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Borel measurable function; Bernoulli sequence of random variables; Strong law of large numbers
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 $P[X^p_i = 1] = p$.
[1] Feller W.: An Introduction to Probability Theory and its Applications. Volume II. John Wiley & Sons, Inc. New York, London and Sydney (1966). MR 0210154
[2] Štěpán J.: Personal communication. (1992).
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