Abstract:Of concern are some simple criteria about the convergence of sequences of positive linear operators and functionals in the framework of spaces of bounded functions which are continuous on a given subset of their domain. Among other things some applications concerning the behaviour of the iterates of Bernstein operators defined both on $[0,1]$ and on the $d$-dimensional simplex and hypercube $(d\geq 1)$ are discussed. A final section treats the behaviour of integrated arithmetic means with respect to a probability Borel measure on a convex compact subset $K$ of a locally convex space. As a consequence a general weak law of large numbers for sequences of $K-$valued random variables is derived.

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中文翻译：

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关于有界函数空间上正线性算子和泛函序列的收敛

摘要：令人关注的是关于在其域的给定子集上连续的有界函数空间框架中正线性算子和泛函序列收敛的一些简单标准。讨论了在 $[0,1]$ 和 $d$ 维单纯形和超立方体 $(d\geq 1)$ 上定义的 Bernstein 算子的迭代行为的一些应用。最后一部分处理关于局部凸空间的凸紧子集 $K$ 上的概率 Borel 测度的积分算术平均值的行为。因此，推导出了 $K-$ 值随机变量序列的一般弱大数定律。

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