On lattice-perfect measures

The general properties of lattice-perfect measures are discussed. The relationship between countable compactness and measure perfectness, and the relationship between lattice-measure tightness and lattice-measure perfectness are investigated and several applications in topological measure theory are given.

Some topologies on the set of lattice regular measures

We consider the general setting of A.D. Alexandroff, namely, an arbitrary setXand an arbitrary lattice of subsets ofX,ℒ.𝒜(ℒ)denotes the algebra of subsets ofXgenerated byℒandMR(ℒ)the set of all lattice regular, (finitely additive) measures on𝒜(ℒ).First, we investigate various topologies onMR(ℒ)and on various important subsets ofMR(ℒ), compare those topologies, and consider questions of measure repleteness whenever it is appropriate.Then, we consider the weak topology onMR(ℒ), mainly whenℒisδand normal, which is the usual Alexandroff framework. This more general setting enables us to extend various results related to the special case of Tychonoff spaces, lattices of zero sets, and Baire measures, and to develop a systematic procedure for obtaining various topological measure theory results on specific subsets ofMR(ℒ)in the weak topology withℒa particular topological lattice.

Fubini and martingale theorems in C* -algebras

The basic integration theory of Radon measures on locally compact spaces, as described in (5), has been developed in various directions both in commutative and non-commutative analysis. Thus if Ω is a compact Hausdorff space, and C(Ω) denotes the space of continuous complex-valued functions on Ω, Kaplan (16) showed how topological measure theory can be performed in the second dual C(Ω)** of C(Ω). Here as usual C(Ω)* is identified with the space of Radon measures on Ω by associating with a measure μ the linear functional φμ whereThus if f is a bounded real-valued function on Ω, there is an affine function f* defined on the set P(Ω) of Radon probability measures μ on Ω, byand f* extends by linearity to a functional in C(Ω)**. Conversely any function x on P(Ω) determines a function x0 on Ω. bywehere εω is the unit point of mass at ω.

Strict Topologies for Vector-Valued Functions

This paper is motivated by work in two fields, the theory of strict topologies and topological measure theory. In [1], R. C. Buck began the study of the strict topology for the algebra C*(S) of continuous, bounded real-valued functions on a locally compact Hausdorff space S and showed that the topological vector space C*(S) with the strict topology has many of the same topological vector space properties as C0(S), the sup norm algebra of continuous realvalued functions vanishing at infinity. Buck showed that as a class, the algebras C*(S) for S locally compact and C*(X), for X compact, were very much alike. Many papers on the strict topology for C*(S), where S is locally compact, followed Buck's; e.g., see [2; 3].