The paper is concerned with the study of the limit behaviour of the sequences of the positive linear functionals and operators associated with integrated generalized means defined with respect to a given probability Borel measure in the framework of Borel convex subsets of a Hilbert space. The main results are easily achieved through some new Korovkin-type theorems for composition operators and for functionals which are established in the context of function spaces defined on a metric space. Several applications are shown in the special cases of bounded and unbounded real intervals which involve the most common integrated means. Furthermore, some consequences concerning the convergence in distribution, and hence stochastic, of generalized means of vector-valued random variables are also presented. Finally the paper ends with an application related to the so-called box integral problem which refers to the problem to evaluate the limit behaviour as $n\to \infty$ of the average distance between two points of ${[0,1]}^n$ randomly chosen according to a given distribution on ${[0,1]}^n$.

本文涉及在Hilbert空间的Borel凸子集框架内，针对给定的概率Borel测度，定义与正则线性泛函和算子有关的综合广义方法相关的极限行为的研究。通过一些新的Korovkin型定理，可以轻松实现主要结果，这些定理适用于合成算符和在度量空间上定义的函数空间的上下文中建立的函数。在有界和无界实数区间的特殊情况下，显示了几种应用，其中涉及最常见的集成方法。此外，还提出了一些与向量值随机变量的广义手段的分布趋同（因此是随机的）有关的后果。$ñ\to \infty$ 两点之间的平均距离的 ${[0，\mathrm{1个}]}^{ñ}$ 根据给定的分布随机选择 ${[0，\mathrm{1个}]}^{ñ}$。