3. Cafiero theorem for additive set functions and SCP

Vol. 33, No. 1, 2003, 117–127

MATRIX METHOD APPROACH TO CAFIERO THEOREM

Paolo de Lucia¹, Endre Pap²

Abstract. We give a simple proof of the Cafiero theorem based on a matrix method approach in the form of Lemma 2.4 in the σ-additive context. Based on a version of Drewnowski lemma for an SCP-ring we obtain an extension of Cafiero theorem for exhaustive finitely additive set functions defined on an SCP-ring. As consequences, the well-known Nikod´ym and Brooks-Jewett convergence theorems are obtained.

AMS Mathematics Subject Classification (2000): 28A33

Key words and phrases: additive set function, exhaustive, matrix method, Nikod´ym convergence theorem

1. Introduction

In a paper of 1952 F. Cafiero [3] has characterizedσ-additive set functions defined on aσ-ring that are uniformly additive, see Cafiero [3, 4]. The notion of the uniform additiveness was introduced by R. Caccioppoli [2] and indepen- dently by Dubrovskii [15], and it was utilized by many authors to obtain a Lebesgue type theorem for the convergence of integrals. Since the time when R.

Rickart [22] introduced the notion of exhaustivity or s-boundedness for additive set functions defined on a ring it has been clear that forσ-additive set functions the uniform additivity of Caccioppoli and Dubrovskii is equivalent to the uni- form exhaustivity. Therefore the Cafiero lemma was reformulated for the case of the uniform exhaustivity (see d’Andrea, de Lucia [7], H. Weber [25]).

In this paper we give a simple proof of the Cafiero theorem based on a matrix method approach (diagonal theorems), as a further extension of sliding hump method, and which was initiated by Mikusi´nski [18]. Many results in measure theory and functional analysis can be found in Antosik and Swartz [1], Pap [19, 20, 21], Swartz [23] (see also Diestel and Uhl Jr [13], Weber [26]), where the matrix method is used instead of the Baire Category Theorem, which is

1Universita ”Federico II”, Dipartimento di Matematica, e Applicazioni ”Renato Cacciop- poli”, via Cinthia, 80126 Napoli, Italy, e-mail: [email protected]

2Institute of Mathematics, University of Novi Sad, Trg D. Obradovi´ca 4, 21000 Novi Sad, Yugoslavia, e-mail: [email protected]

unsuitable for obtaining more general results. We start with a matrix method lemma, Lemma 2.4, which we use in the proof of Cafiero theorem 2.5 in the σ-additive context. After that, by a version of Drewnowski lemma 3.3 for an SCP-ring, we arrive at an extension of the Cafiero theorem 3.4 for exhaustive finitely additive set functions defined on a SCP-ring. Finally, in section 4 we obtain immediately, by the Cafiero theorem 3.4, the well known Nikod´ym and Brooks-Jewett convergence theorems.

For more information on Cafiero theorem see d’Andrea, de Lucia [8], de Lucia [9], de Lucia, Salvati [11], de Lucia, Traynor [12], Traynor [24], Weber [25].

2. Cafiero uniform exhaustivity theorem

LetRbe a ring andMa family of finitely additive set functions defined on R.A finitely additive set functionm:R →Risexhaustive(strongly additive or strongly bounded) if lim_j→∞m(Ej) = 0 for every sequence{Ej}j∈Nof pairwise disjoint elements fromR.A familyMof additive set functions isuniformly ex- haustiveif lim_j→∞m(E_j) = 0 uniformly inm∈ Mfor every sequence{E_j}_j∈N of pairwise disjoint elements fromR. It is obvious that a finite measure on a σ-ring is exhaustive.

The following propositions are well known (see de Lucia, Pap [10]).

Proposition 2.1. LetMbe a family of finite additive exhaustive set functions onR. The following statements are equivalent

(i) Mis uniformly exhaustive,

(ii) for every increasing sequence {Ei}i∈N the sequence {m(Ei)}i∈N is a con- vergent sequence, uniformly form∈ M

(iii) for every disjoint sequence{E_i}_i∈NofRthe series

im(E_i)is convergent uniformly form∈ M.

Proposition 2.2. LetMbe a family of finite additive exhaustive set functions onR. Then

(i) Mis uniformly exhaustive, implies

(ii) for every decreasing sequence{Ei}i∈N the sequence {m(Ei)}i∈N is a con- vergent sequence uniformly form∈ M.

IfRis an algebra then (i) and (ii) are equivalent.

Let Σ be aσ-algebra andMfamily ofσ-additive set functions defined on Σ.

