Completeness of /-groups and of /-seminormsAbstract. We consider the following properties of an /-seminorm on a commutative /-group: completeness, a

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXI (1979) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PR ACE MATEMATYCZNE XXI (1979)

M

W

(Wroclaw)

Completeness of /-groups and of /-seminorms

Abstract. We consider the following properties of an /-seminorm on a commutative /-group: completeness, a-subadditivity and C-completeness. We also separate a class of weakly .c-complete /-groups arising in this context.

1. Introduction. The reader is referred to Birkhoff [1] -or Fuchs [3] for fundamentals of the theory of /-groups (= lattice-ordered groups). Let us fix some notation used consequently throughout the paper. (G, + , ^ ) is a commutative /-group, L is an /-subgroup of G and v is an /-seminorm on L, that is a function of L to [0, oo] satisfying

v( 0 ) = 0 ,

v (a + b ) ^ v(a) + v(b), a , b e L , v(|a|) = v(a), a e L , v(a) ^ v(b) if 0 ^ a ^ b, a , b e L .

All joins and meets in the paper are taken exclusively in G. In conse­

quence, some of the considered properties of L and v (see, e.g., Definitions 1-3) depend also on G. Such a general approach is indispensable for applications in measure and integration theory to /-seminorms on function spaces and to submeasures on rings. Nevertheless, the special case L = G may be also of interest. We consider the following properties of v:

completeness (in the semimetric sense), (i-subadditivity (Definition 1) and C-completeness (Definition 2). We prove that C-completeness implies rr-sub- additivity (Theorem 3). In our previous paper [9] it is shown that if L = G = Rs (all real-valued functions on a set S Ф 0 , with the pointwise addition and ordering), then — conversely — сг-subadditivity implies C-com­

pleteness. Here we separate a larger class of pairs L, G (L is called weakly а-complete (with respect to G) — Definition 3) for which this statement remains true (Theorem 4). This class seems to be of interest in itself; in case L = G we get a new class of /-groups — weakly ^-complete, in­

terrelations of which with conditionally cr-complete /-groups are mentioned

in Section 3. If G is itself weakly er-complete, then the cr-subadditivity of v on L is a necessary and sufficient condition in order that v be extendable to a C-complete /-seminorm on an /-subgroup of G (Theorem 5).

2. C -completeness and a -subadditivity of v. The seminorm v generates, in a natural way, an invariant semimetric for L (g (a ,b ) = v (a — b)). We say that v is complete if this semimetric is complete. Analogous meaning have such terms as “v-Cauchy sequence” or “v-convergent sequence”.

T

1. Let L be a commutative l-group and v an I-seminorm on L.

The following three conditions are equivalent:

(i) v is complete;

(ii) every monotone v-Cauchy sequence is v-convergent;

0 0

(iii) every increasing sequence {a„} c L + with £ v (an + i ~ an) < 00 IS

n =

1

v-convergent.

Proof. It is sufficient to prove that (iii) implies (i). Let {a„} c= L and

00

£ v{an +

— an) < oo. Define for any n e N

n =

1

n

К = X ( a i ~ a i -1 ) + a n d = X ( a i - V i - i T ,

i=

1 i= 1

where a

= 0 and a~ denotes the element ( — a)+. The sequences {bn} and {c„} are_in L +, are increasing and

( 1 ) an = bn — cn,

( 2 ) an + 1 an bn i bn + c„ +

cn,

00 00

(2) implies that £ v(bn + l — bn) < oo and £ v(c„ + 1 —c„) < oo. By (iii), there

n =1 n= 1

are b , c e L with lim v(b — bn) = lim v(c — cn) = 0. Define a = b — c e L . For-

П -+00 n-*o

rnula ( 1 ) gives the inequality \a — an\ ^ \b — bn\ + |c —c„|, which yields lim v(a — an) П~> 00

= 0. It follows that v is complete.

Let S be a non-empty set, d? a ring of subsets of S (cf. Halmos [4]) and r\ a submeasure on -M, that is, a function on M to [ 0 , oo] such that

П(Ф) = 0 ,

q(A kj B) ^ q(A )+ q(B ), A ,B e d t, rj(A) ^ q(B) if A с; B, A ,B e .j$ .

Let L* be the /-subgroup of Rs generated by the collection of all charac­

teristic functions of members of L* consists of all .^-measurable simple integer-valued functions on S. Put v,,(a) = q ({s e S : a(s) Ф 0}) for a e L #; vt]

is an /-seminorm on L *>.

