Publisher Summary This chapter presents the use measures on infinite-dimensional products to prove representation theorems for positive functionals on some rather broad classes of linear topological algebras. It discusses the groups that are not locally compact. Such a group lacks the group algebra that frequently serves as a convenient tool of harmonic analysis; hence, the usual normed algebra techniques used in the study of locally compact groups are ineffective. Consequently, the use of commutative weakly closed operator algebras on Hilbert spaces must be resorted to. The chapter shows the connection between such algebras and multiplication algebras on localizable measure spaces, thus highlighting the properties of localizable measure spaces.