I have been studying Patrick Billingsley's book Probability and Measure. In the literature the author models the coin flipping experiment ( or more generality, arbitrary time dependent experiment with two possible outcomes in each trial) by expanding every number on the unit interval $(0,1]$ into its representation in the binary number system. I have moved beyond this part for a while but every time I look back I am simply awed by the marvelousness of this structure, and therefore have the following questions:
In the construction of the model the author makes use of the dyadic expansion on each real number $ \omega \in (0,1] $. I'm not sure whether this process is well known but here is a brief illustration, just in case: for every $\omega$ as above define a map $$T: (0,1] \rightarrow (0,1]: \omega \mapsto 2\omega \ \text{ if } \ 0<\omega\leq\frac{1}{2}, \omega \mapsto 2\omega - 1 \ \text{ if } \ \frac{1}{2}<\omega \leq 1 .$$ Then if we define $$d_{1}:(0,1] \rightarrow \{ 0,1 \}: \omega \mapsto 0 \ \text{if} \ 0 < \omega \leq \frac{1}{2}, \omega \mapsto 1 \ \text{if} \ \frac{1}{2} < \omega \leq 1$$ and $d_{i}(\omega)=d_{1}\big(T^{i-1}(\omega)\big)$, i > 1. Then it can be shown that $$ \omega = \sum_{i=1}^{ \infty } \frac{d_{i}(\omega)}{2^i} .$$ Of course I recognise this as the definition of binary representation. However so far I only know this by definition, that is every real number admits such a representation. The above is effectively a full process of constructing the number system and so I wonder whether it is the only way of doing this. I have this doubt because geometrically this construction corresponds to dividing the unit interval into two parts, assigning them the notion of 0 and 1, and then for every such interval a new division can be done - the fact that binary representation of real numbers corresponds to coin flipping so well (even with their orders on the real line) is just unnatural and hard to believe. Moreover I think the fact that this system cope so well with conditional probability is also something to be amazed of (all the normalised ratios corresponds well with intuition of coin flipping). How was this idea even developed?