Completeness o f l-groups and of l-seminorms 273

Theorem 1 applied to yields immediately the following result of Drewnowski ([2], Theorem 1.1), proved by the author in a different way.

C orollary 1. Let 01 be a ring of subsets of S and ц a submeasure on 02. The following three conditions are equivalent:

(i) rj is complete;

(ii) every monotone q-Cauchy sequence is q-convergent;

O O

(iii) every increasing sequence {A„} c=

with £ q(An +

\A n) < oo is n=

rj-convergent.

Here the terms “complete”, “//-Cauchy” and “//-convergent” refer, in fact, to the semimetric corresponding to q ( q (A ,B ) = q(A A B )).

00

Let us introduce some conventions. For a ,a ne G we write a v » .

n = l

oo oo • oo m

if a = \J (a a an). For b ,b ne G + we write b £ bn if b \J £ b„.

n=

n= 1 m=

n =

Finally, for c ,c „ eG the relation c ~ {c„} denotes that

(3) |c -c „ | X ki + i-Cfl for all n e N . i = n

oo oo oo

m

In case the join a' = \J an (respectively; the sum b' — £ bn = \J £ b„)

n=

n=

exists in G, the above holds if and only if a ^ a! (resp. b ^ b'). If the sequence {c„} is increasing, then (3) reduces to

c„ + \c — c„\ \J ct for all n e N . i=

The following lemma is an easy generalization of Corollary V.2 (to the Refinement Theorem V.l) of Fuchs [3] and of Lemma 3 of Wilhelm [10].

a

00

Le m m a

1. Let a ,a ne L +. I f a £ an, then there are af„ e L +

00

n —

= Y j a'n and dn ^ an f or n ^ N {and conversely).

n=

with

n n

Proof. Put fli = я л a± and for n = 2, 3, . . . , a'„ = (а л £ а,) —(а л £ at).

i

i

A sequence { an } c= G is said to be order convergent to an element a e G {a = o — lim an) if there are elements b „ eG + such that \an — a\ ^ bn for all

П-* 00

n e N and bnjO (cf. Fuchs [3]).

L emma 2. Let a ,a ne G . I f a = o — lim a„, then a ~ {a„}.

П 00

18 — Prace Matematyczne 21.1

Proof. Fix a positive integer n. Put

n + k - 1

Ьк = \ а - а л\ л £ \ai+

- a t \ e L +

i = n

and

n + J t - 1

ck = (\a — a„\ —

i

g L + for any k e N .

i = n

We are going to prove that b k] \ a — a„\. Since b k + ck = \a — a n\, it suffices to show that cfc|0 . The inequality

n + k - 1

I a - a n\ ^ £ \a i + i - a i\ + \a - a n+k\

i = n

yields the estimation ck ^ \a — a n+k\. It follows that ckj 0, as desired.

The converse statement fails to be true in general.

E xample 1. Consider a , a n s G = Rs. Then the relation a ~ { a n} holds if and only if

00

|a (s) — (s)| ^ ^

+ i (s) ~ ai(s)l for all n e N and s e S ,

i — n

i.e. if and only if

00

a(s) = lim a„(s) whenever £ K + i ( s ) — a„(s)| < oo for any s e S .

n - с о n = 1

Hence, if

lim inf an ^ a ^ lim sup a„ (in particular if a — lim an),

M~GO

И — * 00

then a ~ {an}; the converse implication is of course not true.

E xample 2. Let f be a ring of subsets of S and A ,A neé%. Consider the corresponding /-subgroup of Rs ; the relation 1A ~ {1^ } holds if and only if

00

M «) = lim 1 ah ( s ) whenever £ l ^ n + 1 (s) ~ W s)l < oo, s e S ,

if and only if

lim inf An c= A c lim sup A„.

п - > о о гг-*-oo

In this case we will simply write A ~ {An}. If A = lim A„ (cf. Halmos [4], П -» 00

1.3), then A ~ {An}; not conversely.

Completeness o f l-groups and of l-seminorms 275

D e f i n i t i o n

1. The /-seminorm

is o-subadditive if

GO 00

v(a) <c ^ v(an) whenever a = an, a ,a nGL+.

n = 1 n = 1

Remark. The pair (L, v) is called in [10] a real integrable space provided G = Rs and v is u-subadditive (1).

