The Bartle-Dunford-Schwartz Integral pp 65-116 | Cite as

# 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.

## Keywords

Scalar Function Banach Lattice Vector Measure Convex Space Relative Topology## Preview

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