Measurable Gram-Schmidt and constant-field trivializations on dimension strata

For any sequence $(u_n)$ of measurable sections, set $v_n=u_n-{\sum }_{j<n}⟨u_n,f_j⟩f_j$ and $f_n=v_n\mathrm{/}\mathrm{\parallel }{v}_n\mathrm{\parallel }$ when $\mathrm{\parallel }{v}_n\mathrm{\parallel }>0$, with $f_n=0$ otherwise. Inductively, measurable-section closure [F2] makes each finite coefficient, sum, residual, and residual norm measurable; the reciprocal on $(0,\mathrm{\infty })$ extended by zero at zero is Borel, so each $f_n$ is measurable. If the earlier nonzero $f_j$ are orthonormal, then for each nonzero $f_{\mathrm{\ell }}$ with $\mathrm{\ell }<n$, $⟨v_n,f_{\mathrm{\ell }}⟩=⟨u_n,f_{\mathrm{\ell }}⟩-{\sum }_{j<n}⟨u_n,f_j⟩⟨f_j,f_{\mathrm{\ell }}⟩=0$; normalizing a nonzero residual preserves orthogonality. Also $u_n$ lies in the span of $f_0,\dots ,f_n$, and induction gives equality of the spans of the terms with indices $0,\dots ,n$ of $(u_n)$ and $(f_n)$. Thus the nonzero $f_n$ form an orthonormal family whose closed span equals that of $(u_n)$. Apply this recursion to the fundamental family $(e_n)$ to obtain a complete system in each $H_x$; [F2] also gives Borel $x\mapsto ⟨\xi (x),f_n(x)⟩$ for every measurable section $\xi$.

Let $a_n(x)={\mathbf{1}}_{\{\mathrm{\parallel }{f}_n(x)\mathrm{\parallel }=1\}}$, set $N_{-1}(x)=0$, and for $m\ge 0$ put $N_m(x)={\sum }_{0\le n\le m}{a}_n(x)$. These are Borel by [F2,F7]. For finite $p\ge 1$, $X_p=({\bigcap }_m\{N_m\le p\})\cap ({\bigcup }_m\{N_m=p\})$; also $X_{\mathrm{\infty }}={\bigcap }_{q\ge 1}{\bigcup }_m\{N_m\ge q\}$ and $X_0={\bigcap }_m\{N_m=0\}$. Sigma-algebra closure makes these sets Borel. Since each nonzero $f_n(x)$ has norm one and their family is complete, $N_m(x)$ counts the active vectors at indices $0,\dots ,m$; the formulas therefore give exactly the finite, infinite, and zero dimensions.

Apply the recursion of step 1.1 to $({\xi }_k)$, obtaining measurable $h_n$ whose nonzero values form an orthonormal basis of $S_x=\stackrel{}{\mathrm{span}}\{{\xi }_k(x):k\in \mathbb{N}\}$. Each $S_x$ is a closed Hilbert subspace of $H_x$, and the Borel Gram coefficients and dense span of $(h_n)$ make $(S_x,h_n(x))$ a measurable closed Hilbert subfield by [F1,F2]. Every $S$-measurable section $\zeta$ is $H$-measurable: its finite expansions ${\sum }_{n\le N}⟨\zeta ,h_n⟩h_n$ are measurable $H$-sections and converge pointwise in norm to $\zeta$ by [F4,F5], so [F2] applies. Conversely, every $H$-measurable section taking values in $S_x$ is $S$-measurable because its pairings with the measurable $h_n$ are Borel by [F2]. Thus inclusion induces an isometric embedding ${\mathcal{H}}_S\hookrightarrow {\mathcal{H}}_H$. The direct-integral Hilbert theorem [F3] makes its domain complete, so its image is closed.