Th e o r e m

2. Let G be a commutative l-group, L an l-subgroup of G and

an l-seminorm on L. The following five conditions are equivalent:

(i)

is a-subadditive;

00 00

(ii) if a ,a „ e L + and a £ an, then v(a) ^ £ v(a„);

n= 1 n= 1

00

(iii) if a, aneL , a ~ {an} and У v(an + l — a„) < oo, then lim v(a — an) = 0;

n=l n->0°

(iv) if a, a„eL, a — o — lim an and lim v(a„ — am) = 0 , then lim v(a —

. ~ «-►oo n,m~* oo « “►oo

~a„) = 0 ;

(v) if ane L +, a„j0 (in G) and lim v(a„ — am) = 0, then lim v(a„) = 0.

nym~+ oo «-►oo

Proof. (i)=>(ii) Let {a^} be as in Lemma 1; then

00 00

v(a) ^ Z v (a'„) ^ £

n = l « = 1

00

(ii) =>(iü) By (ii), we have v ( a - a n) < Z v(ai +

- a i ) ^ 0 .

i = n

(iii) =>(iv) A standard induction argument enables us to choose a sub-

00

sequence |a„,} with У v ( a — an ) < oo. We have again a = o — lim a„ ,

к k = l k - t 1 к fc -► oo K

so that, by Lemma 2, a ~ (a„ }. By (iii), lim v(a — a„ ) = 0, which implies

k k-> oo

the assertion.

(iv) =>(v) Evident.

00

(v) =>(i) Let b, b „ eL + and b - £ bn. We may additionally assume

n = 1

00 «

that v(b„) < oo. Put an = b ~ Y , bf, then Ь+ э а п[ 0 and an+k — an

n = 1 i = 1

n + k

= — £ bf. It follows that {a„} is a v-Cauchy sequence. By (v), the

i = n+ 1

sequence {a„} v-converges to 0 , and so

« « oo

V(b) ^ v( X b,) + v(an) ^ X v (b«)+v.(an)„-^ Z v(b£).

i = 1 i = 1 i = 1

l1) There are some unessential differences; v considered in [10] is on L+ into R +.

The submeasure q is called o-subadditive if

oo oo

rj(A) ^ Y whenever A = ( J An, A, Л „ е ^ . и= 1

Thus ц is cr-subadditive if and only if the corresponding /-seminorm is cr-subadditive.

C

2. Let be a ring of subsets of S and g a submeasure on

. The following five conditions are equivalent :

(i) rj is o-subadditive;

00 00

(ii) if A, AneM and A c= (J A„, then rj(A) ^ I ri(A„);

n = 1 n= 1

00

(iii) if A ,A ne P 0 ,A ~ {An} and Y п(Ап +

А А п) < oo, then n= 1

lim q ( A A A n) = 0 ;

n-* oo

(iv) if A , Ane $ , A = lim An and lim rj(AnA A m) = 0, then n~*

lim 1 )(ЛДЛ„) = 0 ;

«“► 00

(v) if A „ e $ , Ап[Ф and lim rj(AnA A m) = 0, then lim rj(An) = 0.

n , m ~ > oo и -*-oo

The equivalence of (i) and (v) is stated in Drewnowski [2] (prop­

erty {a")).

D

2. The /-seminorm v is C-complete if v is complete and (4) for any a ,a ns L + with an] a and v(a„) = 0 for all n e N , v(a) = 0.

First let us consider separately condition (4). It is equivalent to

00 00

(4') for any a ,a ne L + with a = Y an and X v (an) = 0, v(a) = 0.

n= 1 n= 1

Thus (4) holds provided v is c-subadditive; not conversely.

T

4. Let G be a commutative l-group, L an l-subgroup of G and v an l-seminorm on L. Condition (4) holds (if and) only if for any a, an,b e L satisfying

(5) \a — an\ ^ b„lO for some bne L +

and lim v(b — an) = 0 , v(a — b) = 0 .

«-► 00

Proof. Put c„ = (\a — b\ — bn)+ 6 L+ ; then c x ^ c

^ ... and

c„ ^ (\a — b\ — \a — an\)+

\b-a„\.

Completeness of l-yroups and of l-seminorms 277

Hence the sequence {v(c„)} c= [0, oo] is non-decreasing and v(c„) ^ v(b — an) P •

It follows that v(c„) = 0 for all n e N . Since c„ j \a — b\, (4) implies v(a — b) = 0.

Remark 2. If G is conditionally cr-complete and L closed under countable joins in G, then (5) holds if (and only if) a = o — lim an in G.

« - ► o o

Proof. If dne G + and \a — an\ ^ d„ jO, then \a — at\ ^ dn for i ^ n.

00

Define bn = \J \a — a ^ e L f .

i = «

Now we can compare C-completeness and cr-subadditivity of v.

