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On quasi-invariant measures in topological vector spaces. (French) Zbl 0935.46047
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 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 I. M. Gel’fand and N. Ya. Vilenkin [“ Applications of Harmonic Analysis”, Academic Press (1964; Zbl 0136.11201)]. Another proof was also given by 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.
MSC:
46G12 Measures and integration on abstract linear spaces
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
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