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> Let $(\Omega, \mathcal{F}, \mathbb{P})$ be the unit interval with Lebesgue measure on the Borel subsets. Then we can find independent random variables $X\_1, X\_2, X\_3, \dots$ defined on $(\Omega, \mathcal{F}, \mathbb{P})$, each normal mean zero, variance $1$.
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This is a special case of the Borel isomorphism theorem, but I was wondering if anyone could supply/refer me to a proof which does not invoke it, since it seems like overkill here. Thanks!



How to construct i.i.d. standard normal random variables on $\Omega = [0, 1]$ with the Lebesgue measure
If we can find independent random variables $U\_j$ uniform on $[0,1]$ we can transform them to $\mathcal N(0,1)$. If $Y$ is uniform on $[0,1]$, let $D\_j$ be its $j$'th decimal place, i.e. $Y = \sum\_{j=1}^\infty 10^{1-j} D\_j$ with 
$D\_j \in \{0,1,\ldots, 9\}$. If $p\_j$ is the $j$'th prime, let
$U\_j = \sum\_{k=1}^\infty 10^{1-k} D\_{p\_j^k}$. Then $U\_j$ are iid uniform on $[0,1]$.