T

4. Let G be a commutative l-group, L an l-subgroup of G and

v

am l-seminorm on L. I f v is C-complete, then y is o-subadditive.

Proof. Let ane L +, a „ | 0 and lim v{an — am) = 0. Since v is complete,

п , т ~ > o o

lim v(b — an) = 0 for some b e L . By Theorem 3 applied to a = 0, v(b) = 0,

П - * o o

and so lim v(an) = 0. The assertion is now established by implication

«-*■ 00

(v) => (i) of Theorem 2.

A submeasure rj is said to be C-complete (Orlicz [7], 2.4.1) if ц is complete and

( 6 ) for any A , AneM with An\ A and v(A„) = 0 for all ne N, r ] ( A) = 0.

Evidently, ( 6 ) holds if and only if the corresponding /-seminorm v,, satisfies (4); rj is C-complete if and only if v„ is C-complete. Thus we get

C

3. Let M be a ring of subsets of S and rj a submeasure on Ж. Then

(i) Condition ( 6 ) holds (if and) only if for any A , A n, BG. f? satisfying

(7) A A A n ΠBn 1 0 for some BnG,M

and lim t ] ( BAA„) = 0 , r j ( AAB) =

.

«->00

(ii) If ц is C-complete, then rj is o-subadditive.

For a cr-ring M part (ii) is due to Orlicz ([7], 2.4.2), for a <5-ring follows from Theorem 1.4 of Drewnowski [2].

Re ma rk 3. If $ is a c-ring, then (7) holds if (and only if) A = lim An.

«->00

3. Weak ^-completeness of L. We are interested in a converse to The­

orem 2. Namely, under what completeness-type hypothesis of L (x-sub- additivity implies C-completeness (for any /-seminorm on L). Corollary 1 of Wilhelm [9] shows that for L = G = Rs the implication is satisfied.

The /-group Rs is conditionally complete (as a lattice) and o-complete

(cf. Fuchs [3], V.ll). Unfortunatelly, the last two properties of an /-group L = G are not sufficient in general.

E

3. Let L = G be the /-subgroup of Rs consisting of all functions with finite support; L is conditionally complete and o-complete.

Let v(a) = max \a(s)\ for a e L ; v is a cr-subadditive /-norm on L, but v is

seS

not complete, unless S is finite.

On the other hand, if L = G is cr-complete (as a lattice), then L = {0}

(cf. Birkhoff [1], Chapter XIII or Fuchs [3], V.9). As things are, let us introduce another completeness-type property of L (depending also on G L), which is suitable to our purposes.

D

3. The /-subgroup L of G is weakly o-complete (with respect to G) if for any increasing sequence {an} c= L + there exists an element a e L satisfying a ~ {an}.

It can be shown that if L is weakly cr-complete, then for any sequence {an} c= L there exists an a e L with a ~ |a„}. From Example 1 it follows that Rs (= L = G) is weakly cr-complete, from Example 2 that L.# c= Rs is weakly cr-complete if and only if J? is a o-ring. If Л is a cr-ring, then also the /-subgroup L of Rs consisting of all .^-measurable functions (cf.

Halmos [4], § 18) is weakly cr-complete (with respect to G = Rs, or with respect to G = L).

Now let us consider the case L = G. We recall that if L is conditionally cr-complete, then L is Archimedean (cf. Birkhoff [1], XIII. 1 or Fuchs [3], V.9). It can be proved that

1° If L is weakly cr-complete and Archimedean, then L is conditionally cr-complete.

2° There exists a conditionally cr-complete L which is not weakly cr-complete (Example 3).

3° There exists a non-Archimedean L which is weakly cr-complete.

4° There exists a non-Archimedean L which is not weakly cr-complete.

The above facts are not necessary here, and we intend to prove them, and to discuss the related property of conditional weak cr-completeness in another paper.

4. С-completeness and cr-subadditivity of v (continued).

T

5. Let G be a commutative l-group, L an l-subgroup of G and v an l-seminorm on L. Let L be weakly o-complete (with respect to G).

I f v is о-subadditive, then v is C-complete.

00

Proof. Let {an} c= L + be an increasing sequence with £ v(an +

— an) n= 1

< oo. By Definition 3, there exists an element a in L such that a ~ {a„}.

Implication (i) => (iii) of Theorem 2 shows that lim v(a — an) = 0. Finally,

Completeness o f l-groups and of l-seminorms 279

implication (iii) => (i) of Theorem 1 proves that v is complete, and so C-complete.

Similarly to the preceding corollaries we infer the following one, which states a well-known result of Orlicz ([ 6 ], 1.6.).

