Explicit construction of a an infinite-dimensional Gaussian Hilbert Space A collection of lecture notes for a course on probability given by Tsirelson, mentions that there exists an infinite sequence of independent standard normal random variables in $L_2(0,1)$ (with the Lebesgue measure), and it has to do with "tricks with binary digits".
Can anyone point out a reference where such a construction is carried out?
 A: I don't know a reference, but it goes roughly like this.
Let $b_i(x)$ be the $i$th binary digit of $x \in [0,1]$.  (Values of $x$ with multiple binary expansions can be ignored, as they form a set of measure zero.)  Then the $b_i$ are iid Bernoulli random variables, see How can I show that the "binary digit maps" $b_i : [0,1) \to \{0,1\}$ are i.i.d. Bernoulli random variables?.
Partition $\mathbb{N}$ into infinitely many disjoint infinite sets $\{n_{1,1}, n_{1,2}, \dots\}, \{n_{2,1}, n_{2,2}, \dots\}, \dots$, and set $$U_k(x) = \sum_{i=1}^\infty 2^{-i} b_{n_{k,i}}(x).$$
It is a standard exercise to check that each $U_k$ has a uniform $U(0,1)$ distribution, and that the $U_k$ are independent because they are constructed from disjoint sets of the $b_i$.
Finally let $\Phi$ be the standard normal cdf, and $\Phi^{-1}$ its inverse.  Let $Z_k = \Phi^{-1}(U_k)$.  It's easy to check that each $Z_k$ has a standard normal distribution, and they are independent because the $U_k$ were.
Relating this to How to construct i.i.d. standard normal random variables on $\Omega=[0,1]$ with the Lebesgue measure, the map $x \mapsto (U_1(x), U_2(x), \dots)$ is (almost) a Borel isomorphism between $[0,1]$ and the Hilbert cube $[0,1]^\omega$.  (I say "almost" because some sets of measure zero are left out.)
