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 . In Chap. II, we shall prove Minlos' theorem for nuclear spaces. The original Minlos' theorem  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".
Cite this article
Yasuo Umemura, Measures on infinite dimensional vector spaces. Publ. Res. Inst. Math. Sci. 1 (1965), no. 1, pp. 1–47DOI 10.2977/PRIMS/1195196433