Integration With Respect to lcHs-valued Measures

Abstract

Let X be an lcHs and let m :P → X be σ-additive. The concepts of m-measurable functions and (KL) m-integrable functions given in Chapters 1 and 2 are suitably generalized here to lcHs-valued σ-additive measures. Theorem 4.1.4 below plays a key role in the subsequent theory of (KL) m-integrability. While (i)(iv) and (viii) of Theorem 2.1.5 of Chapter 2 are generalized in Theorem 4.1.8 to an arbitrary lcHs-valued σ-additive measure m on P, the remaining parts of Theorem 2.1.5, Theorem 2.1.7 and Corollaries 2.1.8, 2.1.9 and 2.1.10 of Chapter 2 are generalized in Theorems 4.1.9 and 4.1.11 and in Corollaries 4.1.12, 4.1.13 and 4.1.14, respectively, when X is quasicomplete. Finally, the above mentioned results are also generalized to σ(P)-measurable functions in Remark 4.1.15 when X is sequentially complete.