On the error of approximation by means of scenario trees with depth 1

Let $\Lambda_n$ denote the set of scenario trees with depth 1 and $n$ scenarios. Let $X=(0\le x_1<\dots<x_n\le1)$ and let $\Lambda_n(X)$ denote the set of all scenario trees of depth 1 with the scenarios $X=(0\le x_1<\dots<x_n\le1)$. Let $G$ be a probability distribution defined on $[0,1]$ and $H$ be a subset of measurable functions defined on $[0,1]$. Let $d_{H,X}(G)=\inf_{\tilde G\in\Lambda_n(X)}d_H(G,\tilde G)$ and $d_H(G)=\inf_{\tilde G\in\Lambda_n}d_H(G,\tilde G)$, where $d_H(G,\tilde G):=\sup_{h\in H}\left|\int h\,dG-\int h\,d\tilde G\right|$. The main goal of the paper is to estimate $d_H(G,X)$ and $d_H(G)$ in the case when the set $H$ is a subset of all algebraical polynomials of degree $\leq n$. Thus, the paper is examined the error of approximation of a continuous distribution $G$ by means of scenario trees with depth 1 and matching the first $n$ moments.