How to construct i.i.d. standard normal random variables on $\Omega=[0,1]$ with the Lebesgue measure

Question

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be the unit interval with Lebesgue measure on the Borel subsets. Can we find independent random variables $X_1, X_2, X_3, \dots$ defined on $(\Omega, \mathcal{F}, \mathbb{P})$, each normal mean zero, variance $1$?

Answer 1

Yes, we can: this is Borel isomorphism theorem. Namely, you have a range space $Y = \Bbb R^{\Bbb N}$, and some distribution $\mathsf Q$ over that space: in general it does not matter which as it will work for anything, in particular for your case of i.i.d. r.v.'s with zero mean and unit variance. The question is whether there exists a measurable map $f:[0,1] \to Y$ such that $\mathsf Q = \mathsf P\circ f^{-1}$. The answer is yes, but the proof is quite technical. You should start here.