Denumerability conditions and Orlicz-Pettis type theoremsDedicated to Professor Wladyslaw Orlicz

ANNALES SOCIETATIS MATHEMATICAE POLONAE Seriez I : COMME NT ATIONES MATHEMATICAE X V III (1971) ROCZNIKI POLSK1EGO TOWARZYSTWA MATEMAT YCZNE G O

Séria I : PRACE MATEMAT Y CZNE XV III(1974)

Iwo L

(Poznan)

Denumerability conditions and Orlicz-Pettis type theorems

Dedicated to Professor Wladyslaw Orlicz on Ms 70th birthday

Abstract. In the present note we give some Orlicz-Pettis type theorems on measures with values in topological groups and vector spaces. The results on non- locally convex non-comparable topologies are obtained.

0. In this paper 0t is a <r-ring of sets, m an additive set function, N a set of natural numbers 1 2, . . . , G a commutative topological group.

If e is a set, we denote 0>(e) its power set. m : 01 -> G is said to be exhaustive if m ( E n)-* Q whenever E n are disjoint, if m is countably additive it is

called a measure.

Generally we will have several group topologies on Gr ; let a be one of them ; in that case instead of saying m : 0 —>(G, a) is a measure (resp.

exhaustive) we will say simply that m is an а-measure (resp. a-exhaustive).

Everywhere below theorems are formulated for set functions on 01 but in proofs M = 0>(N). This is obvious that we can do that because we can restrict ourselves to the u-ring generated by an (arbitrary) sequence (E n) of disjoint sets from and then the formula

(*) m {e ) = m ( E n), e € 0>(N)

nee *

defines an “isomorphic” measure on 0>(N). As we are interested in the

oo

behaviour of the se rie s^ m ( E n) only (for any sequence of disjoint sets n = ]

{En)) we do not loose any generality by taking 0t — 0>(N).

We denote by the a-field 0 {N) endowed with the Fréchet-NIkodym metric g(a, b) — v (aA b), v{a) = £ l/2?\ a e ^ ( N ) .

' nea

ZN is a compact metric space ; m : 0> (N) -> G is a measure iff m : £ N->G

OO

is continuous; £ xn subseries convergent in G iff m : 0>(N)->G (where

n= 1

ш on ^(iV) is defined by (*) taking m ( E n) = m ({n }) = xn) is a measure (cf. [4], n° 3).

1. We will say that the two topologies a and ft are consistent ([8]) with each other if, when xx and x 2 are any two distinct points, x x has an а-neighbourhood U1 and x 2 a /^-neighbourhood U2 such that Ux and

U2 are disjoint.

L

1. Let a, ft be two group topologies on G. They are consistent i f f y = inf (a, ft), a group topology, is H ausdorff.

Though the proof is easy, we prefer to give it here. Let ( U) be a basis of neighbourhoods of zero in (G, a), and ( V) in (G, ft). Then ( U + V ) is a basis of neighbourhoods of zero in y. Suppose a and ft consistent, y not Hausdorff. Let x Ф 0 be in the closure of zero. There exist U and V such that U n x + V = 0 . x e U — V so x = u — v, и = x-\-v hence u e U n x + V , a contradiction.

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2. Let (B n) be a sequence o f sequentially closed, symmetric subsets o f G which cover G, m : 0>(N)-> G a measure. Then:

(A) there is m e N such that m [éP(N)] cz Am + Am-f A m, where

m

i = 1

(B) there is k e N and n e N such that i f e c {1c, k + 1 , ...} , then m ( e ) e B n-\-Bn, that is, m [0>({k, fc-f-1, ...})] <= B n+ B n.

P ro o f, n i: T N 6r being continuous, the inverse image of B n by

OO

m , Sn say, is closed in 2 ^ and ( J £ n = ZN. By the Baire category theorem

n — 1

there is n e N and an open ball K °(a 0,r ) in such that m [K °(a0, r)]

c= B n. We can suppose that a0 is a finite subset of N (see [6], Theorem 1, for details). The definition of the metric in ZN implies the existence of k e N such th at к > т п х {п : n e a 0] and v({к, к + 1 , . . . } ) < r. Thus a 0u a e K ° (a 0, r) for each a cz [к, к + 1 , ...} , a 0e K ° (a 0, r) and we have therefore B n * m ( a 0u a ) — m ( a 0) + m (a) => rn (a)e B n A-Bn => ш\_^({к, к -f 1, ...} )] cz

cz B n Jr B n. This proves (B).

