Editing Dyadic transformation (section)

===Borel space===
A vast amount of simplification results if one instead works with the [[Cantor space]] <math>\Omega=\{0,1\}^\mathbb{N}</math>, and functions <math>\rho:\Omega\to\mathbb{R}.</math> Some caution is advised, as the map <math>T(x)=2x\bmod 1</math> is defined on the [[unit interval]] of the [[real number line]], assuming the [[natural topology]] on the reals.  By contrast, the map <math>T(b_0, b_1, b_2, \dots)=(b_1, b_2, \dots)</math> is defined on the [[Cantor space]] <math>\Omega = \{0,1\}^{\mathbb{N}}</math>, which by convention is given a very different [[topological space|topology]], the [[product topology]]. There is a potential clash of topologies; some care must be taken. However, as presented above, there is a homomorphism from the Cantor set into the reals; fortunately, it maps [[open set]]s into open sets, and thus preserves notions of [[continuous function (topology)|continuity]].

To work with the Cantor set <math>\Omega=\{0,1\}^{\mathbb{N}}</math>, one must provide a topology for it; by convention, this is the [[product topology]]. By adjoining set-complements, it can be extended to a [[Measurable space|Borel space]], that is, a [[sigma algebra]]. The topology is that of [[cylinder set]]s. A cylinder set has the generic form
:<math>(*,*,*,\dots,*,b_k,b_{k+1},*,\dots, *,b_m,*,\dots)</math>
where the <math>*</math> are arbitrary bit values (not necessarily all the same), and the <math>b_k, b_m, \dots</math> are a finite number of specific bit-values scattered in the infinite bit-string. These are the open sets of the topology. The canonical measure on this space is the [[Bernoulli measure]] for the fair coin-toss. If there is just one bit specified in the string of arbitrary positions, the measure is 1/2. If there are two bits specified, the measure is 1/4, and so on. One can get fancier: given a real number <math>0 < p < 1</math> one can define a measure
:<math>\mu_p( *,\dots,*,b_k,*,\dots) = p^n(1-p)^m</math>
if there are <math>n</math> heads and <math>m</math> tails in the sequence. The measure with <math>p=1/2</math> is preferred, since it is preserved by the map

:<math>(b_0, b_1, b_2, \dots) \mapsto x = \sum_{n=0}^\infty \frac{b_n}{2^{n+1}}.</math>
So, for example, <math>(0,*,\cdots)</math> maps to the [[interval (mathematics)|interval]] <math>[0,1/2]</math> and <math>(1,*,\dots)</math> maps to the interval <math>[1/2,1]</math> and both of these intervals have a measure of 1/2. Similarly, <math>(*,0,*,\dots)</math> maps to the interval <math>[0,1/4]\cup[1/2,3/4]</math> which still has the measure 1/2. That is, the embedding above preserves the measure.

An alternative is to write
:<math>(b_0, b_1, b_2, \dots) \mapsto x = \sum_{n=0}^\infty \left[b_n p^{n+1} + (1-b_n)(1-p)^{n+1}\right]</math>
which preserves the measure <math>\mu_p.</math> That is, it maps such that the measure on the unit interval is again the Lebesgue measure.