Helmut H. Schaefer

On The Vitali-Hahn-Saks Theorem

In this paper we generalize the classical Vitali-Hahn-Saks theorem to sets of countably additive vector measures which are compact in the strong operator topology. The main result asserts that a set of countably additive vector measures which is compact in the strong operator topology is uniformly countably additive. We accomplish this by first studying the properties of linear operators from Y *, the dual of a Banach space Y, into a Banach space X which are continuous with respect to the Mackey topology τ(Y *,Y) on Y* and the norm topology on X, and then applying the results to the special case where Y = L 1(μ) and Y * = L ∞(μ). Other related results on vector measures are also included.

Extension properties of order continuous functionals and applications to the theory of Banach lattices

Abstract Let E be a Banach lattice and E 0 an ideal of E . Let f be a positive norm-bounded order continuous functional on E 0 . Then f has a norm preserving positive order continuous extension to E . Moreover, there exists a minimum extension among all such extensions. This result enables us to obtain an isometric lattice homomorphism from ( E 0 ) ∗ n into E ∗ n . As an application, we prove that if F is a Banach lattice and if E 0 is the ideal generated by F in E (= F ∗∗ ), then ( E 0 ) ∗ n can be identified with F ∗ in the canonical sense. Certain extension properties of order continuous functionals thus follow. Banach lattices with strictly monotone norm are also investigated.

Dixmier’s theorem for sequentially order continuous Baire measures on compact spaces

We prove that a Baire measure (or a regular Borel measure) on a compact Hausdorff space is sequentially order continuous as a linear functional on the Banach space of all continuous functions if and only if it vanishes on meager Baire subsets, a result parallel to a much earlier theorem of Dixmier. We also give some results on the relation between sequentially order continuous measures on compact spaces and countably additive measures on Boolean algebras.

Order Continuity of Locally Compact Boolean Algebras

The main purpose of this paper is to exhibit the decisive role that order continuity plays in the structure of locally compact Boolean algebras as well as in that of atomic topological Boolean algebras. We prove that the following three conditions are equivalent for a topological Boolean algebra B: (1) B is compact; (2) B is locally compact, Boolean complete, order continuous; (3) B is Boolean complete, atomic and order continuous. Note that under the discrete topology any Boolean algebra is locally compact.