On the approximation of measurable linear functionals
Let \(\mu\) be a probability Radon measure on a locally convex space E. \(\mu\) is said to have the Riesz property if for every \(\mu\)-measurable linear functional f on E there exists a sequence of elements of E' converging to f in \(\mu\). It is proved that every Gaussian measure has the Riesz property. This extends the result of \textit{M. Kanter} [J. Funct. Anal. 9, 441-459 (1972; Zbl 0232.60023)] obtained in the symmetric case. It is also proved that a shift of the law on \(R^{\infty}\) of an infinite Bernoulli sequence has the Riesz property iff it is not square summable or belongs to \(R_ 0^{\infty}.\) In a forthcoming paper of the author [to appear in Bull. Pol. Acad. Sci., Math.] the following generalization of the results mentioned above is proved: A shift of a symmetric Radon probability measure with the Riesz property inherits this property iff it is not contained in \(J_{\mu}\setminus SJ_{\mu}\), \(where\) SJ\({}_{\mu}=\cap \{L:\mu (L)=1\) and L is a linear \(set\};\) J\({}_{\mu}=\cap \{L:\mu (L)=1\), L is a linear set and \(L=\cup^{\infty}_{n=1}K_ n\), \(K_ n\) convex, compact\(\}\).
scientific article; zbMATH DE number 3932076