Differentiable shifts for measures on infinite dimensional spaces
The purpose of this paper is to develop differentiation theory of measures on infinite dimensional vector space. Here, the following definition is adopted: Let X be a real vector space, the measure \(\mu\) is said to be differentiable for the shift a in X if: \(1/t(\mu_{ta}-\mu)\) (where \(\mu_{ta}\Doteq \mu (\cdot +ta))\) converges as \(t\to 0\) in \({\mathbb{R}}\), with respect to the so-called \(\tau_ s\)-topology, that is the one defined by the family of semi-norms \(\{| \mu (A)| /A\) measurable\(\}\). If the limit is denoted by \(\partial_ a\mu\), properties of the differentiable operator \(\partial\) are studied in a first chapter, for instance in relation with conditional and product measures. In a second chapter, the authors apply these theories to product measures on \({\mathbb{R}}^{\infty}\), equipped with the usual weak topology: characterization of \(R_ 0^{\infty}\)-differentiability is given and differentiability with respect to \(\ell^ 2\) is discussed in detail.
differentiation theory of measures on infinite dimensional vector space
scientific article; zbMATH DE number 4026127