A N N A L E S S O C I E T A T I S M A T H E M A T I C A E P O L O N A E S e r i e s I : C O M M E N T A T I O N E S M A T H E M A T I C A E X Y I I ( 1 9 7 3 )
R O C Z N I K I P O L S K I E G O T O W A R Z Y S T W A M A T E M A T Y C Z N E G O S é r i a I : P R A C E M A T E M A T Y C Z N E X Y I I ( 1 9 7 3 )
J. Di e s t e l (Kent, Ohio)
On the essential uniqueness oî an example oî С. E. Rickart
The notion of strong boundedness for a finitely additive vectorvalued set function was introduced by С. E. Bickart in [3]; the finitely additive [л: 27 -» X (27 an algebra, X a Banach space) is strongly bounded whenever \\/л{Ап)\\ -> 0 for any pairwise disjoint sequence (A n) of members of 27. This notion, which simultaneously generalizes both countable ad
ditivity and finiteness of variation (and for which much of the structure of these notions extends), was shown by Bickart to have been (in general, strictly) stronger than the notion of boundedness for a finitely addi
tive map /л: 27 -> X. The reader is referred to [2] for appropriate references.
In [2], it was shown that whenever (a) X is separable and 27 is a <r-alge- bra or (b) X does not contain c0, then every bounded, finitely-additive X -valued map [л in 27 is strongly bounded. It was also shown that Bickart’s counterexample to the implication “bounded implies strongly bounded”
possessed some particularly singular features ; because we propose to show that Bickart’s counterexample is “essentially” the only counterexample we repeat the main features of that example: 27 = algebra generated ЬУ {[0, i) , [ b f h •••}> x = £ o o [ 0 , l ] , and 27-» X is given by p{A)
= cA — characteristic function of A e 27. It was pointed out in [2] that /л’’s range has a closed linear span isomorphic to c0; in fact, the sequence ц[0, ! ) , / / [ ! , f), ... serves as a basis for this space equivalent to the unit vector basis of c0.
Kow let [л: 27 -> X be a bounded, finitely additive but non-strongly bounded map. Let x'e X' ( =dual of X). Then given any pairwise disjoint sequence (A n) of members of 27 we have Xn\x'([л{Ап))\ < oo {x'• [л is bounded, finitely additive, scalar valued hence of finite variation), whereby f*{An) -> 0 weakly. On the other hand, there exists a pairwise disjoint sequence (A n) in 27 with ||^(J.J|| ^>0; we can assume 0 < <5 < ||^(J-„)t|
< M < oo for all n (by choosing an appropriate subsequence). Thus, we have [л(Ап) ^ 0 weakly yet inf \\[л(Ап)^ > d > 0, and the selection principle of Bessaga and Pelczyhski [1] yields a basic subsequence which we still denote by ([л{Ап]). But now we have infJ|/j(J.n)|| > 0 yet
264 J . D i e s t e l
27п|ж'(^(Ап))| < oo for each x' eX' . Thus by Theorem 5 of [1], the basic sequence (/i { An)) is equivalent to the unit vector basis in c0. This completes our “proof” of the essential uniqueness of Rickart’s counterexample.
A d d e d i n p ro of . The author and B. Faires have recently shown that if E is a sigma-algebra and there exists /л: E -> X which is bounded, additive but non-strongly bounded, then X contains a copy of L^ [0, 1].
References
[1] C. B e s s a g a and A. P e l c z y n s k i , On bases an d unconditional convergence o f series in B anach spaces, Studia Math. 17 (1958), p. 151-164.
[2] J. D i e s t e l , A p p lica tio n s o f weak compactness and bases to vectorm easures and vectorial integration, R evue Roum. Math, (to appear).
[3] С. E. R ic k a r t , D ecompositions o f additive set fun ctio n s, Duke Math. J. 10 (1943), p. 653-665.
K E N T S T A T E U N IV E R S IT Y K E N T , O H IO
