Strongly Mackey topologies and Radon vector measures
Summary: Let \(X\) be a topological Hausdorff space and \(\mathcal{B}o\) be the \(\sigma\)-algebra of Borel sets in \(X\). Let \(B(\mathcal{B}o)\) be the space of all bounded \(\mathcal{B}o\)-measurable scalar functions on \(X\), equipped with the Mackey topology \(\tau(B(\mathcal{B}o),M(X))\), where \(M(X)\) denotes the Banach space of all scalar Radon measures on \(X\). It is proved that \((B(\mathcal{B}o),\tau(B(\mathcal{B}o),M(X)))\) is a strongly Mackey space. For a sequentially complete locally convex Hausdorff space \((E,\xi)\), let \(M(X,E)\) denote the space of all Radon measures \(m:\mathcal{B}o\to E\), equipped with the topology \(\mathcal{T}_s\) of setwise convergence. It is proved that a subset \(\mathcal{R}\) of \(M(X,E)\) is relatively \(\mathcal{T}_s\)-compact if and only if \(\mathcal{R}\) is uniformly regular and for each \(A\in\mathcal{B}o\), the set \(\{m(A):m\in\mathcal{R}\}\) in \(E\) is relatively \(\xi\)-compact, if and only if the family \(\{T_m:m\in\mathcal{R}\}\) of corresponding integration operators \(T_m:B(\mathcal{B}o)\to E\) is \((\tau(B(\mathcal{B}o),M(X)),\xi)\)-equicontinuous and for each \(v\in B(\mathcal{B}o)\), the set \(\{\int_X v\,dm:m\in\mathcal{R}\}\) in \(E\) is relatively \(\xi\)-compact. As an application, we get a Nikodym theorem and a Dieudonné-Grothendieck type theorem on the setwise sequential convergence in \(M(X,E)\).
scientific article; zbMATH DE number 7376802
scientific article