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On integrable functions in complete bornological locally convex spaces. (English) Zbl 1246.46043

This paper is a continuation of the first author’s paper: [“On integration in complete bornological locally convex spaces”, Czech. Math. J. 47, No. 2, 205–219 (1997; Zbl 0926.46037)]. Here, \(X, Y\) are Hausdorff complete bornological locally convex spaces with filtering upwards bases of bornologies \(\mathcal{U}, \mathcal{W}\) respectively; here each \( U \in \mathcal{U}\) is a closed, absolutely convex bounded subset of \(X\) and \( U \supset U_{0}\), a fixed closed, absolutely convex bounded subset of \(X\). The subspace of \(X\) generated by \(U\), with Minkowski functional of \(U\), is a Banach space \(X_{U}\). The topology of \(X\) is the inductive limit topology of the Banach spaces \(\{ X_{U}: U \in \mathcal{U} \}\); similar properties for the topology of \(Y\), arising from \(\mathcal{W}\). \(L(X, Y)\) is the space of all linear continuous functions from \(X\) to \(Y\). The study of measures is kind of reduced to the Banach spaces \(X_{U}, Y_{W}\).
\(T\) is a set, \(\Delta\) is a \(\delta\)-ring of subsets of \(T\), and \(m: \Delta \to L(X, Y)\) is a finitely additive measure. Starting with \((U, W) \in \mathcal{U} \times \mathcal{W}\), the \((U, W)\)-semivariation \(\Hat{m}_{U, W}\) of \(m\) is defined as: \(\Hat{m}_{U, W}(E)= \sup p_{W} (\sum_{i} m(E \cap E_{i}) x_{i}\) (here \(p_{W}\) is the Minkowski functional for \(W\), \(\{ x_{i} \}\) is a finite collection of elements from \(U\), \(\{ E_{i} \}\) a mutually disjoint collection from \(\Delta\), and \(E\) is in the \(\sigma\)-algebra generated by \(\Delta\)).
On \(\Phi = \{ \varphi: \mathcal{U} \to \mathcal{W} \}\), the order is defined as: \( \varphi_{1} \leq \varphi_{2}\) if \( \varphi_{1}(U) \subset \varphi_{2}(U) \; \forall U \in \mathcal{U}\). With the help of these, the authors define \(\sigma_{\varphi}\)-finite \((\mathcal{U}, \mathcal{W})\)-variation, \(\sigma_{\varphi}\)-additivity, and \(\Hat{m}_{\mathcal{U}, \mathcal{W}}\)-a.e. of the measure \(m\). For \((U, W) \in (\mathcal{U}, \mathcal{W})\), first the integration of \(X\)-valued, \(\Delta_{U, W}\)-simple functions is defined. Then, using \(\{ \Hat{m}_{U, W}: U \in \mathcal{U}, W \in \mathcal{W} \) }, \(\Phi\), and sequences of simple functions, \(\Delta_{(\mathcal{U}, \mathcal{W})}\)-integrable functions are defined. This is denoted by \(\mathcal{I}_{\mathcal{U}, \mathcal{W}, \Delta}\). Many results about \(\mathcal{I}_{\mathcal{U}, \mathcal{W}, \Delta}\) are proved.

MSC:

46G10 Vector-valued measures and integration
28B05 Vector-valued set functions, measures and integrals

Citations:

Zbl 0926.46037
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References:

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