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 $P[X^p_i = 1] = p$.
@article{118586,
author = {Petr Vesel\'y},
title = {Bernoulli sequences and Borel measurability in $(0,1)$},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {34},
year = {1993},
pages = {341-346},
zbl = {0777.60003},
mrnumber = {1241742},
language = {en},
url = {http://dml.mathdoc.fr/item/118586}
}
Veselý, Petr. Bernoulli sequences and Borel measurability in $(0,1)$. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) pp. 341-346. http://gdmltest.u-ga.fr/item/118586/
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