Summary: The necessary and sufficient condition for a function \(f:(0,1)\to[0,1]\) to be Borel measurable (given in this paper) provides a technique to prove the existence of a Borel measurable map \(H:\{0,1\}^ \mathbb{N}\to\{0,1\}^ \mathbb{N}\) such that \({\mathfrak L}(H(\mathbf{X}^ p))={\mathfrak L}(\text\textbf{X}^{1/2})\) holds for each \(p\in(0,1)\), where \(\mathbf{X}^ p=(X^ p_ 1,X^ p_ 2,\dots)\) denotes Bernoulli sequence of random variables with \(P[X_ i^ p=1]=p\).