N \ {k , k + 1, ...} is a finite set so there is m eN , m > n, such that m \ f?{N \ {k, k + 1, ...} ) ] cz A m. Let e be an arbitrary subset of iV; then e = exu e 2, where ex c N \ {k, k + 1, e2 cz {к, к + 1, . ..} ; hence m (e)

= m ( e x)-\-m(e2)e A m-{-Am-\-Am (since B m cz A m) which completes the proof of (A).

B e m a r k . If 6? is a non-commutative group (with multiplication as a group operation), then the analogon of the subseries convergence is

OO

still meaningful. The product } ] xn is said to be subproduct convergent

n = l

Orlicz-Pettis type theorems

if for every increasing sequence (k{) c N lim [ ] xk. exists. Put n(e) = f ] xk. ,

• m i = 1 i 1

where кг, к г, . . . is the arrangement of elements of e in the increasing order, e c A ; n { 0 ) — the unit of G. One can prove (cf. [1]) that the “non- commutative measure” n is continuous on UN, and of course the continuity

00

of n on Рту implies the subproduct convergence of [ f xn.

n = l

The lemma above is still valid in this more general setting (the proof remains the same).

2. Parts (A) and (B) of Lemma 2 show us two aspects of the same thing. In some situations G can be covered by a countable number of arbitrarily small “neighbourhoods”, then the inclusion in (B) implies the subseries convergence in a corresponding topology. This point of view was studied by Drewnowski [1] while giving the direct proofs of Kalton’s theorems. Some consequences of (A) will be presented now.

T

1. Let (G, fi) be a group which is a union o f a countable number o f symmetric compact sets K n {in particular a countable at infinity locally compact group), and a an arbitrary group topology on G consistent with ($.

I f m : -» G is an a-m easure, then it is a /5-measure.

P ro o f. B y Lemma 1 y = inf (a, /1) is Hausdorff ; of course m is a y- measure and we can suppose K n to be increasing. K n are у-closed since y being Hausdorff coincides with /9 on K n. B y Lemma 2 m [^ (jV )] с V

= K m + K m^rKm. As y — fi on V the result follows.

R e m a rk s . 1. If G is non-commutative the theorem remains valid for subproduct convergence (y = inf (a, (3) is still Hausdorff, so in view of the remark after Lemma 2 the same proof holds).

2. A separable locally compact group G is necessarily countable at infinity, for let (xn) be a dense sequence in G and V a compact (symmetric)

00

neighbourhood of zero in G; then G = xn + V.

71=1

A locally convex Hausdorff (topological vector) space is said to be x-space if it can be represented as a union of countably many weakly compact subsets K nC). A closed vector subspace, the quotient (if it is Hausdorff), a countable inductive limit (if it is Hausdorff) of ^-spaces are ^-spaces. The class of ^-spaces in “usual cases” reduces to the strong duals of reflexive i^-spaces (see [3]), hence usually they are Incom plete exhaustive ([2]) thus the result below is partially covered by [2], 2.9.

(1) We can always suppose that K n <= K n+1, n e N , and are balanced, since the finite union and balanced cover of a compact subset of a linear Hausdorff topo­

logical vector space are again compact.

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2. Let X be a x-space, a a linear topology on X such that K n, n e N , are а -closed. I f m : f f l X is an а -measure, then it is exhaustive (in the initial topology o f X).

P r o o f. Denote by Sf a set of finite elements of gP(N). B y Lemma 2 there is n e N snch that m [^ ( N )] <= K n + K n + K n; this set being weakly compact, is relatively weakly compact hence ([7], Theorem 2) the

oo

series m ({n }) is weakly snbseries convergent thus snbseries convergent

71 = ]

by the Orlicz-Pettis theorem. This implies m ({ n }) 0 (n -» oo) and ends the proof.

R e m a rk s . 1. The analogon of the theorem above is valid for a topo­

logical group which is a union of a countable number of symmetric compact sets K n, and which has no non-trivial compact subgroup. This is a conse­

quence of [7], Proposition 3.

