In this paper we continue the study of the concepts of pairwise Borel and Baire measures in a bispace, recently introduced in [10]. We investigate some of its consequences including the problem of a pairwise regular Borel extension of a pairwise Baire measure.

For $i=(1,2)$, let $X_i$ be completely regular Hausdorff spaces, $E_i$ quasi-complete locally convex spaces, $E=E_1\breve{\otimes }{E}_2$, the completion of the their injective tensor product, $C_b\left(X_i\right)$ the spaces of all bounded, scalar-valued continuous functions on $X_i$, and ${\mu }_i$$E_i$-valued Baire measures on $X_i$. Under certain...

We investigate the products of topological measure spaces, discussing conditions under which all open sets will be measurable for the simple completed product measure, and under which the product of completion regular measures will be completion regular. In passing, we describe a new class of spaces on which all completion regular Borel probability measures are τ-additive, and which have other interesting properties.

A necessary and sufficient condition for the existence of the projective limit of measures with values in a locally convex space is given. A similar theorem for measures with values in different locally convex spaces (under certain conditions) is given too (in this case, the projective limit is valued in the projective limit of these spaces). Finally, a result about the projective limit of vector measures is stated.

We investigate properties of the class of compact spaces on which every regular Borel measure is separable. This class will be referred to as MS. We discuss some closure properties of MS, and show that some simply defined compact spaces, such as compact ordered spaces or compact scattered spaces, are in MS. Most of the basic theory for regular measures is true just in ZFC. On the other hand, the existence of a compact ordered scattered space which carries a non-separable (non-regular) Borel measure...