C

4. Let M be a a -ring of subsets of S and rj a submeasure on I f ц is о-subadditive, then rj is C-complete.

L

3. Let a, a„, a„ke L +. I f a = Y an and an = Y ank for all n e N ,

n= 1 *= l

00

then a = Y ank.

n , k = 1

m

Proof. Let b e L + and b ^ Y

n , k =

show that b ^ a.

ank for all m e N ; it is sufficient to

m p

Since b — Y ank ^{^} Y ^{a i} к f°r all m , p e N ,

n = 2 k = 1

k = 1

ъ- Y ank > «1

n = 2 k = 1

for m e N .

m p m

Since b — al — Y ank ^ Z ^a

k f°r w , p e N , b — al — Y ank ^ ai- Continu-

n = 3 к — 1 n = 3

k= 1 k = l

ing this process we see that

r m

b ~

Z

Z

i = l n = r + 2 k = 1

in particular,

r + 1

b > Y ai f ° r апУ

i = 1

It follows that b ^ a.

0 0 0 0

L

4. Let a, an, anke L +. I f a Y an and an ^ ' Z ^{a*k f or ab}

n = 1 fc = l

00

n e N , then a ^ .

n , k ~ 1

00

Proof. By Lemma 1, there are a'n, a'nke L + such that a = Z < ’ < <

n = 1

00 00

a„ = Y a'nk and ^ (n , k e N ). Lemma 3 shows that a = Y a'nk->

k = 1 n , k = 1

which implies the assertion.

Let us extend to /-seminorms on /-subgroups an important construction

of integration theory (see, e.g., Stone [ 8 ] or Wilhelm [9]): define

oo oo

v*(a) = inf [ X v (a„): {a„} c L + and |a| £ a„J

n = 1 n = 1

for all a e G (inf 0 = oo). Evidently, v* is an /-seminorm on G.

L

5. I f v is G-subadditive , then so is v*.

00

Proof. Let a , a ne L +, a = £ an and e > 0. There are anke L + such

GO 00 n= 1 00

that an J ank and £ v(anfc) ^ v*(a„)+e-2“". By Lemma 3, a < ' £ ank.

к — 1 к 1 и,/с = 1

00 oo

Hence v*(a) ^ У v(a„k) < У v*(a„)+e. This proves that v* is rr-sub-

n , k = 1

n= 1

additive.

From Theorem 4, Lemma 5 and Theorem 5 we infer

T

6. Let G be a commutative l-group, L an l-subgroup of G and v an l-seminorm on L. Let G be weakly а -complete. The following two conditions are equivalent:

(i) there exists an l-subgroup L x of G extending L and a C-complete l-seminorm Vj on L x extending v;

(ii) v is a-subadditive.

In this case we may also take L x = G and Vj = v* in (i).

C

5. Let M be a ring of subsets of S and r\ a submeasure on

ft.

The following two conditions are equivalent:

(i) there exists a ring of subsets of S extending 2ft and a C-complete submeasure r

on 2ft,k extending rj;

(ii) ц is G-subadditive.

In this case we may also take 2ft г = 2s and = rj*, where

00 00

q*(A) = inf { £ q(A n): {An} cz УЛ and A c (J An} for A cz 2s. ~

n = 1

/1= 1

Notice that (v^)*0) = q*(supp a) for a e R s.

Implication (ii) => (i) is given in Kluvanek [5] (Lemma 2).

References

[1] G. B irk h o ff, Lattice theory, Third Edition, Providence 1967.

[2] L. D re w n o w s k i, On complete submeasures, Comm. Math. 18 (1975), p. 177-186.

[3] L. F u c h s, Partially ordered algebraic systems, Budapest 1963.

[4] P. R. H alm o s, Measure theory, New York 1950.

[5] I. K lu v a n e k , Completion o f vector measure spaces, Rev. Roumaine Math. Pures Appl.

12 (1967), p. 1483-1488.

Completeness o f l-groups and of l-seminorms

[6] W. O r liez, Linear operations in Saks spaces. I, Studia Math. 11 (1950), p. 237-272.

[7] — On space L*,p based on the notion of a finitely additive integral, Prace Mat. 12 (1968), p. 99-113.

[8] M. H. S tone, Notes on integration. 1, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), p. 336-342.

[9] M. W ilhelm , Integration of functions with values in a normed group, Bull. Acad. Polon.

Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), p. 911-916.

[10] — Real integrable spaces, Colloq. Math. 32 (1975), p. 233-248.

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