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...

This work is devoted to the concept of statistical solution of the Navier-Stokes equations, proposed as a rigorous mathematical object to address the fundamental concept of ensemble average used in the study of the conventional theory of fully developed turbulence. Two types of statistical solutions have been proposed in the 1970’s, one by Foias and Prodi and the other one by Vishik and Fursikov. In this article, a new, intermediate type of statistical solution is introduced and studied. This solution...

The first part of the paper presents results on Gaussian measures supported by general Banach sequence spaces and by particular spaces of Besov-Orlicz type. In the second part, a new constructive isomorphism between the just mentioned sequence spaces and corresponding function spaces is established. Consequently, some results on the support function spaces for the Gaussian measure corresponding to the fractional Brownian motion are proved. Next, an application to stochastic equations is given. The...

Nous donnons des conditions permettant de vérifier que l’image d’une mesure cylindrique $\mu$ sur un espace vectoriel topologique $E$, par une application linéaire continue dans un autre espace vectoriel topologique $F$, est une mesure de Randon. Dans une première partie, nous donnons des résultats généraux qui portent, soit sur des propriétés géométriques de l’espace $F$, soit sur la mesure cylindrique $\mu$. Dans une seconde partie, nous donnons des conditions plus précises quand $\mu$ est une mesure cylindrique...

The main concern of this paper is to present some improvements to results on the existence or non-existence of countably additive Borel measures that are not Radon measures on Banach spaces taken with their weak topologies, on the standard axioms (ZFC) of set-theory. However, to put the results in perspective we shall need to say something about consistency results concerning measurable cardinals.

For a Banach space $E$ and a probability space $(X,𝒜,\lambda )$, a new proof is given that a measure $\mu :𝒜\to E$, with $\mu \ll \lambda$, has RN derivative with respect to $\lambda$ iff there is a compact or a weakly compact $C\subset E$ such that ${\left|\mu \right|}_C:𝒜\to [0,\infty ]$ is a finite valued countably additive measure. Here we define ${\left|\mu \right|}_C\left(A\right)=sup\{{\sum }_k|\langle \mu \left(A_k\right),f_k\rangle \left|\right\}$ where $\left\{A_k\right\}$ is a finite disjoint collection of elements from $𝒜$, each contained in $A$, and $\left\{f_k\right\}\subset E^{\text{'}}$ satisfies ${sup}_k\left|f_k\left(C\right)\right|\le 1$. Then the result is extended to the case when $E$ is a Frechet space.