Theorem

dyadic expansion of uniformly distributed r.v.

Let $\omega \sim \text{Uniform}(0,1)$ and write $\omega$ in binary as $\omega = \sum_{n=1}^{\infty} \frac{d_{n}(\omega)}{2^n} = 0.d_{1}(\omega)d_{2}(\omega)d_{3}(\omega) \dots$, where each $d_{n}(\omega) \in \{0, 1\}$. We write $1$ as $0.11111 \cdots$, and if a number has two possible expansions we agree to use the non-terminating one (the one that ends in $1$s). Then the $d_{n}$ are iid random variables with common distribution $\text{Bernoulli}(0.5)$.