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On weakly convergent nets in spaces of non-negative measures. (English) Zbl 0727.28011

Let \({\mathcal M}^+({\mathcal E})\) be the set of all nonnegative finite measures defined on the \(\sigma\)-algebra \({\mathcal E}\) of Borel subsets of a topological space X. (Actually the setting may be more general: \({\mathcal E}\) is the \(\sigma\)-algebra generated by a nonempty family of “open” sets, which is closed under the formation of finite unions and intersections.) Denote by \({\mathcal M}^+({\mathcal E},{\mathcal K})\) the subset of \({\mathcal M}^+({\mathcal E})\) consisting of all measures regular with respect to compact sets. The author proves several characterizations of the following notion of weak convergence: A net \(\{\mu_{\alpha}\}\subset {\mathcal M}^+({\mathcal E})\) weakly converges to a \(\mu \in {\mathcal M}^+({\mathcal E})\) iff \(\lim \mu_{\alpha}X=\mu X\) and \(\liminf \mu_{\alpha}G\geq \mu G\) for each open \(G\subset X.\) The paper contains also many other results concerning properties of nets in \({\mathcal M}^+({\mathcal E})\) and \({\mathcal M}^+({\mathcal E},{\mathcal K}).\) In particular, the reader can find necessary and sufficient conditions for a weak convergent net \(\{\mu_{\alpha}\}\subset {\mathcal M}^+({\mathcal E})\) to be weakly tight, which means that for any \(\epsilon >0\) there are a compact K and an \(\alpha_ 0\) such that \(\sup \{\mu_{\alpha}(X\setminus K):\;\alpha \geq \alpha_ 0\}<\epsilon.\) It turns out that the locally compact spaces are the only regular and Hausdorff spaces in which every net \(\{\mu_{\alpha}\}\subset {\mathcal M}^+({\mathcal E},{\mathcal K})\) weakly convergent in \({\mathcal M}^+({\mathcal E},{\mathcal K})\) is weakly tight.

MSC:

28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
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References:

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