Finite-cell sums converge on a compact frequency shell

The norm inequality in [F14] gives $\mid \mathrm{\mid }k\mathrm{\mid }-\mathrm{\mid }\mathrm{\ell }\mathrm{\mid }\mid \le \mathrm{\mid }k-\mathrm{\ell }\mathrm{\mid }$, so the norm is continuous; each coordinate is continuous because $\mathrm{\mid }{k}_i-{\mathrm{\ell }}_i\mathrm{\mid }\le \mathrm{\mid }k-\mathrm{\ell }\mathrm{\mid }$. Since $\beta \omega >0$, [F9] gives $e^{\beta \omega }>e^0=1$. Thus the inline definition makes $N_{\beta }$ positive and continuous on $(0,\mathrm{\infty })$, and strict increase of the denominator makes $N_{\beta }$ decreasing. In particular $N_{\beta }(c\mathrm{\mid }k\mathrm{\mid })\le N_{\beta }(a)$ on the shell. Its points have $\mathrm{\mid }k\mathrm{\mid }\ge a\mathrm{/}c>0$, so $Q_{ij}$ is continuous there. Also $\mathrm{\mid }{Q}_{ii}\mathrm{\mid }\le 1$ and, when $i\mathrm{\ne }j$, $\mathrm{\mid }{Q}_{ij}\mathrm{\mid }=\mathrm{\mid }{k}_i{k}_j\mathrm{\mid }\mathrm{/}\mathrm{\mid }k{\mathrm{\mid }}^2\le 1$. By [F10] the phase factors are continuous and have modulus one. Consequently $G_t(k)=Q_{ij}(k)[N_{\beta }(c\mathrm{\mid }k\mathrm{\mid })e^{\mathrm{i}c\mathrm{\mid }k\mathrm{\mid }t}+(N_{\beta }(c\mathrm{\mid }k\mathrm{\mid })+1)e^{-\mathrm{i}c\mathrm{\mid }k\mathrm{\mid }t}]$ is continuous on the shell and satisfies $\mathrm{\mid }{G}_t(k)\mathrm{\mid }\le M_{\beta }:=2N_{\beta }(a)+1$.

At a fixed tag put $u=k_C\mathrm{/}\mathrm{\mid }{k}_C\mathrm{\mid }$. Complete $u$ to an independent list of three vectors by successively taking a standard coordinate vector outside the span already obtained; this is possible because the three coordinate vectors span ${\mathbb{R}}^3$. Apply [F7] to this finite list. Its first orthonormal vector is $u$ and its remaining two vectors ${\epsilon }_1,{\epsilon }_2$ form an orthonormal basis of $u^{\perp }$. Applying [F8] first to $\mathrm{span}(u)$ and then to its complement gives ${\epsilon }_1{\epsilon }_1^{\mathsf{T}}+{\epsilon }_2{\epsilon }_2^{\mathsf{T}}=I-u{u}^{\mathsf{T}}$. Thus ${\sum }_{\lambda =1}^2{\epsilon }_{\lambda ,i}{\epsilon }_{\lambda ,j}=Q_{ij}(k_C)$, independently of the transverse basis chosen.

The norm continuity in step 1.1 makes $A=A_{a,b}$ closed, hence Borel by [F12] and Lebesgue measurable by [F13]. Set $\mathcal{A}=\{E\cap A:E\in \mathcal{L}({\mathbb{R}}^3)\}$ and $\mu (D)={\lambda }_3(D)$ for $D\in \mathcal{A}$. Since $A$ is Lebesgue measurable, every such $D$ is Lebesgue measurable; the sigma-algebra axioms on $A$, disjoint countable additivity and the empty-set value are inherited from [F13]. Thus $(A,\mathcal{A},\mu )$ is the restricted Lebesgue measure space. By the subspace identity in [F12] and the Borel inclusion in [F13], $\mathcal{B}(A)\subseteq \mathcal{A}$, so every Borel function on the shell is measurable on this measure space. Countable Choice is used here to apply the measure and Borel-inclusion results in [F13].