2. Let us still note that in Theorem 2 a is not assumed to be comparable with some topology compatible with the dual pair (X, X') so we do not

OO

know whether m ({n }) — m (N ).

n = l

3. In particular, a can be consistent with the weak topology of X (then K n are automatically a-closed [8]), then there exists y —

— inf (a, cr(X, X')) therefore in that case, as in Theorem 1, we can state that m is a measure.

4. The theorem could be formulated for a topology a compatible with the additive structure of X only, but I do not know any example of such a topology on a %-space which fulfils the assumptions of the theorem and is not linear.

5. A good example of a is the topology of convergence in measure on L p spaces for 1 < p < oo (see, however, [5]).

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3. Let X be a vector space, a and (5 linear topologies on X such that (3 has a basis o f a-sequentially closed neighbourhoods o f zero. I f m : £?(N) -> X is an a-measure, then its range m [^a(iV)] is (3-bounded.

P r o o f. We can suppose that this basis is formed by symmetric neighbourhoods. L et U be an arbitrary ^-neighbourhood of zero, and V such that V + V -f V c U and V is а-sequentially closed. We have

oo oo

A = U w 7 (= ( J Vn)- By Lemma 2 there is m e N such that m [^ (N )]

71 = 1 71 = 1

c m V A-mV + m 7 = m (P + V + V) <=. m V which implies the result.

As an immediate consequence of the preceding theorem, we have the following generalization of [5], Theorem 1.

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4. Let X be a vector space ; a, (3 H ausdorff linear topologies

on X such that (X , (3) is an (0)-space and (3 has a basis o f a-sequentially

Orlicz-Pettis type theorems

closed neighbourhoods at 0. I f m : M X is an a-measure, then it is a fi- measure. Moreover, i f a is metrizable, a-exhaustivity o f m im plies its ft- exhaustivity.

P ro o f. B y the theorem above m is /^-bounded (on 0>(N)) thus the proof is the same as in [5] (we remark that it is sufficient to suppose the basis of (3 to be formed by the a-sequentially closed neighbourhoods since we are working with Cauchy sequences only).

Theorem 1 of [5] was used to prove the fact that in L p, 0 < p < oo, over an arbitrary positive measure space (T, 38, v) the topology со of convergence in r-measure on every set of finite v-measure and the norm topology are ^-equivalent ([6]). We could not establish in [5] this result for generalized zl2 Orlicz spaces L *9 since they are not pseudo-convex in general. Consequently, we did not know whether m [ 0>(N)] is bounded in L *9 provided m is an co-measure. Now we can state:

T

5. Let L *9 be generalized Л% Orlicz space over (T , v).

m : & - + L *9 is a measure i f f it is co-measure. Moreover, i f v is a-finite, co-exhaustivity o f m im plies its exhaustivity.

References

[1] L. D rew n ow sk i, On the Orlicz-Pettis type theorems of Kalton, Bull. Acad. Polon.

Sci., Sér. Sci. math. astr. phys. 21 (6) (1973), p. 51 5 -5 1 8 .

[2] — et I. L a b u d a , S ur quelques théorèmes du type d ’Orlicz-Pettis I I , ibidem 21(2) (1973), p. 119-126.

[3] D. J . H. G arlin g , Locally convex spaces with denumerable systems of wealdy compact subsets, Proc. Cambridge Phil. Soc. 60 (4) (1964), p. 813-815.

[4] I. L a b u d a , S u r quelques généralisations des théorèmes de Nikodym et de Vitali- -H ahn-Salcs, Bull. Acad. Polon. Sci., Sér. Sci. math. astr. phys. 20 (6) (1972), p. 447-456.

[5] — S u r quelques théorèmes du type d ’Orlicz-Pettis I , ibidem 21 (2) (1973), p. 127- 132.

[6] — S u r quelques théorèmes du type d ’Orlicz-Pettis I I I , ibidem 21(7)(1973), p. 599-606.

[7] A. P. R o b e rts o n , On unconditional convergence in topological vector spaces, Proc.

Royal Soc. of Edinburgh 68 (2) (1969), p. 145-157.

[8] J . D. W e sto n , On the comparison of topologies, J . London Math. Soc. 32 (3) (1957), p. 342-354.

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i — R oczniki PTM — P r a c e M atem atyczn e X V in .