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 is quite a subtle topic. A naive approach ...