On $X_p$, put $J_p=\{1,\dots ,p\}$ for finite $p$ and $J_{\mathrm{\infty }}={\mathbb{N}}_{>0}$. Enumerate the active frame indices increasingly: for $j\in J_p$, let ${\nu }_j(x)$ be the $j$th $n\ge 0$ with $a_n(x)=1$, and put $g_j(x)=f_{{\nu }_j(x)}(x)$. Using the convention $N_{-1}=0$, the fibers are $\{{\nu }_j=n\}=X_p\cap \{N_{n-1}=j-1\}\cap \{a_n=1\}$ for $n\ge 0$; hence each piece is Borel and the sections $g_j$ are measurable. Their values form an orthonormal basis of $H_x$. For a fixed separable $K_p$ of dimension $p$, take an at most countable dense set, enumerate it using [F5], and apply the dense-sequence Gram--Schmidt theorem to obtain an orthonormal basis $(b_j{)}_{j\in J_p}$. The map $g_j(x)\mapsto b_j$ on finite linear combinations is well-defined and isometric because both families are orthonormal. For any $v\in H_x$, choose finite combinations $v_n$ converging to $v$; their images are Cauchy, so completeness of $K_p$ defines $U_{x}v:={\lim }_n{U}_x{v}_n$, independently of the approximating sequence and preserving linearity and norm. If $U_x{v}_n\to w$ for a sequence in the range, isometry makes $(v_n)$ Cauchy; completeness of $H_x$ gives $v_n\to v$, whence $w=U_{x}v$ and the range is closed. It contains the dense span of $(b_j)$, so $U_x$ is onto. Parseval [F5] gives $⟨U_x\xi (x),b_j⟩=⟨\xi (x),g_j(x)⟩$ and convergence of the corresponding partial expansions.

Let $L$ be the closed span in ${\mathcal{H}}_H$ of all $f{\mathbf{1}}_E{\xi }_k$, where $E=D_j\cap \{x:\mathrm{\parallel }{\xi }_k(x)\mathrm{\parallel }\le r\}$, $(D_j)$ ranges over a countable finite-measure Borel cover, $r\in \mathbb{R}$ with $r\ge 1$, and $f$ ranges over bounded Borel scalar functions supported in $E$. It is enough to use integer radii: for any real $r\ge 1$, [F12] gives an integer $R\ge r$, so $E_r\subseteq E_R$; since $f$ is supported in $E_r$, $f{\mathbf{1}}_{E_r}{\xi }_k=f{\mathbf{1}}_{E_R}{\xi }_k$. Thus the real-radius and integer-radius generating families coincide. Each generator is an $S$-measurable section by step 2.2 and is square-integrable because $\mathrm{\parallel }f{\mathbf{1}}_E{\xi }_k\mathrm{\parallel }\le r\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}{\mathbf{1}}_{D_j}$; hence $L\subseteq {\mathcal{H}}_S$. Let $\eta \in L^{\perp }$ and fix $k,j$ and an integer $r\ge 1$. The pairing $h(x)=⟨{\xi }_k(x),\eta (x)⟩$ is Borel by [F2], and ${\mathbf{1}}_{E}h$ is integrable: ${\mathbf{1}}_E{\xi }_k$ and ${\mathbf{1}}_E\eta$ are square-integrable, so [F11] supplies the direct-integral Cauchy--Schwarz estimate. Define $f={\mathbf{1}}_E\bar{h}\mathrm{/}\mathrm{\mid }h\mathrm{\mid }$ where $h\mathrm{\ne }0$, and $f=0$ where $h=0$. This is bounded Borel and supported in $E$ by [F7]. Since the inner product is linear in its first variable, $0=⟨[f{\xi }_k],[\eta ]⟩={\int }_E\mathrm{\mid }h\mathrm{\mid }d\mu$, so [F6] gives that the Borel set $N_{k,j,r}:=E\cap \{x:h(x)\mathrm{\ne }0\}$ is null. For each $k$, the sets $E$ with $j$ varying and integer $r\ge 1$ cover $X$: the $D_j$ cover $X$, and every finite norm is bounded by some integer. There are countably many triples $(k,j,r)$ by iterating the pairing in [F9], so their null sets have a null union by [F6]. Off that union, $\eta (x)\perp {\xi }_k(x)$ for every $k$, hence $\eta (x)\perp S_x$. Thus $\eta \perp {\mathcal{H}}_S$, so $L^{\perp }\subseteq {\mathcal{H}}_S^{\perp }$. Since $L\subseteq {\mathcal{H}}_S$ and both are closed, [F10] yields ${\mathcal{H}}_S={\mathcal{H}}_S^{\perp \perp }\subseteq L^{\perp \perp }=L$. Therefore $L={\mathcal{H}}_S$.

