Francisco J. Freniche

The Vitali-Hahn-Saks theorem for Boolean algebras with the subsequential interpolation property

It is shown that the Vitali-Hahn-Saks theorem holds for a new class of Boolean algebras which are defined by a separation property of its disjoint sequences: the Subsequential Interpolation Property. It is also proved that this property is strictly weaker than the Interpolation Property, the (f)Property and the Subsequential Completeness Property. In [2, p. 36] the problem is raised of finding an algebraic characterization of those Boolean algebras for which the Vitali-Hahn-Saks theorem holds. Some results in this direction are known: in [6 and 3] it is proved that this theorem is true for Boolean algebras with the Interpolation Property, in [5] the same result is proved for those which have the (f)-Property, and in [4] it is announced that the theorem is true for the class of Boolean algebras with the Subsequential Completeness Property. In this paper we define a new class of Boolean algebras (Definition 1), strictly containing the above-mentioned classes (Proposition 2 and Theorem 7), and such that the Vitali-Hahn-Saks theorem holds for measures on these Boolean algebras (Theorem 4). DEFINITION 1. Let A be a Boolean algebra. We say that A has the Subsequential Interpolation Property if, for every disjoint sequence (an)n(C in A and for every infinite M c w, there exist a E A and an infinite N C M such that an < a if n E N and an A a = 0 if n E w N. PROPOSITION 2. If a Boolean algebra A has one of the following properties: (1) Interpolatzon Property, (2) (f)-Property, (3) Subsequential Completeness Property, then A also has the Subsequential Interpolation Property. PROOF. We first suppose that A has the (f)-Property. Let (an)nEc be a disjoint sequence in A and M be an infinite subset of w. We apply (2) to the sequences (an)neM and (an)nCw-M, taking an infinite N, c M and b, c A such that an < b, if n c N, and an A bl = 0 if n e w M, and for each N C N,, aN E A such that an < aN if n E N and an A aN = 0 if n c N, N. We consider the sequences (an)nEM-Nl and (an)nEN,. By (2) we can choose an infinite N C N, and b2 c A such that an < b2 if n C N and an A b2 = 0 if Received by the editors June 27, 1983. 1980 Mathematics Subject Classification. Primary 28A33; Secondary 06E10, 28A60.

EMBEDDING C0 IN THE SPACE OF PETTIS INTEGRABLE FUNCTIONS

Abstract We show that the normed space of μ-measurable Pettis integrable functions on a probability space with values in a Banach space X contains a copy of the sequence space c0 if and only if X contains a copy of c0. In this case, if the probability μ has infinite range, a copy of c0 consisting of μ-measurable functions can be found, such that it is complemented in the bigger space of all weakly μ-measurable Pettis integrable functions.

TENSOR PRODUCTS AND OPERATORS IN SPACES OF ANALYTIC FUNCTIONS

Let X be an infinite dimensional Banach space. The paper proves the non-coincidence of the vector-valued Hardy space H p ([ ], X ) with neither the projective nor the injective tensor product of H p ([ ]) and X , for 1 p L p . A characterization is given of when every approximable operator from X into a Banach space of measurable functions [Fscr ]( S ) is representable by a function F : S → X as x [map ] 〈 F (·), x 〉. As a consequence the existence is proved of compact operators from X into H p ([ ]) (1 [les ] p < ∞) which are not representable. An analytic Pettis integrable function F :[ ] → X is constructed whose Poisson integral does not converge pointwise.

Correction to the paper “Embedding c 0 in the space of pettis integrable functions”

A complete argument is given for the proof of Theorem 5 in [F].