Measurable sections have measurable pointwise inner products

Fix a bijection $\rho :\mathbb{Q}\to \mathbb{N}$ from [F8] and a bijection $\beta :{\mathbb{N}}^2\to \mathbb{N}$ from [F7]. Encode a term $(n,a,b)$, representing $(a+ib)e_n$, by $\kappa (n,a,b)=\beta (n,\beta (\rho (a),\rho (b)))$, and encode a finite list of such term-codes by the finite-sequence code [F6]. To decode $j$, first write ${\beta }^{-1}(j)=(k,m)$ and recursively apply ${\beta }^{-1}$ to recover a length-$k$ list from $m$; accept it only if the final remainder is $0$, and otherwise use the empty list. The recursion takes exactly $k$ steps, so this defines $q_j$ for every $j$, and every rational-complex finite combination occurs since each finite list has its code. Only the two single bijections [F7,F8] are fixed. The scalar set $\mathbb{Q}+i\mathbb{Q}$ is dense in $\mathbb{C}$: for $z=a+ib$ and $\epsilon >0$, choose rationals $p,q$ strictly between $a-\epsilon \mathrm{/}3,a+\epsilon \mathrm{/}3$ and $b-\epsilon \mathrm{/}3,b+\epsilon \mathrm{/}3$ using [F9], and [F10] gives $\mathrm{\mid }z-(p+iq)\mathrm{\mid }\le \mathrm{\mid }a-p\mathrm{\mid }+\mathrm{\mid }b-q\mathrm{\mid }<\epsilon$. Given $v\in H_x$, first approximate it within $\epsilon \mathrm{/}2$ by a finite complex combination $u={\sum }_{r=1}^N{z}_r{e}_{n_r}(x)$ using [F1]. With $C={\sum }_r\mathrm{\parallel }{e}_{n_r}(x)\mathrm{\parallel }$, choose rational-complex $c_r$ with $\mathrm{\mid }{z}_r-c_r\mathrm{\mid }<\epsilon \mathrm{/}(2(1+C))$; the fibre norm triangle inequality and homogeneity give $\mathrm{\parallel }u-{\sum }_r{c}_r{e}_{n_r}(x)\mathrm{\parallel }\le {\sum }_r\mathrm{\mid }{z}_r-c_r\mathrm{\mid }\mathrm{\parallel }{e}_{n_r}(x)\mathrm{\parallel }<\epsilon \mathrm{/}2$. Thus $(q_j(x){)}_j$ is dense in every fibre. These finitely many existential instantiations are not an axiom-of-choice use. For decoded $q_j={\sum }_{r=1}^N{c}_r{e}_{n_r}$, the coefficient $⟨q_j(x),e_m(x)⟩={\sum }_r{c}_r⟨e_{n_r}(x),e_m(x)⟩$ is a finite Gram sum. The real and imaginary coordinate maps are continuous for [F11]'s metric and Borel by [F13]; finite tuples are measurable in ${\mathbb{R}}^{2N}$ by the countable rational-box basis [F12]; and the finite-sum map is continuous and Borel [F13], so composition [F14] makes each $q_j$ measurable. The same argument makes $\mathrm{\parallel }{q}_j(x){\mathrm{\parallel }}^2$ measurable from its finite Gram expansion. The square-root map is continuous on $[0,\mathrm{\infty })$ because for $r\ge s\ge 0$, $(\sqrt{r}-\sqrt{s}{)}^2\le r-s$; hence it is Borel [F13], and [F14] makes $\mathrm{\parallel }{q}_j(x)\mathrm{\parallel }$ measurable.

If $a,b$ are measurable complex scalar functions and $\xi ,\eta$ have measurable coefficients, then $⟨a\xi +b\eta ,e_n⟩=a⟨\xi ,e_n⟩+b⟨\eta ,e_n⟩$ by [F2]. The finite tuple of inputs is measurable in ${\mathbb{C}}^4\cong {\mathbb{R}}^8$ because its real coordinates are Borel [F11,F13,F14] and rational boxes [F12] form a countable basis. The map $(u,z,v,w)\mapsto uz+vw$ is continuous and Borel [F13], so composition [F14] makes each displayed coefficient measurable. Thus measurable sections are closed under measurable scalar combinations.

