Given a probability space \( (\Omega , {\mathcal {A}}, P) \), a complete and separable metric space X with the \( \sigma \)-algebra \( {\mathcal {B}} \) of all its Borel subsets, a \( {\mathcal {B}} \otimes {\mathcal {A}} \)-measurable and contractive in mean \( f: X \times \Omega \rightarrow X \), and a Lipschitz F mapping X into a separable Banach space Y we characterize the solvability of the equation

in the class of Lipschitz functions \(\varphi : X \rightarrow Y\) with the aid of the weak limit \(\pi ^f\) of the sequence of iterates \(\left( f^n(x,\cdot )\right) _{n \in {\mathbb {N}}}\) of f, defined on \( X \times \Omega ^{{\mathbb {N}}}\) by \(f^0(x, \omega ) = x\) and \( f^n(x, \omega ) = f\left( f^{n-1}(x, \omega ), \omega _n\right) \) for \(n \in {\mathbb {N}}\), and propose a characterization of \(\pi ^f\) for some special rv-functions in Hilbert spaces.

给定的概率空间\（（\欧米茄，{\ mathcal {A}}，P）\） ，一个完整的，可分离的度量空间X与\（\西格玛\） -代数\（{\ mathcal {B}} \ ）的所有Borel子集\（{\ mathcal {B}} \ otimes {\ mathcal {A}} \）-可测量且收缩的均值\（f：X \ times \ Omega \ rightarrow X \），以及Lipschitz F将X映射到可分离的Banach空间Y中，我们表征了方程的可解性

在Lipschitz函数类\（\ varphi：X \ rightarrow Y \）的帮助下，通过迭代序列\（\ left（f ^ n（x，\ cdot ）的弱极限\（\ pi ^ f \））\ right）的f的_ {n \ in {\ mathbb {N}}} \）在\（X \ times \ Omega ^ {{{\ mathbb {N}}} \\）上由\（f ^ 0（的x，\欧米加）= X \）和\（F ^ N（X，\欧米加）= F \左（F ^ {N-1}（X，\欧米加），\欧米加_n \右）\）为\ （n \ in {\ mathbb {N}} \）中，并为希尔伯特空间中的某些特殊rv函数提出\（\ pi ^ f \）的表征。