A family Mof countable additive measures µ: Σ→R isuniformly countable additiveif

n→∞lim ∞ j=n

µ(E_j) = 0

uniformly in µ ∈ Mfor every sequence {E_j}_j∈N of pairwise disjoint elements from Σ.We have by Propositions 2.1 and 2.2

Proposition 2.3. Let Σ be a σ-algebra and M family of σ-additive set func- tions defined on Σ. The following statements are equivalent

(i) Mis uniformly exhaustive, (ii) Mis uniformly countable additive,

(iii) for every decreasing sequence{E_i}_i∈N of Σ the sequence{µ(E_i)}_i∈N is a convergent sequence uniformly for µ∈ M.

First we shall give an elementary matrix type lemma.

Lemma 2.4. Let [xni]_n,i∈N be an infinite matrix of real numbers such that 1) for every n∈Nand every subsetI ofN there exists

i∈Ix_ni;

2) for every sequence{Ik}k∈Nof pairwise disjoint subsets ofNand for every ε >0 there exists¯k∈N andn0∈Nsuch that

i∈I¯k

x_ni

< εfor everyn > n₀. Then

i→∞lim xni= 0uniformly in n∈N.

Proof. We note that if the matrix [xni]_n,i∈Nhas the properties 1) and 2) then also every its submatrix has the same properties. By 1) we have

i→∞lim xni= 0 (n∈N).

(1)

Then we need to prove that: for everyε > 0 there existk, m ∈ Nsuch that

| xni |< ε for everyi > kand everyn > m. Suppose that this is not true.

Then there existsσ >0 such that for everyk, m∈Nthere existi > k andn >

msuch that |xni|≥σ.By induction we can then construct two strictly increas- ing sequences {ik}k∈Nand {nk}k∈N of natural numbers such that|xnkik |≥σ

for everyk∈N.Therefore there exists a submatrix of the starting matrix, which we denote by the same symbol [x_ni]_n,i∈N, such that it satisfies 1), 2) and for someσ >0

3) |xkk|≥σfor everyk∈N.

Let{σ_r}_r∈N be a decreasing sequence of real numbers such that

r∈N

σr<σ 2,

from 2) we have that for everyr∈Nthere existir andmr such that

|xnir|< σr

for everyn≥mr.We can suppose that the sequences{ir}and{mr}are strictly increasing and so to construct a new submatrix of [x_ni], which we denote by the same symbol [x_ni]_n,i∈N,such that it satisfies 1), 2), 3) and

for every i∈N, |x_ni|< σ_ifor alln≥m_i. (2)

Now we construct by induction two strictly increasing sequences{ρ_n}_n∈N and {r_n}_n∈Nof natural numbers with the property

ρ_n> r_n>max{ρ_n−1, m_rn−1}for everyn∈N, whereρ0=m0= 0,such that we have for everyn∈N,

|xrki|< σrn for every i > ρn and k= 1, . . . , n.

(3)

Suppose thatr1, . . . , rn−1,;ρ1, . . . , ρn−1are determined and let bernan element ofNsuch that

rn >max{ρn−1, mrn−1}, by (1) we can findρn such thatρn > rn and (3) is true.

Consider now the submatrix [x_rnri]_n,i∈N,forn∈Nandi= 1, . . . , n−1,we havern> mri and then, by (2)

|xrnri |< σri, and fori≥n+ 1 it results

ri> ρi−1, n≤i−1 and then, by (3)

|xrnri|< σri−1.

Hence for an infinite subsetI ofNandn∈I we have by 3)

i∈I

x_rnri

≥ |x_rnrn| −

i∈I,i=n

|x_rnri|

≥ σ−

i∈N

σi

= σ

2.

If{Ik}k∈Nis a disjoint sequence of infinite subsets ofNwe have for everyk∈N

i∈Ik

xrnri

> σ

2 for alln∈Ik

and this contradicts 2).

Theorem 2.5. [Cafiero]LetΣbe aσ-algebra. A sequence{µn}n∈Nof countable additive real measures defined on Σ is uniformly exhaustive if and only if the following condition holds

α) for every sequence {En}n∈N of pairwise disjoint elements of Σ and every ε >0there exist ¯k, n0∈Nsuch that

|µ_n(E¯k)|< εfor every n≥n₀.

Proof. Let {E_n}_n∈N be a sequence of pairwise disjoint elements of Σ. Then (µ_n(E_i))_n,i∈N is an infinite matrix of real numbers. By countable additivity of every µn we have that for everyn∈Nand I⊂N there exists

i∈Iµn(Ei).If {Ik}k∈N is a sequence of pairwise disjoint subsets ofN, then {∪i∈IkEi}k∈N is a sequence of pairwise disjoint elements of Σ and then byα) for everyε >0 there exist ¯k, n0∈Nsuch that

µ_n





i∈I¯k

E_i



 =

i∈I¯k

µ_n(E_i)

< εfor everyn≥n₀.

