Abstract:The necessary and sufficient condition for a function $f : (0,1) \to [0,1] $ to be Borel measurable (given by Theorem stated below) provides a technique to prove (in Corollary 2) the existence of a Borel measurable map $H : \{ 0,1 \}^\Bbb N \to \{ 0,1 \}^\Bbb N$ such that $\Cal L (H(\text {\bf X}^p)) = \Cal L (\text {\bf X}^{1/2})$ holds for each $p \in (0,1)$, where $\text {\bf X}^p = (X^p_1 , X^p_2 ,...)$ denotes Bernoulli sequence of random variables with $P[X^p_i = 1] = p$.
Keywords: Borel measurable function, Bernoulli sequence of random variables, Strong law of large numbers
AMS Subject Classification: 60A10, 28A20