On quasi-invariant measures in topological vector spaces
On an infinite-dimensional locally convex topological vector space \(X\) over \(\mathbb{R}\) there exists no \(\sigma\)-finite Borel measure which is quasi-invariant with respect to translations by the elements of \(X\). This theorem was proved by \textit{V. N. Sudakov} [Dokl. Akad. Nauk SSSR 127, 524-525 (1959; Zbl 0100.11001)], and some special case is discussed in Chap. IV, Sect. 5, of \textit{I. M. Gel'fand} and \textit{N. Ya. Vilenkin} [`` Applications of Harmonic Analysis'', Academic Press (1964; Zbl 0136.11201)]. Another proof was also given by \textit{Y. Umemura} [Publ. Res. Inst. Math. Sci., Kyoto Univ., Ser. A 1, 1-47 (1965; Zbl 0181.41502)]. The note under review concerns this theorem.
scientific article; zbMATH DE number 1266842