Measures on the product of compact spaces (Q1849693)

In his paper ``Remarks on a theorem of Taskinen on spaces of continuous functions'' [Math. Nachr. 250, 98-103 (2003; Zbl 1026.47023)], the author proves the following result: Let \(K\) be a metrizable, uncountable compact space, \(X\) a Banach space and \( n \in {\mathbb N}\). Then there exist isomorphic injections \(A_{m} : C(K) \to C(K)\) \((1 \leq m \leq n)\) such that for every operator \( T: {\widehat \otimes}_{\varepsilon}^{n} C(K) \to X \), there exists an operator \(B: C(K) \to X \) such that \[ T(f_{1} \otimes f_{2} \dots \otimes f_{n}) =B\Biggl( \prod_{m=1}^{n} A_{m}(f_{m}) \Biggr). \] In this paper, two measure-theoretic versions of the above result are proved. For a compact Hausdorff space \(K\), \( \Sigma_{K} \) denotes the sigma-algebra of all the Borel subsets of \(K\) and \( B(\Sigma_{K})\) stands for all bounded scalar-valued Borel measurable functions on \(K\). For a Banach space \(X\), \(X^{*}\) and \(X^{**}\) denote its topological dual and bidual, respectively. The two main results are: I. Let \(K\) be a metrizable, uncountable compact space, \(X\) a Banach space and \( n \in {\mathbb N}\). The there exist \(n\) linear and multiplicative isometric inclusions \(I_{m} : B(\Sigma_{K}) \to B(\Sigma_{K})\) \((1 \leq m \leq n)\) (taking characteristic functions to characteristic functions) such that for every operator \( T: {\widehat \otimes}_{\varepsilon}^{n} C(K) \to X \) with associated measure \( \mu : \Sigma_{K^{n}} \to X^{**}\) there exists an operator \( B: C(K) \to X \) with representing measure \( \mu_{B} : \Sigma_{K} \to X^{**}\) satisfying the relation \[ \mu(A_{1} \times A_{2} \dots \times A_{n}) = \mu_{B}\Biggl( \bigcap_{m=1}^{n} I_{m}( \chi_{A_{m}})\Biggr). \] Moreover, \(\mu\) is \(X\)-valued if and only if \( \mu_{B}\) is \(X\)-valued. II. Let \(K\) be a metrizable, uncountable compact space, \(X\) a Banach space and \( n \in {\mathbb N}\). Then there exist \(n\) different linear and multiplicative isometric inclusions \(I_{m} : B(\Sigma_{K}) \to B(\Sigma_{K})\) \((1 \leq m \leq n)\) (taking characteristic functions to characteristic functions) such that for every operator \( T: \Sigma_{K^{n}} \to X \) there exists an operator \( B: B(\Sigma_{K}) \to X^{**}\) so that for any \( g_{1} \otimes g_{2} \dots \otimes g_{n} \in {\widehat \otimes}_{\varepsilon}^{n}B(\Sigma_{K}) \subset B(\Sigma_{K^{n}}) \), one gets \[ T(g_{1} \otimes g_{2} \dots \otimes g_{n})=B\left(\prod_{m=1}^{n} I_{m} (g_{n})\right). \] If \(T\) is weakly compact then \(B\) is also weakly compact and \(X\)-valued.