4 Standard Borel Spaces (\(\mathbb {R} \cong \mathbb {R} \times \mathbb {R}\))
This chapter constructs the measurable bijection \(\mathbb {R} \cong \mathbb {R} \times \mathbb {R}\), proving that the product of standard Borel spaces is standard Borel. The construction composes three layers: (1) a bijection \(\mathbb {R} \leftrightarrow (0,1)\) via arctangent, (2) a bijection \((0,1)^2 \leftrightarrow (0,1)\) via binary digit interleaving, and (3) the composed encode/decode pair.
4.1 R to (0,1) Bijection
\(\phi (x) = \frac{\arctan x}{\pi } + \frac{1}{2}\) maps \(\mathbb {R} \to (0,1)\). \(\psi (y) = \tan (\pi (y - 1/2))\) is the inverse.
\(0 {\lt} \phi (x) {\lt} 1\) for all \(x \in \mathbb {R}\).
\(\psi (\phi (x)) = x\) unconditionally. \(\phi (\psi (y)) = y\) for \(y \in (0,1)\).
\(\phi \) is measurable on all of \(\mathbb {R}\). \(\psi \) is measurable on \((0,1)\) and (via an approximation argument) on all of \(\mathbb {R}\).
4.2 Binary Digit Machinery
\(\mathrm{bin\_ digit}(x, n)\) extracts the \(n\)-th binary digit of \(x \in [0,1)\) via the doubling map. \(\mathrm{bin\_ partial\_ sum}(d, n) = \sum _{i{\lt}n} d(i) \cdot 2^{-(i+1)}\) and \(\mathrm{bin\_ sum}(d) = \lim _n \mathrm{bin\_ partial\_ sum}(d, n)\).
Partial sums converge, are non-decreasing, and the limit satisfies \(0 \leq \mathrm{bin\_ sum}(d) \leq 1\).
\(\sum _{i{\lt}n} (1/2)^{i+1} = 1 - (1/2)^n\).
For \(0 \leq x {\lt} 1\): \(\mathrm{bin\_ sum}(\mathrm{bin\_ digits}(x)) = x\).
Show \(x - \mathrm{bin\_ partial\_ sum}(n) = \mathrm{rem}(n) \cdot (1/2)^n\) with \(0 \leq \mathrm{rem}(n) {\lt} 1\), so the error converges to \(0\).
A digit sequence \(d\) has no trailing ones if for every \(N\), there exists \(n \geq N\) with \(d(n) = \mathrm{false}\).
For \(0 \leq x {\lt} 1\): \(\mathrm{no\_ trailing\_ ones}(\mathrm{bin\_ digits}(x))\).
If \(\mathrm{no\_ trailing\_ ones}(d)\), then \(\mathrm{bin\_ digits}(\mathrm{bin\_ sum}(d)) =_1 d\).
\(\mathrm{bin\_ sum}(d) = d(0) \cdot \tfrac {1}{2} + \mathrm{bin\_ sum}(d \circ S) \cdot \tfrac {1}{2}\).
4.3 Digit Interleaving
\(\mathrm{interleave}(d_1, d_2)(n) = \begin{cases} d_1(n/2) & \text{if } n \text{ even} \\ d_2(n/2) & \text{if } n \text{ odd} \end{cases}\). Deinterleaving extracts even-indexed and odd-indexed subsequences.
Interleaving then deinterleaving (and vice versa) recovers the original sequences.
If both \(d_1\) and \(d_2\) have no trailing ones, then so does \(\mathrm{interleave}(d_1, d_2)\).
4.4 Pairing Functions
\(\mathrm{pair\_ to\_ unit}(x, y) = \mathrm{bin\_ sum}(\mathrm{interleave}(\mathrm{bin\_ digits}(x), \mathrm{bin\_ digits}(y)))\). \(\mathrm{unit\_ to\_ pair}\) deinterleaves and reconstructs both components.
For \(0 \leq x, y {\lt} 1\): \(\mathrm{unit\_ to\_ pair}(\mathrm{pair\_ to\_ unit}(x, y)) = (x, y)\).
For \(0 \leq r {\lt} 1\) with no-trailing-ones on deinterleaved subsequences: \(\mathrm{pair\_ to\_ unit}(\mathrm{unit\_ to\_ pair}(r)) = r\).
4.5 Composed Bijection \(\mathbb {R} \times \mathbb {R} \to \mathbb {R}\)
\(\mathrm{encode}(x, y) = \psi (\mathrm{pair\_ to\_ unit}(\phi (x), \phi (y)))\) and \(\mathrm{decode}(r) = (\psi (u_1), \psi (u_2))\) where \((u_1, u_2) = \mathrm{unit\_ to\_ pair}(\phi (r))\).
\(\mathrm{decode}(\mathrm{encode}(x, y)) = (x, y)\) for all \(x, y \in \mathbb {R}\).
\(\mathrm{encode}(\mathrm{decode}(r)) = r\) under additional no-trailing-ones hypotheses (not needed for is_standard_borel).
4.6 Measurability
Both \(\mathrm{encode}\) and \(\mathrm{decode}\) are measurable functions.
\(\mathrm{pair\_ to\_ unit}\) is measurable, established by showing each partial sum of interleaved digits is measurable and applying \(\mathrm{measurable\_ fun\_ cvg}\).
Both projections of \(\mathrm{unit\_ to\_ pair}\) are measurable.
4.7 The Punchline
There exist measurable \(f : \mathbb {R}^2 \to \mathbb {R}\) and \(g : \mathbb {R} \to \mathbb {R}^2\) with \(g(f(x,y)) = (x,y)\).
Take \(f = \mathrm{encode\_ RR}\), \(g = \mathrm{decode\_ RR}\). Measurability is by measurable_encode_RR and measurable_decode_RR; the round-trip is encode_RRK.