Measures on infinite dimensional vector spaces

Abstract

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".