JournalsprimsVol. 1 , No. 1pp. 1–47

Measures on infinite dimensional vector spaces

  • Yasuo Umemura

    Kyoto University, Japan
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This paper consists of three chapters. In Chap. I, using a modification of Weil's theorem on invariant measures on groups, we shall give another proof of non-existence of translationally quasi-invariant measure on infinite dimensional vector spaces, which was firstly proved by Sudakov [6]. In Chap. II, we shall prove Minlos' theorem for nuclear spaces. The original Minlos' theorem [10] required more restrictions, but actually only the nuclearity condition is necessary. In Chap. Ill, we shall discuss infinite dimensional Gaussian measures, and prove that we can characterize a rotationally invariant measure as a superposition of Gaussian ones. For this fact, infinite dimensionality is essential. Some applications are stated in "Introduction".

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Yasuo Umemura, Measures on infinite dimensional vector spaces. Publ. Res. Inst. Math. Sci. 1 (1965), no. 1 pp. 1–47

DOI 10.2977/PRIMS/1195196433