Any real number \(x\in (0,1)\) can be written as a convergent series \[ x=\frac{1}{g_1}-\frac{1}{g_1(g_1+g_2)}+\cdots +\frac{(-1)^{n-1}}{g_1(g_1+g_2)\cdots (g_1+g_2+\cdots +g_n)}+\cdots \] where \(g_1,g_2,\dots ,g_n,\dots\) are natural numbers called the \(\bar{O}^1\)-symbols of \(x\). This is denoted as \(x=\bar{O}^1(g_1,g_2,\dots ,g_n,\dots )\). Let \(\{ V_n\}\) be a sequence of subsets of \(\mathbb N\). The authors study metric properties of the set \(C[\bar{O}^1,\{ V_n\}]\) of all real numbers whose \(\bar{O}^1\)-symbols are such that \(g_n(x)\in V_n\) for all \(n\). Conditions are found for the Lebesgue measure of this set to be positive or zero under various assumption regarding \(\{ V_n\}\). These results are used for the study of distributions of the random variables \( \xi =\bar{O}^1(\xi_1,\xi_2,\dots ,\xi_k,\dots ) \) where \(\xi_k\) are random variables with values from \(\mathbb N\).