Therefore the conditions 1) and 2) of Lemma 2.4 are satisfied and we have that

i→∞lim µ_n(E_i) = 0 uniformly inn∈N.

2

3. Cafiero theorem for additive set functions and SCP

To extend the previous results to the finitely additive case we will prove a generalization of a very useful lemma obtained by Drewnowski [14] for aσ- ring, now for the case of an SCP-ring. We have the following definition by Constantinescu [5, 6] and Haydon [17], see also Freniche [16] and H. Weber [25].

Definition 3.1. A ringRhas the Sequential Completeness Property, and will be called SCP-ring, if each disjoint sequence{E_n}_n∈NfromRhas a subsequence {E_nj}_j∈N, whose union is inR.

Lemma 3.2. [Drewnowski lemma with SCP] Let R be an SCP-ring. If m : R →R is an exhaustive monotone set function with m(∅) = 0 and {En}n∈N

is a sequence of pairwise disjoint sets from R, then there exist a subsequence {Ekn}n∈N of {En}n∈Nand a SCP-ring R with SCP such that R⊆ R, Ekn∈ R for everyn∈N, andm is order continuous on the ringR.

Proof. Let{J_i}_i∈Nbe a sequence of pairwise disjoint infinite subsets ofN.Then for everyi∈Nthere exists an infinite subset J_i of J_i such that {∪_k∈JiE_k}_i∈N is a sequence of pairwise disjoint subsets of R. By the exhaustivity of mit is possible to find an infinite subsetN₁ ofNsuch that

m

k∈N1

Ek

< 1 2.

In the same way, if{Ji}i∈Nis a sequence of pairwise disjoint infinite subsets of N1\ {minN1},it is possible to find an infinite subsetN2 ofN1\ {minN1}such that

m

k∈N2

E_k

< 1 2².

In this way, we construct, by induction, a decreasing sequence{N_i}_i∈Nof infinite subsets ofNsuch that

Ni+1⊆Ni\ {minNi}, m

k∈Ni

Ek

< 1

2ⁱ for everyi∈N.

Letki= minNi and letR be the set

X ∈ R: there existsI⊆Nsuch thatX =

i∈I

Eki

. (4)

We claim that R is a ring with the SCP. It is clear that the supremum of two elements ofR belongs toR.LetX1, X2 be two elements of R such that X2⊆X2.Then there existI1 andI2 subsets ofN such that

Xp=

j∈Ip

Ekj for everyp∈N.

Then we have

X1\X2=

j∈I1\I2

Ekj ∈ R,

i.e.,X₁\X₂ ∈ R. ThereforeR is a ring. Let{X_p}_p∈Nbe a disjoint sequence of elements ofR, then there exists a disjoint sequence{I_p}_p∈N of subsets ofN such that

X_p=

j∈Ip

E_kj.

By the property of SCP ofRthere exists a subsequence{X_pr}_r∈Nof {X_p}_p∈N

such that

r∈N

X_pr =

j∈∪r∈NI_pr

E_kj

belongs to R and then also to R that the restriction of m to R is order continuous is proved in the same way as it was proved in the case of theσ-ring,

see de Lucia, Pap (2002).

We shall call the ringR defined by (4) the SCP-ring generated by{Ekn}. Lemma 3.3. [Drewnowski lemma for additive set functions] Let Rbe a SCP- ring. Ifµ_i:R →Ris a sequence of exhaustive additive set functions and{E_n} is a sequence of pairwise disjoint sets from R, then there exists a subsequence {E_kn}_n∈N of {E_n}_n∈N such that µ_i is countable additive on the SCP-ring R generated by{E_kn}_n∈N.

Proof. Let for everyn∈Ndenote by|µn|the total variation ofµn.We introduce the set functionm:R →Rby

m=

n∈N

1 2ⁿ

|µ_n| 1 +Mn,

whereMn = sup{|µn|(A) :A∈ R}.Since the functionmsatisfies the conditions of Drewnowski lemma 3.2 there exists a SCP-ringRgenerated by a subsequence {Ekn}n∈Nof{En}n∈N such that the restriction ofmtoR is order continuous.

Then it easily follows that the restrictions of|µn| to R are order continuous and therefore everyµn is countable additive onR.

We will generalize Theorem 2.5 to the additive case.

Theorem 3.4. [Cafiero theorem for additive set functions] Let Rbe an SCP- ring. A sequence{µn}n∈Nof finite additive bounded set functions defined on R is uniformly exhaustive if and only if the following condition holds

α) for every sequence {En}n∈N of pairwise disjoint elements of R and every ε >0there exist ¯k, n0∈Nsuch that

|µ_n(E¯k)|< εfor every n≥n₀.