Suppose ${\xi }_k(x)\to \xi (x)$ in norm pointwise and each ${\xi }_k$ is measurable. For every $n$, Cauchy--Schwarz [F3] gives $\mathrm{\mid }⟨{\xi }_k(x)-\xi (x),e_n(x)⟩\mathrm{\mid }\le \mathrm{\parallel }{\xi }_k(x)-\xi (x)\mathrm{\parallel }\mathrm{\parallel }{e}_n(x)\mathrm{\parallel }\to 0$. The complex coefficients converge pointwise; their real and imaginary coordinates are Borel by [F11,F13] and composition [F14] preserves measurability. The pointwise-limit theorem [F4] makes both real limits measurable, so every fundamental coefficient of $\xi$ is measurable.

For a coefficient-measurable $\xi$, $⟨\xi ,q_j⟩$ is a finite linear combination of its fundamental coefficients by [F2] and step 1.1. Put $b_j(x)=\mathrm{\mid }⟨\xi (x),q_j(x)⟩\mathrm{\mid }\mathrm{/}\mathrm{\parallel }{q}_j(x)\mathrm{\parallel }$ when $\mathrm{\parallel }{q}_j(x)\mathrm{\parallel }>0$, and $b_j(x)=0$ otherwise. Modulus is continuous by the reverse triangle inequality from [F10,F11]; division is continuous where the denominator is positive, and extending by zero on the Borel set where it vanishes gives a Borel map. The input tuple is measurable by the rational-box basis [F12], so composition [F13,F14] makes $b_j$ measurable. Cauchy--Schwarz [F3] gives $b_j(x)\le \mathrm{\parallel }\xi (x)\mathrm{\parallel }$. On a nonzero fibre, the normalized nonzero $q_j(x)$ are dense in the unit sphere: for any unit $u$, take the least index $j_m$ with $\mathrm{\parallel }{q}_{j_m}(x)-u\mathrm{\parallel }<1\mathrm{/}m$; then normalization converges to $u$. If $\xi (x)\mathrm{\ne }0$, apply this to $u=\xi (x)\mathrm{/}\mathrm{\parallel }\xi (x)\mathrm{\parallel }$ and use [F3] to obtain ${\sup }_j{b}_j(x)=\mathrm{\parallel }\xi (x)\mathrm{\parallel }$; for $\xi (x)=0$ both sides vanish. On a zero fibre every test and the norm are zero. Thus $\mathrm{\parallel }\xi (x)\mathrm{\parallel }={\sup }_j{b}_j(x)$ in every fibre, and the countable-supremum clause [F4] makes the norm measurable.

Fix a measurable section $\eta$ and $k\ge 1$. By step 1.2, $\eta -q_j$ is measurable; step 2.1 then makes $E_{j,k}=\{x:\mathrm{\parallel }\eta (x)-q_j(x)\mathrm{\parallel }<1\mathrm{/}k\}$ measurable. Density from step 1.1 gives ${\bigcup }_j{E}_{j,k}=X$. At each $x$, take the least $j_k(x)$ with $x\in E_{j_k(x),k}$. The sets $E_{j,k}\setminus {\bigcup }_{i<j}{E}_{i,k}$ form a measurable partition, so ${\eta }_k(x)=q_{j_k(x)}(x)$ is measurable coefficient by coefficient and $\mathrm{\parallel }{\eta }_k(x)-\eta (x)\mathrm{\parallel }<1\mathrm{/}k$. Least-index selection is canonical and uses no choice.

For each fixed $j$, $⟨\xi ,q_j⟩$ is measurable by the finite-combination argument of step 1.1, so $⟨\xi ,{\eta }_k⟩$ is measurable on the partition of step 3.1. Cauchy--Schwarz [F3] gives $\mathrm{\mid }⟨\xi (x),{\eta }_k(x)-\eta (x)⟩\mathrm{\mid }\le \mathrm{\parallel }\xi (x)\mathrm{\parallel }\mathrm{/}k\to 0$, so the pairings converge pointwise to $⟨\xi ,\eta ⟩$. The real and imaginary coordinate maps are continuous for [F11]'s metric and Borel [F13]; composition [F14] and pointwise-limit measurability [F4] show the limit is measurable. This proves coefficient measurability implies measurability of pairing with every measurable section.

Conversely, if pairing with every measurable section is measurable, each $e_n$ is a measurable section by its Gram coefficients [F1]. Testing against $e_n$ gives every fundamental coefficient measurable, which is the coefficient criterion. Together with step 4.1 this proves both directions of the equivalence.