ABSTRACT
This chapter considers relations between abstract and topological measures and some applications. Utilizing the lattice properties of measures, a topological measurable space is associated with each such measure space through the Stone isomorphism theorem. The chapter includes an application to a Stone-Weierstrass theorem on some function spaces, and discusses the relations between the abstract and topological ideas of measures. A general method of connecting an abstract measure space with a topological one is by means of a measurable mapping of the first to the second with respect to its Borel (or Baire) σ-algebra. Through such a mapping, one induces an image measure, which turns out to be regular or of Radon type. The chapter presents theorems, proofs and lemmas for the lattice and homomorphism properties. It also includes multiple exercises that help students try themselves and perform complements.
