Geodesic
Bernoulli sequences and Borel measurability in $(0,1)$
Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) no. 2, pp. 341-346

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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 $\text{\bf X}^p = (X^p_1 , X^p_2 , \ldots )$ denotes Bernoulli sequence of random variables with $P[X^p_i = 1] = p$.
Classification : 28A20, 60A10
Keywords: Borel measurable function; Bernoulli sequence of random variables; Strong law of large numbers 
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     title = {Bernoulli sequences and {Borel} measurability in $(0,1)$},
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Veselý, Petr. Bernoulli sequences and Borel measurability in $(0,1)$. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) no. 2, pp. 341-346. http://geodesic.mathdoc.fr/item/CMUC_1993__34_2_a15/