows that there is an(o)-sequence (qk)k with the property that

(14) v(mj)(F 3 {k + 1, k + 2, . . . }) , qk

whenever k # N, j # N ' {0} and F 0 N. Moreover, thanks to (RO)-convergence of (mj)j to m0, an (o)-sequence (pl)l can be found, such thatfor each l and k # N there exists j0 # N such that

(15) |mj(F 3 {1, . . . , k}))m0(F 3 {1, . . . , k})| , pl for all F 0 N.

(Thanks to Lemmas 2.2 and 2.3, proceeding analogously as in [5, Corollary5.6], we get that the sequence (pl)l can be found independently of F 0 N).In particular (15) holds even if we take l = k, and so to each k # N wecan associate a positive integer jk > k with

|mj(F ))m0(F )| , |mj(F 3 {1, ..., k}))m0(F 3 {1, ..., k})|

+ v(mj)(F 3 {k + 1, k + 2, . . . })+ v(m0)(F 3 {k + 1, k + 2, . . . })

, pk + 2 qk

for all j / jk. Set j0 := 0, p0 := p1, q0 := q1. Without loss of generality,we can suppose jk!1 < jk for any k # N. To each j there corresponds oneinteger k = k(j) # N ' {0} with jk , j < jk+1. Set zj := pk(j) + 2 vk(j),j # N. It is easy to see that (zj)j is an (o)-sequence and that

|mj(F ))m0(F )| , zj

for all j # N and F 0 N. Therefore we obtain (UO)-convergence of (mj)j

to m0. This concludes the proof.

We now formulate the Vitali-Hahn-Saks theorem. From now on, weassume that R is any Dedekind complete (l)-group.

Theorem 3.8. (Vitali-Hahn-Saks theorem for *-additive measures)Let (mj : A " R)j be an (RO)-convergent sequence of equibounded 9-absolutely continuous set functions, where 9 : A" [0,+1] is a *-additivemeasure. Then the mj’s are uniformly 9-absolutely continuous.

Sketch of proof. Let (Hn)n be any fixed sequence in A withlimn 9(Hn) = 0, and W := 3n Hn. Set Cl := Hl \ Hl+1, l # N: thenCl # A. It is easy to check that W = H1 \ (''l=1 Cl), and so W # A. Wenow claim that v(mj)(Hn \W ) = v(mj)(Hn) for every j, n # N. Indeed,since 9 is positive and monotone, then for every K # A, K 0 W , we get9(K) = 0, and hence mj(K) = 0 for all j, by 9-absolute continuity. Thusv(mj)(W ) = 0 for any j # N, and so the claim follows. From this, arguinganalogously as in Theorem 3.6 and replacing [uniform] *-additivity with[uniform] 9-absolute continuity, we get the assertion.


Theorem 3.9. (Vitali-Hahn-Saks theorem for finitely additive mea-sures) Let (mj : A " R)j%N/{0} be an equibounded sequence of positivefinitely additive measures, 9 : A " [0,+1] be a finitely additive mea-sure, such that (RO) limj mj = m0 and each mj, j # N, is 9-absolutelycontinuous on A.

Then the set functions mj, j # N, are uniformly 9-absolutely contin-uous.

Proof. Observe that, by Theorem 3.5, up to the complement ofmeager sets the maps mj(·)($) are uniformly (s)-bounded.

We now use the so-called Stone isomorphism technique. Given any setG and an algebra A 0 P(G), there exists a compact, totally disconnected,Hausdor# topological space E such that A is isomorphic to the field F ofthe clopen sets of E: we denote by 2 : F " A such an isomorphism. Ewill be called the Stone space associated with A (see also [3, 26]).

The measures mj := mj + 2 : F " R, 9 := 9 + 2 : F " [0,+1]and the set functions mj(·)($) := mj(·)($)+2 are *-additive, and so theyadmit *-additive extensions to A(F), which we denote by Amj , "9 and #mj,,

respectively. By [12, Corollario 3.15] used with % = R, the maps #mj,, areuniformly *-additive. This implies that the measures Amj are uniformly*-additive too. Indeed, let (Hk)k be any decreasing sequence in A(F)with 3'k=1 Hk = %. By [3, Lemma 2.7] there exists a meager set N$ 0 !such that [Amj(Hk)]($) = #mj,,(Hk) for all j, k # N and $ (# N$. So, in thecomplement of a meager sets N$$ (without loss of generality N$$ < N$)we get:

{infk

Amj(Hk)}($) = infk

[Amj(Hk)($)] = infk

#mj,,(Hk) = 0.

Thus we obtain uniform *-additivity of the Amj ’s on A(F).We now turn to absolute continuity. Let (Hk)k be a decreasing se-

quence in F , with 9(Hk) = 0, and set Bk := Hk \ Hk+1, k # N. Let Lbe the *-algebra generated by the Hk’s, and choose a decreasing sequence(As)s in L, with "9(As) = 0. Note that every As is a union of some Bk.It is easy to check the existence of a subsequence (Ask)k)2 with Amj(Ask) ,mj(Hk) for any j and k # N.

Indeed, if there is q2 # N with As < B1 for s > q2, then 9(B1) = 0and mj(B1) = 0 for all j # N. So, in this case, Amj(As) = Amj(As \ B1) ,mj(H2) for all s > q2 and j # N. Otherwise an integer l2 > 1 can befound, with Al2 0 H2, and thus Amj(Al2) , mj(H2) for all j # N. In anycase there is s2 # N such that Amj(As2) , mj(H2) for all j # N.If there exists q3 # N such that As < B1 ' B2 for any s > q3, then9(B1'B2) = 0 and so mj(B1'B2) = 0 for each j # N. So for each s > q3


and j # N,Amj(As) = Amj(As \ (B1 'B2)) , mj(H3).

If not, then there is l3 > s2 with Al3 0 H3, and so Amj(Al3) , mj(H3) forevery j # N. In any case there is N A s3 > s2 with Amj(As3) , mj(H3) foreach j # N. By induction, for all k / 2 we get a set Ask with Amj(Ask) ,mj(Hk) for all j # N, and sk < sk+1 for any k / 2.

So, by 9-absolute continuity of mj , j # N, we have

0 , infk

Amj(Ak) , infk

Amj(Ask) , infk

mj(Hk) = 0, j # N,

and therefore the maps Amj are "9-absolutely continuous on L. Since "9 is *-additive and the Amj ’s are uniformly *-additive, then by Theorem 3.8 theyare uniformly "9-absolutely continuous on L. Thus infk[supj mj(Hk)] =infk[supj Amj(Hk)] = 0, and hence the mj ’s are uniformly 9-absolutelycontinuous. “Coming back” to the given *-algebra A, we get that the setfunctions mj , j # N, are uniformly 9-absolutely continuous on A. Thisconcludes the proof.

Open problems:

a) It is interesting to examine similar problems using the definitions (1)and (2).

b) It is also interesting to derive analogous results considering (D)-convergence, which has been defined in [23].

References

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A. Boccuto:Dipartimento di Matematicae InformaticaUniversita di Perugiavia Vanvitelli 106123 Perugia, [email protected]@dmi.unipg.it

X. Dimitriou:Department of MathematicsUniversity of AthensPanepistimiopolisAthens 15784, [email protected]@math.uoa.gr

N. Papanastassiou:Department of MathematicsUniversity of AthensPanepistimiopolisAthens 15784, [email protected]

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Coun tably Additiv e Restrictions and Limit Theorems in l ... fileIn this pap er w e deal with * -additiv e restrictions of (l)-group-v alued Þnitely additiv e measures and with Nik