Abstract Various equicontinuity properties for families of Markov operators have been – and still are – used in the study of existence and uniqueness of invariant probability for these operators, and of asymptotic stability. We prove a general result on equivalence of equicontinuity concepts. It allows comparing results in the literature and switching from one view on equicontinuity to another, which is technically convenient in proofs. More precisely, the characterisation is based on a ‘Schur-like property’ for measures: if a sequence of finite signed Borel measures on a Polish space is such that it is bounded in total variation norm and such that for each bounded Lipschitz function the sequence of integrals of this function with respect to these measures converges, then the sequence converges in dual bounded Lipschitz norm to a measure.

中文翻译：

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从测度空间的类 Schur 性质导出的马尔可夫算子的等连续性概念的等价性

摘要 马尔可夫算子族的各种等连续性特性已经——而且仍然——用于研究这些算子的不变概率的存在性和唯一性，以及渐近稳定性。我们证明了等连续性概念等价的一般结果。它允许比较文献中的结果并从关于等连续性的一种观点切换到另一种观点，这在证明上在技术上很方便。更准确地说，该表征基于测度的“类 Schur 性质”：如果波兰空间上的有限符号 Borel 测度序列是这样的，它在总变异范数中是有界的，并且对于每个有界的 Lipschitz 函数，序列该函数的积分相对于这些度量收敛，然后序列以对偶有界 Lipschitz 范数收敛到一个度量。