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 $M(X,E)$.

中文翻译：

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强Mackey拓扑和Radon矢量测度

假设$ X $是Hausdorff拓扑空间，而$ \ mathcal B o $是$ X $中Borel集的$ \ sigma $代数。设$ B（\ mathcal B o）$为$ X $上所有有界$ \ mathcal B o $可测量标量函数的空间，并配备Mackey拓扑$ \ tau（B（\ mathcal B o），M（ X））$，其中$ M（X）$表示$ X $上所有标量Radon量度的Banach空间。证明$（B（\ mathcal B o），\ tau（B（\ mathcal B o），M（X）））$是一个强Mackey空间。对于顺序完整的局部凸Hausdorff空间$（E，\ xi）$，令$ M（X，E）$表示所有Radon度量$ m：\ mathcal B o \ to E $的空间，并配备了拓扑$ \ mathcal T_s $ setwise收敛。证明了当且仅当$ \ mathcal R $一致地规则且对于每个$ A \ in \ mathcal时，$ M（X，E）$的子集$ \ mathcal R $相对$ \ mathal T_s $ -compact B o $，集合$ \ {m（A）：当且仅当相应积分运算符$ T_m：B的族$ \ {T_m：m \ in \ mathcal R \} $时，$ E $中的m \ in \ mathcal R \} $相对$ \ xi $-紧凑（\ mathcal B o）\ to E $是$（\ tau（B（\（mathcal B o），M（X）），\ xi）$）-连续的，对于B（\ mathcal B o）中的每个$ v \ $，$ E $中的集合$ \ {\ int_X v \，dm：m \ in \ mathcal R \} $相对紧凑。作为应用程序，我们获得了关于$ M（X，E）$中的按序顺序收敛的Nikodym定理和Dieudonné-Grothendieck型定理。