Properties of the Integral

This chapter demonstrates that the integral is additive with respect to disjoint domains, that it depends continuously on the domain of integration, that the contribution from negligible sets is zero, and that it commutes with continuous linear mappings and differentiation. It shows that the domain where the function vanishes does not contribute to its integral. The chapter also shows that the integral is additive with respect to disjoint measurable sets. It focuses on continuity with respect to the domain of integration. The chapter states some of the consequences, starting with the continuity of the integral with respect to increasing or decreasing sequences of the domain of integration. It defines the image of a measure under a linear mapping and then shows that the image of an integrable measure under a continuous linear mapping is integrable.