Extension: Examples of the Infinite-Dimensional Case

Abstract

In this chapter we consider to some extent the transfer of the material of Chapter 15 to the infinite-dimensional case, i.e. to the case in which infinitely many expectation values are specified. The two issues to be faced remain those of consistency and extension. These issues are too large for us to treat systematically, and we shall in fact consider only some particular cases of extension, of interest either in that they make a point or in that they have already loomed into sight. So, in Sections 2-4 we indicate how the expectation approach ties in with the usual one, based on the concepts of σ-fields of subsets of Ω. Interestingly, this idea of a σ-field generalizes to the more attractive concept of a linear lattice of r.v.s.