Let $D\subseteq K_p$ be a countable dense set and enumerate it, and enumerate ${\mathbb{Q}}_{>0}$. The balls $B(d,q)$ for $d\in D$ and $q\in {\mathbb{Q}}_{>0}$ form a countable base: given $x\in B(y,\epsilon )$, put $\delta =(\epsilon -\mathrm{\parallel }x-y\mathrm{\parallel })\mathrm{/}3>0$ and choose $d\in D$ with $\mathrm{\parallel }x-d\mathrm{\parallel }<\delta$. Then $\mathrm{\parallel }d-y\mathrm{\parallel }\le \mathrm{\parallel }d-x\mathrm{\parallel }+\mathrm{\parallel }x-y\mathrm{\parallel }<\epsilon -2\delta$, so $\mathrm{\parallel }x-d\mathrm{\parallel }<\epsilon -\mathrm{\parallel }d-y\mathrm{\parallel }$. Choose rational $q$ strictly between these two bounds. It follows that $x\in B(d,q)\subseteq B(y,\epsilon )$. For the measurable section $\xi$, write $c_j(x)=⟨\xi (x),g_j(x)⟩$ for $j\in J_p$. If $\xi$ is measurable, each $c_j$ is Borel by [F2]; for every fixed $y\in K_p$, Parseval gives $\mathrm{\parallel }{U}_x\xi (x)-y{\mathrm{\parallel }}^2={\lim }_{N\to \mathrm{\infty }}{\sum }_{\begin{array}{c}j\in J_p\\ j\le N\end{array}}\mathrm{\mid }{c}_j(x)-⟨y,b_j⟩{\mathrm{\mid }}^2,$ a Borel function by [F7]. Hence inverse images of the countable basic balls are Borel, proving $x\mapsto U_x\xi (x)$ Borel. Conversely, if this map is Borel, continuity of its coordinate functionals follows from Cauchy--Schwarz [F10] and makes each $c_j$ Borel; the partial sections ${\sum }_{\begin{array}{c}j\in J_p\\ j\le N\end{array}}{c}_j{g}_j$ are measurable and converge pointwise in norm to $\xi$ by [F4,F5], so [F2] makes $\xi$ measurable. This proves both directions of the section criterion.

Let $T_x\in \mathcal{B}(H_x)$ be weakly measurable and set ${\tilde{T}}_x=U_x{T}_x{U}_x^{-1}$ on $K_p$. For basis indices $i,j\in J_p$, $⟨{\tilde{T}}_x{b}_i,b_j⟩=⟨T_x{g}_i(x),g_j(x)⟩$ is Borel by [F8] and the measurable sections of step 3.1. On the radius-$C$ operator ball, the basis matrix coefficients generate the WOT: if ${\xi }_m,{\eta }_m$ are finite basis expansions converging to $\xi ,\eta$, then uniformly for $\mathrm{\parallel }T\mathrm{\parallel }\le C$, Cauchy--Schwarz and the operator norm give $\mathrm{\mid }⟨T\xi ,\eta ⟩-⟨T{\xi }_m,{\eta }_m⟩\mathrm{\mid }\le C(\mathrm{\parallel }\xi -{\xi }_m\mathrm{\parallel }\mathrm{\parallel }\eta \mathrm{\parallel }+\mathrm{\parallel }{\xi }_m\mathrm{\parallel }\mathrm{\parallel }\eta -{\eta }_m\mathrm{\parallel })\mathrm{.}$ The matrix coefficients separate operators by density of the finite basis spans, and every WOT coefficient is a uniform limit on the ball of finite linear combinations of these coordinates; conversely each matrix coordinate is WOT-continuous. Thus the ball's WOT topology is its subspace topology from ${\mathbb{C}}^{J_p\times J_p}$, with the index set $J_p$ from step 3.1. This is a countable product of second-countable copies of $\mathbb{C}$ by [F9], using AC through [F4]. Since all coordinate maps are Borel, $x\mapsto {\tilde{T}}_x$ is Borel into the WOT ball whenever $\mathrm{\parallel }{T}_x\mathrm{\parallel }\le C$ for every $x\in X_p$.