By Thiruvaiyaru V. Panchapagesan
This quantity is an intensive and accomplished treatise on vector measures. The features to be built-in will be both [0,infinity]- or actual- or complex-valued and the vector degree can take its values in arbitrary in the community convex Hausdorff areas. additionally, the area of the vector degree doesn't must be a sigma-algebra: it might probably even be a delta-ring. The publication includes not just a large number of new fabric but additionally corrects numerous mistakes in famous effects to be had within the literature. it is going to attract a large viewers of mathematical analysts.
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Extra resources for The Bartle-Dunford-Schwartz integral: integration with respect to a sigma-additive vector measure
In T and such that limn E sn dm = E f dm, E ∈ σ(P), the limit being uniform with respect to E ∈ σ(P). e. e. in T . See Remark 9 of [DP2] and [P5]. 9 holds for m-integrable vector functions f if and only if f ∈ L1 (m). See [P5]. 12. 3 of [L2] gives LDCT (with the hypothesis of (KL) mintegrability for the dominated σ(P)-measurable functions) for σ-additive vector measures μ deﬁned on a δ-ring τ of subsets of a set S with values in a sequentially complete lcHs E. 4 of [L2] which states as follows.
8. In Chapter 4 we study the (BDS)-integral of scalar functions with respect to a σ-additive quasicomplete (resp. 5 of Chapter 3 in Chapter 4. 2. (BDS) m-integrability 27 Before concluding the chapter we give some examples of σ-additive Banach space-valued measures deﬁned on a δ-ring of sets. 16 below. 1. Let m : P → X be σ-additive and let f : T → K or [−∞, ∞] be m-measurable. Then there exists N ∈ σ(P) with ||m||(N ) = 0 such that f χT \N is σ(P)-measurable. 5(i). ) n N (f )\N N (f )\N sn dm.
1. 5(v), bounded m-measurable functions are m-integrable in T . 1, without loss of generality we shall assume that all the functions fn and f are further S-measurable. Let E ∈ S and let limn E fn dm = xE ∈ X. 21. Therefore, f is (KL) m-integrable in T and hence m-integrable in T . Moreover, E f dm = xE = limn E fn dm for E ∈ σ(P). Let γ(E) = E f dm. 5(ii), γ n (·) = (·) fn dm, n ∈ N, are σ-additive on σ(P). Moreover, as observed above, limn γ n (E) = γ(E) for E ∈ σ(P). Hence by VHSN γ n , n ∈ N, are uniformly σ-additive on σ(P).
The Bartle-Dunford-Schwartz integral: integration with respect to a sigma-additive vector measure by Thiruvaiyaru V. Panchapagesan