On the Radon-Nikodým property in Banach lattices

In his recent Memoirs, Talagrand showed that under Axiom L, the weak*-Radon-Nikodym property is equivalent to the Radon-Nikodym property in a Banach lattice. In this note we prove this result by a simple method without assuming Axiom L. In [3] Talagrand introduced the following notion of the weak*-Radon-Nikody'm property: DEFINITION 1 (TALAGRAND [3]). A Banach space E has the weak*-Radon-Nikod4m property if for every bounded linear operator T: L1[0, 1] -> E, there exists a Pettis integrable function 4: [0,1] E** such that T( f ) = Pettisff4 dX for everyf E L1[0, 1]. Let us recall the following DEFINITION 2. A Banach space E has the Radon-Nikodym property (resp., the weak-Radon-Nikodym property) if for every bounded linear operator T: L'[0, 1] -> E there exists a function 4: [0, 1] E-* such that 4 is Bochner integrable (resp., Pettis integrable) and T(F) = Bochnerf f 4 dX (resp., T(f) = Pettis-j f ) dX) for every f E L'[0, 1]. In [3, pp. 88-91] Talagrand showed that under Axiom L a Banach lattice E has the Radon-NikodSym property if and only if it has the weak*-Radon-Nikod'ym property. His proof, besides the fact that it uses Axiom L, is quite involved. (Axiom L states that [0,1] cannot be covered by less than the continuum of closed negligible sets.) Received by the editors October 15, 1984. 1980 Mathematics Subject Classification. Primary 46B22, 28B05, 46G10. C)1985 American Mathematical Society 0002-9939/85 $1.00 + $. 25 per page 736 This content downloaded from 157.55. 39.92 on Wed, 22 Jun 2016 06:46:24 UTC All use subject to http://about.jstor. org/terms ON THE RADON-NIKODYM PROPERTY 737 The following more general theorem is true and its proof is quite simple. THEOREM 3. For any Banach lattice E, the following statements are equivalent: (1) The space E has the Radon-Nikodym property. (2) The space E has the weak-Radon-Nikodym property. (3) The space E has the weak*-Radon-Nikodym property. PROOF. (1) is equivalent to (2) by Ghoussoub and Saab [1]. So it is enough to show that (3) -* (2). (3) implies that every vector measure of bounded variation with values in E has relatively compact range by a result of Stegall, and therefore co does not embed in E (see [1]). Hence E is weakly sequentially complete and therefore E is complemented in its bidual E** by a projection P: E** -E (see [2, p.34]). Let T: L1[0, 1] E-* be a bounded linear operator. By (3) there is a function f: [0,1] EF ** , Pettis integrable such that T( f ) = Pettisff d X for every f in L1[0, 1]. The function P.: [0, 1] -* E is Pettis integrable and T(F) = P(T(f )) = Pettis-| f P4, dX. So T has a Pettis kernel in E and therefore E has the weak-Radon-Nikod'ym property. REFERENCES 1. N. Ghoussoub and E. Saab, On the weak Radon-Nikodym property, Proc. Amer. Math. Soc. 81 (1981), 81-84. 2. J. Lindenstrauss and L Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 97, Springer-Verlag, Berlin, 1979. 3. M. Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc. No. 307 (1984). DEPARTMENT OF MATHEMATICS, UNI'VERSITY OF MISSOURI-COLUMBIA, COLUMBIA, MISSOURI 65211 This content downloaded from 157.55. 39.92 on Wed, 22 Jun 2016 06:46:24 UTC All use subject to http://about.jstor. org/terms