If $f_1,\dots ,f_m$ are measurable finite real functions on a measurable space, the tuple $(f_1,\dots ,f_m)$ is Borel measurable: preimages of rational open boxes are finite intersections of measurable coordinate preimages, and the family of sets with measurable preimage is a sigma-algebra, so [F16] and [F12] apply. Continuous real and imaginary coordinate projections give the same assertion for tuples of complex functions after identification with real coordinates. By [F15], composing such tuples with the ordinary continuous finite sums, products, or squared-modulus map preserves measurability. In particular, the continuous map $k\mapsto c\mathrm{\mid }k\mathrm{\mid }$ from the shell into $(0,\mathrm{\infty })$ is Borel measurable; composition with the given Borel $\kappa$ and with $z\mapsto \mathrm{\mid }z{\mathrm{\mid }}^2$ makes $h$ Borel measurable. The continuous $G_t$ in step 1.1 is Borel measurable as well.

Such partitions exist: intersect the bounded Borel shell with disjoint half-open grid cubes of side $1\mathrm{/}(n+1)$, each of diameter at most $\sqrt{3}\mathrm{/}(n+1)$, retaining the finitely many positive-volume intersections and omitting the zero-volume ones. For any partitions and tags in the Statement, let $E_n$ be the omitted Borel null part and define the Borel simple function ${\varphi }_n(k)={\sum }_C{\mathbf{1}}_C(k)G_t(k_C)$ on the shell, equal to zero on $E_n$. The grid cubes are Borel since their coordinate intervals are Borel and coordinate maps are continuous. The positive-volume cells are nonempty and bounded, so their diameters in [F14] are defined. For any open subset of $\mathbb{C}$, its preimage under ${\varphi }_n$ is the union of the cells whose constant values lie in that open set, together with $E_n$ if the open set contains zero; this proves the asserted Borel measurability by [F12]. By [F11], the measurable set $E={\bigcup }_n{E}_n$ has measure at most ${\sum }_{n}0=0$. For every $k\mathrm{\notin }E$, its cell tag satisfies $\mathrm{\mid }{k}_C-k\mathrm{\mid }\le {\delta }_n\to 0$, so continuity in step 1.1 gives ${\varphi }_n(k)\to G_t(k)$. Everywhere $\mathrm{\mid }{\varphi }_n\mathrm{\mid }\le M_{\beta }$. No tags on omitted null cells are needed.

By step 2.2 and the measurability of ${\varphi }_n$ in step 3.1, the functions $h{\varphi }_n$ and $h{G}_t$ are Borel measurable and $\mathrm{\mid }h{\varphi }_n\mathrm{\mid }\le M_{\beta }h$, where the nonnegative majorant is integrable by the hypothesis and [F4]. Step 3.1 gives almost-everywhere convergence to $h{G}_t$. On the shell with its restricted Lebesgue measure from step 2.1, [F1] therefore gives ${\int }_{A_{a,b}}h{\varphi }_n{d}^{3}k\to {\int }_{A_{a,b}}h{G}_t{d}^{3}k$, with all these complex integrals well-defined. Each cell weight is finite and nonnegative: $h{\mathbf{1}}_C$ is measurable and bounded in modulus by $h$, so the same integrability conclusion of [F1], applied to the constant sequence $h{\mathbf{1}}_C$, applies.

By [F5] and finite linearity [F4], ${\int }_{A_{a,b}}h{\varphi }_n{d}^{3}k={\sum }_C{G}_t(k_C){\int }_{C}h{d}^{3}k={\sum }_C{w}_C{G}_t(k_C)$. The omitted cells have no integral contribution: [F6] gives zero integral over $E_n$ for the nonnegative majorant $M_{\beta }h$, for $h$, and for $\mathrm{\mid }h{G}_t\mathrm{\mid }$; then [F3] gives ${\int }_{E_n}h{G}_t{d}^{3}k=0$. Combining this identity with step 4.1 proves the stated convergence.

Since $w_C\ge 0$, multiplying each polarization's coefficient by $\sqrt{w_C}$ makes its quadratic contribution $w_C{\epsilon }_{\lambda ,i}{\epsilon }_{\lambda ,j}$. Step 1.2 sums these to $w_C{Q}_{ij}(k_C)$, exactly the weight used in step 5.1. Only finitely many basis choices are required for each partition; no measurable transverse frame over the continuum is assumed. The weight is the specified integral of $\mathrm{\mid }\kappa (c\mathrm{\mid }k\mathrm{\mid }){\mathrm{\mid }}^2$, with no factor $\gamma$ inserted.