Basic Properties of the Bartle-Dunford-Schwartz Integral
In [L1,L2] Lewis studied a Pettis type weak integral of scalar functions with respect to an lcHs-valued σ-additive vector measure. As noted on p. 27 of [KK], this type of definition has also been considered by Kluvánek in [K2]. In honor of these mathematicians, we call the integral introduced in [L1, L2] the Kluvánek-Lewis integral or briefly, the (KL)-integral. For a Banach space-valued σ-additive vector measure m defined on a δ-ring P of sets, we define the (KL) m-integrability and the (KL) m-integral of a scalar function f, study the basic properties of the (KL) m-integral and show that the Lebesgue dominated convergence theorem (briefly, LDCT) is valid for the integral. When P is a σ-ring S, the Lebesgue bounded convergence theorem (briefly, LBCT) is also valid. The reader can note that the version of LDCT given here is much stronger than the Banach space versions of Theorem 2.2(2) of [L1] and of Theorem 3.3 of [L2] (whose proof is incorrect see Remark 2.2.12 below) and of Theorem II.5.2 of [KK]. The version of LDCT given in [Ri5] is the same as ours except that the domain of m in [Ri5] is a σ-algebra Σ and the functions considered there are Σ-measurable. (See Remark 2.1.12.) In Chapter 4 we define (KL) m-integrability in T of m-measurable functions (resp. σ(P)- measurable functions) when m is a quasicomplete (resp. sequentially complete) lcHs-valued σ-additive measure on P and the arguments and results of this section play a key role in generalizing them in Chapter 4.
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