Integral representations of linear functionals on function modules (Q1100402)

Let X be a compact space, \(E_ x\) be a Banach space for each \(x\in X\) and E be a uniformly separable function module on X defined by spaces \(E_ x\) (that is closed linear subspace of \[ \{f\in \prod_{x\in X}E_ x:\sup_{x\in X}\| f(x)\| <\infty \} \] for which a) the functions \(x\to \| f(x)\|\) are upper semicontinuous for each \(f\in E\), b) \(\{f(x):f\in E\}=E_ x\), c) \(\alpha\) \(f\in E\) for each \(\alpha\in C(X)\) and \(f\in E\) and d) E containes a countable subspace F such that \(F_ x=\{f(x):f\in F\}\) is dense in \(E_ x\) for each \(x\in X)\). It is proved that for each \({\mathbb{R}}\)-valued bounded linear functional \(\phi\) on E there exists a regular Borel measure \(\mu_{\phi}\) on X and a family \(\xi_{\phi}(x)\) of linear functionals on \(E_ x\) of norm at most 1 such that for each \(f\in E\) the function \(x\to \xi_{\phi}(x)(f(x))\) is Borel-measurable and \[ \phi (f)=\int_{X}\xi_{\phi}(x)(f(x))d\mu_{\phi}(x) \] for each \(f\in E\). This result has been applied to spaces of (weighted) vector-valued functions and Grothendieck's G-spaces.