Chacon three cut one spacer towers

Work on $[0,1)$ with the Lebesgue measurable sets and restricted set function introduced in Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$. Under the countable-choice consequence of The Axiom of Choice, Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra containing every elementary set and ${\lambda }_n$ is a complete measure extending elementary volume makes this a complete measure space. The same assumption supplies the countable choice in A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included used for the actual interval measures below. The finite interval recursion itself is choice-free.

The stage-zero tower is the ordered list consisting of $L_{0,0}=[0,2\mathrm{/}3)$; its reservoir is $R_0=[2\mathrm{/}3,1)$. Suppose the stage-$r$ list is $(L_{r,0},\dots ,L_{r,h_r-1})$, with equal width $w_r$. Cut each physical half-open interval $L_{r,j}$ into its left, middle and right thirds $L_{r,j}^{(0)},L_{r,j}^{(1)},L_{r,j}^{(2)}$. The next ordered list is $(L_{r,0}^{(0)},\dots ,L_{r,h_r-1}^{(0)},L_{r,0}^{(1)},\dots ,L_{r,h_r-1}^{(1)},J_r,L_{r,0}^{(2)},\dots ,L_{r,h_r-1}^{(2)}),$ where $J_r=[1-3^{-(r+1)},1-3^{-(r+2)})$ is taken from the left end of $R_r$ and retained as a new level. Put $R_{r+1}=[1-3^{-(r+2)},1)$.

The height and width obey $h_0=1$, $h_{r+1}=3h_r+1$ and $w_r=2\mathrm{/}{3}^{r+1}$. Indeed the initial width is $2\mathrm{/}3$; taking thirds divides it by three, and $\mathrm{\mid }{J}_r\mathrm{\mid }=3^{-(r+1)}-3^{-(r+2)}=2\mathrm{/}{3}^{r+2}=w_{r+1}$. The spacer and the remaining reservoir partition the previous reservoir; all old levels partition into their thirds. Thus by finite induction the new intervals are pairwise disjoint and, together with $R_{r+1}$, partition $[0,1)$.

Induction also gives $h_r=(3^{r+1}-1)\mathrm{/}2$: the initial value is one, and $3(3^{r+1}-1)\mathrm{/}2+1=(3^{r+2}-1)\mathrm{/}2$. Consequently the tower union $C_r={\bigcup }_{j<h_r}{L}_{r,j}$ has measure $h_r{w}_r=1-3^{-(r+1)}$, and the reservoir has measure $3^{-(r+1)}$. Every interval is left-closed and right-open; no endpoint belongs to two levels.

Define $T_r$ on $C_r\setminus L_{r,h_r-1}$ by the unique translation taking $L_{r,j}$ to $L_{r,j+1}$. If their left endpoints are $a_j,a_{j+1}$, that formula is $T_r(x)=x+a_{j+1}-a_j$ on $L_{r,j}$. At stage zero this is the empty partial map. Its range is $C_r\setminus L_{r,0}$. The existence of an invertible limiting probability transformation is a separate lemma, not part of the finite definition.