Proof. Suppose that the {µn}n∈N are not uniformly exhaustive. Then there exists a disjoint sequence{Ei}i∈N of Rsuch that lim_n→∞µn(Ei) = 0 but not

uniformly inn.Then there existsε >0 such that we can construct a subsequence of {E_i}_i∈N and a subsequence of {µ_n}_n∈N, that for simplicity, we will denote yet by the same symbols so that

|µ_n(E_n)|> εfor every n∈N.

(5)

By Drewnowski lemma 3.3 there exists a subsequence{E_ik}_k∈Nof{E_i}_i∈Nsuch that ifR is a SCP-ring generated by{E_ik}_k∈N the restriction ofµ_n to R are σ-additive. From Theorem 3.4 it follows that these restrictions are uniformly exhaustive but by (5) we have

|µ_ik(E_ik)|> εfor everyk∈N.

A contradiction.

4. Applications

4.1. Nikod´ym convergence theorem

Theorem 4.1. [Nikod´ym convergence theorem] Let R be an SCP-ring. Let {µ_n}_n∈N be a pointwise convergent sequence of countable additive measures de- fined onR, i.e.,

n→∞lim µ_n(E) =µ(E), E∈Σ, (6)

then

(i) {µn}n∈N converges to a countable additive measureµ, (ii) {µ_n}_n∈N is uniformlyσ-additive.

Proof. We consider first a special case. If{µn}n∈N is pointwise convergent to zero then conditionα) in Theorem 2.5 is satisfied. Then {µn}n∈N is uniformly exhaustive and, by Proposition 2.3, it is uniformlyσ-additive.

The general case for (ii), i.e., under the condition (6), it easily follows by the fact that by (6){µn(E)}n∈N is a Cauchy sequence. Namely, suppose that (ii) does not hold for{µn}n∈N, i.e., that there is a sequence of pairwise disjoint sets{En}n∈Nfrom Σ, a subsequence{µkn}n∈Nof{µn}n∈Nandε >0 such that

| µkn(Ekn) |≥ 2ε. By exhaustivity of µkn there exists a subsequence {pn}n∈N

of {kn}n∈N such that | µpn(Epn+1) |≤ ε. Taking mn = µpn+1 −µpn we ob- tain a sequence{mn}n∈Nof countable additive set functions which is pointwise convergent to zero and therefore by the previously proved part it is uniformly countable additive, but this is in contradiction with

|mn(Epn+1)| ≥ |µpn+1(Epn+1)| − |µpn(Epn+1)| ≥ ε

for alln∈N.

To prove (i), we have to use (ii). Namely, by (ii) we have µ



^∞

j=1

E_j



 = lim

i→∞µ_i



^∞

j=1

E_j





= lim

i→∞ lim

n→∞

n j=1

µ_i(E_j)

= lim

n→∞µ



ⁿ

j=1

Ej





= ∞ j=1

µ(Ej).

4.2. Brooks-Jewett theorem and related results

A generalization of the Nikod´ym convergence theorem was obtained by Brooks and Jewett.

Theorem 4.2. [Brooks-Jewett] Let R be an SCP-ring. A pointwise conver- gent sequence{mn}n∈N of finitely additive scalar and exhaustive set functions (strongly additive) defined on anR, i.e.,lim_n→∞m_n(E) =m(E), E∈ R,

(i) converges to an additive and exhaustive set functionm, (ii) {m_n}_n∈Nis uniformly exhaustive.

Proof. If{m_n}_n∈Nis pointwise convergent to 0 then the conditionα) of Theorem 3.4 is verified and so in this special case (ii) is true. The general case for (ii) follows in the same manner as in the proof of Nikod´ym theorem.

Then (i) follows by (ii)

j→∞lim m(E_j) = lim

j→∞ lim

i→∞m_i(E_j)

= lim

i→∞ lim

j→∞m_i(E_j)

= 0.

Theorem 4.3. [Nikod´ym boundedness theorem for additive case]Let Rbe an SCP-ring. A familyMof finitely additive bounded set functions m, defined on

an R, which is pointwise bounded, i.e., for each E ∈ R there exists M_E > 0 such that

|m(E)|< ME (m∈ M), is uniformly bounded, i.e., there existM >0 such that

|m(E)|< M (m∈ M, E∈ R)

The proof is the same as the proof of Nikod´ym boundedness theorem, only using Brooks-Jewett theorem 4.2 instead of Nikod´ym convergence theorem (see

de Lucia, Pap [10]).

Acknowledgement.The second author wants to acknowledge partial financial support of the Project MNTRS-1866.

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Received by the editors December 12, 2002