5 - Measures

Summary

The first section of this chapter is devoted to a review of basic definitions of measure theory. Among other topics, we recall basic properties of positivity preserving operators, which provide tools useful in constructive quantum field theory.

The rest of this chapter is devoted to measures on infinite-dimensional Hilbert spaces. It is well known that there are no Borel translation invariant measures on infinite-dimensional vector spaces. However, one can define useful measures on such spaces which are not translation invariant. In particular, the notion of a Gaussian measure has a natural generalization to the infinite-dimensional case.

Measures on an infinite-dimensional Hilbert space X is quite a subtle topic. A naive approach to this subject leads to the notion of a weak distribution, which is a family of measures on finite-dimensional subspaces satisfying a natural compatibility condition. It is natural to ask whether a weak distribution is generated by a measure on X. In general, the answer is negative. In order to obtain such a measure, one has to consider a larger measure space containing X. Many choices of such a larger space are possible. A class of such choices that we describe in detail are Hilbert spaces BX for a self-adjoint operator B satisfying certain conditions.

Measures on Hilbert spaces play an important role in probability theory and quantum field theory. One of them is the Wiener measure, used to describe Brownian motion.