Remarks on Integrals

This is an introductory chapter to Part II, dealing with basic properties of integrals with respect to a positive, signed, and complex measure that will be required in the sequel. It does not yet deal with measures on topological spaces. We will not equip the underlying set X with a topology in this chapter, but we will do it from the next chapter onwards. However, to avoid trivialities, we assume that the underlying set X is nonempty. By a positive measure we simply mean a measure $$\mu: \mathcal{X}\!\rightarrow \mathbb{R}$$ on a $$\sigma$$ -algebra $$\mathcal{X}$$ of subsets of a nonempty set X (so that $$\mu (X) \geq 0$$ ). (Sometimes this is used to specify a nonzero measure; that is, a measure $$\mu$$ such that $$\mu (X)> 0$$ , but we allow the zero measure here.) The term positive measure is employed just to distinguish it from signed measure (also called real measure) and complex measure. In this section we summarize the basic properties of integrals with respect to a positive measure, as discussed in Chapters 3, 4, and 5. These basic properties will be extended to integrals with respect a signed measure and with respect to a complex measure in the forthcoming sections.