Hilbert cube has a bimeasurable real coding

For $0\le t\le 1\mathrm{/}2$ put $m_k=\lfloor 2^{k}t\rfloor$ for $k\ge 0$ and $b_k=m_k-2m_{k-1}$ for $k\ge 1$. Since $m_{k-1}\le 2^{k-1}t<m_{k-1}+1$, we have $b_k\in \{0,1\}$. Each $m_k$ is Borel: $\{m_k=l\}=[l{2}^{-k},(l+1)2^{-k})\cap [0,1\mathrm{/}2]$. Thus each digit is Borel.

Here $m_0=0$. Telescoping gives ${\sum }_{k=1}^n{b}_k{2}^{-k}=2^{-n}{m}_n$ and $0\le t-2^{-n}{m}_n<2^{-n}$, so $t={\sum }_{k\ge 1}{b}_k{2}^{-k}$. The digits cannot be eventually all ones: such a tail would make $t$ dyadic, whereas for dyadic $t$ the integers $2^{k}t$ are exact for all sufficiently large $k$, giving $b_k=0$. Thus there are infinitely many zeros, including when $t=0$ or $t=1\mathrm{/}2$.

Interleave cube digits by $a_{J(i,k-1)}=b_k(x_i\mathrm{/}2)$. The bijection in [F3] assigns exactly one digit to each positive position. For any binary sequence $a$, define $T(a)={\sum }_{j\ge 1}2a_j{3}^{-j}$. Its values lie in $[0,1]$. Agreement through position $n$ gives $\mathrm{\mid }T(a)-T(a^{\ast })\mathrm{\mid }\le 3^{-n}$. If the first differing position is $n$, its contribution has magnitude $23^{-n}$ and the remaining tail has magnitude at most $3^{-n}$, so $\mathrm{\mid }T(a)-T(a^{\ast })\mathrm{\mid }\ge 3^{-n}>0$. Thus $T$ is continuous and injective.

Conversely let $(b_k)$ be a binary sequence with infinitely many zeros and sum $t\le 1\mathrm{/}2$. Its tail after $n$, multiplied by $2^n$, lies in $[0,1)$: the all-one tail sums to one, and at least one digit is zero. Hence $\lfloor 2^{n}t\rfloor ={\sum }_{k=1}^n{b}_k{2}^{n-k}$, recovering exactly the digits of step 1.1. In binary sequence space $D=\{0,1{\}}^{{\mathbb{N}}_{\ge 1}}$, the allowable row set is $R={\bigcap }_{N\ge 1}{\bigcup }_{k\ge N}\{b:b_k=0\}\cap \{b:{\sum }_{k\ge 1}{b}_k{2}^{-k}\le 1\mathrm{/}2\}$. It is Borel: cylinders are clopen and the sum is continuous because the tail is at most $2^{-n}$.

For a binary word $w$ of length $n$, let $s_w={\sum }_{j=1}^{n}2w_j{3}^{-j}$ and $I_w=[s_w,s_w+3^{-n}]$. Distinct words of the same length give disjoint intervals separated by a positive gap. The closed set $K={\bigcap }_{n\ge 1}{\bigcup }_{\mathrm{\mid }w\mathrm{\mid }=n}{I}_w$ equals $T[D]$: each point of the intersection has a unique word at each length; nesting forces consistent prefixes; the resulting sequence has sum equal to the point since interval lengths tend to zero. Conversely each sum lies in every prefix interval. The inverse digits are continuous on $K$ because the finitely many cylinders at each length are separated. Hence $T:D\to K$ is a homeomorphism, without an appeal to product compactness.

The deinterleaving row maps $D\to D$, $a\mapsto (a_{J(i,k-1)}{)}_{k\ge 1}$, are continuous: a finite row-cylinder condition is a finite cylinder condition on $a$. Therefore $H=\{a:\text{every deinterleaved row belongs to }R\}$ is Borel in $D$. The homeomorphism gives $C=T[H]=(T^{-1}{)}^{-1}[H]$ Borel in $K$. Since $K$ is closed in $[0,1]$, a trace Borel set in $K$ is Borel in $[0,1]$: the trace sets form a sigma-algebra, and relative opens are traces of ambient opens.

The map $c(x)=T(a(x))$ is measurable: every interleaved digit is Borel by step 1.1 and every finite sum has finite range with Borel level sets (finite unions of intersections of digit level sets), so [F4] applies. Its inverse on $C$ is $x_i=2{\sum }_{k\ge 1}(T^{-1}z{)}_{J(i,k-1)}{2}^{-k}$. These coordinates are measurable by [F4]. They lie in $[0,1]$ and recover both compositions by the row characterization; thus $c$ is a bijection onto precisely $C$.

For completeness, rational intervals restricted to $[0,1]$ form a countable basis by density. Finite coordinate boxes from these intervals are countable explicitly. Enumerate rational endpoints by [F6]. A coordinate condition $x_i\in (p_j,p_k)\cap [0,1]$ has code $J(i,J(j,k))$. A list of condition codes $(z_0,\dots ,z_{l-1})$ has code $J(l,e_l)$, where $e_0=0$ and $e_{r+1}=J(e_r,z_r)$. Inverting the injective J recovers the length and every entry, so this encodes lists injectively. Each box is represented by such a finite list; assigning the least code of its representations injects the family of boxes into the naturals. The empty list represents the whole cube. Every open subset of the cube is a union of a subfamily of this countable basis. Thus the Borel sigma-algebra equals the coordinate-generated sigma-algebra, and coordinate measurability in step 5.1 proves measurability of the whole inverse. This establishes all assertions.