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<p>We study equivalent descriptions of the vague, weak, setwise and total-variation (TV) convergence of sequences of Borel measures on metrizable and non-metrizable topological spaces in this work. On metrizable spaces, we give some equivalent descriptions on the vague convergence of sequences of measures, and some equivalent descriptions on the TV convergence of sequences of measures. There is usually some hierarchy structure on the equivalent descriptions of convergence in different modes, but not always. On non-metrizable spaces, we give examples to show that these conditions are seldom enough to guarantee any convergence of sequences of measures. There are some remarks on the attainability of the TV distance and more modes of sequential convergence proposed at the end of the work.</p>

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Sequential Convergence on the Space of Borel Measures

  • 期刊论文

  • 发表日期：

    2025年7月2日

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  • Liangang Ma

摘要

We study equivalent descriptions of the vague, weak, setwise and total-variation (TV) convergence of sequences of Borel measures on metrizable and non-metrizable topological spaces in this work. On metrizable spaces, we give some equivalent descriptions on the vague convergence of sequences of measures, and some equivalent descriptions on the TV convergence of sequences of measures. There is usually some hierarchy structure on the equivalent descriptions of convergence in different modes, but not always. On non-metrizable spaces, we give examples to show that these conditions are seldom enough to guarantee any convergence of sequences of measures. There are some remarks on the attainability of the TV distance and more modes of sequential convergence proposed at the end of the work.