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Bernoulli sequences and Borel measurability in ( 0 , 1 )

Petr Veselý — 1993

Commentationes Mathematicae Universitatis Carolinae

The necessary and sufficient condition for a function f : ( 0 , 1 ) [ 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 } { 0 , 1 } such that ( H ( X p ) ) = ( X 1 / 2 ) holds for each p ( 0 , 1 ) , where X p = ( X 1 p , X 2 p , ... ) denotes Bernoulli sequence of random variables with P [ X i p = 1 ] = p .

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